# Harnessing the Quantum Behavior of Spins on Surfaces

###### Abstract

The desire to control and measure individual quantum systems such as atoms and
ions in a vacuum has led to significant scientific and engineering
developments in the past decades that form the basis of today’s quantum
information science. Single atoms and molecules on surfaces, on the other
hand, are heavily investigated by physicists, chemists, and material
scientists in search of novel electronic and magnetic functionalities. These
two paths crossed in 2015 when it was first clearly demonstrated that
individual spins on a surface can be coherently controlled and read out in an
all-electrical fashion. The enabling technique is a combination of scanning
tunneling microscopy (STM) and electron spin resonance (ESR), which offers
unprecedented coherent controllability at the Angstrom length scale. This
review aims to illustrate the essential ingredients that allow the quantum
operations of single spins on surfaces. Three domains of applications of
surface spins, namely quantum sensing, quantum control, and quantum
simulation, are discussed with physical principles explained and examples
presented. Enabled by the atomically-precise fabrication capability of STM,
single spins on surfaces might one day lead to the realization of quantum
nanodevices and artificial quantum materials at the atomic scale.

###### keywords:

Quantum sensing, spins on surfaces, quantum simulation, quantum manipulation,
scanning tunneling microscopy, quantum nanoscience

Yi Chen† Yujeong Bae†,∗ Andreas J. Heinrich

Center for Quantum Nanoscience, Institute for Basic Science (IBS), Seoul
03760, Korea
Department of Physics, Ewha Womans University, Seoul 03760, Korea
† These authors contributed equally.
∗ Email Address<EMAIL_ADDRESS>
##  1 Introduction

The last half-century has witnessed tremendous advances in the control and
detection of individual quantum systems. Ions electromagnetically trapped in
vacuum, for example, provide effective two-level systems that enable
sophisticated quantum protocols via high-fidelity optical initialization,
manipulation, and readout [1]. Single photons with quantum information encoded
in their internal degrees of freedom such as polarization and positions can be
individually generated, controlled, and detected [2]. In the solid state,
Josephson-junction-based superconducting circuits provide a highly competitive
platform [3] where two singled-out energy levels can be controlled and read
out via microwave photons [4]. Spins in solid-state materials constitute
another family of individual quantum systems where spin states with naturally
quantized energy levels can be manipulated via magnetic fields. Physical
realizations of solid-state spin systems include gate-defined quantum dots,
single dopants in semiconductors such as phosphorus donors in silicon, and
single defects in insulators such as nitrogen-vacancy centers in diamond [5,
6, 7]. Harnessing quantum resources at the nanoscale gives birth to the
discipline of quantum-coherent nanoscience that may bring forth useful quantum
nanodevices “at the bottom” [8, 9].

In this review, we focus on a new class of quantum spin systems, individual
atomic and molecular spins on surfaces, which has the potential to produce
quantum functionalities at the atomic scale. An STM is used to access this
tiny length scale, where a sharp metallic tip is positioned in nanometer
proximity to spin carriers on a surface to probe their properties through
tunneling electrons [10]. Experiments with single atomic spins on material
surfaces date back to the early days of low-temperature STM, when Yu-Shiba-
Rusinov states and a Kondo resonance were measured in individual magnetic
atoms on bulk superconductors [11] and noble metals [12, 13], respectively.
STM has also been extensively used on single molecular spins to characterize
molecular structures [14, 15, 16], probe and modify their electronic and
magnetic properties [17, 18, 19], and investigate their classical and quantum
applications [19, 20, 21]. Following early STM experiments on single spins,
the desire to further control and magnetically image individual spins has
prompted the developments of new local probe techniques such as spin-polarized
STM [22], inelastic-tunneling-based spin-flip spectroscopy [23], electrical
pump-probe measurements [24], and various innovative forms of force and
magnetic microscopy [25, 26, 27, 28].

An exciting development in recent years is the incorporation of coherent spin
control in atomic-scale microscopy. The need to coherently manipulate and
measure individual spins at this length scale necessitates all-electrical
protocols with Angstrom precision. These stringent requirements are met by
drawing on powerful methodologies from material and quantum sciences, i.e.,
STM’s abilities to build nanostructures atom-by-atom and selectively sense
individual spin-carrying atoms, as well as ESR’s ability to coherently control
electron spin states via electromagnetic waves. This unique combination has so
far enabled the quantum control of single electron spins of atoms [29, 30, 31,
32, 33] and molecules [34], as well as the manipulation of single nuclear
spins through hyperfine interactions [35]. Quantum coherence can be increased
using singlet-triplet states of a coupled spin system [36], allowing the
observation of a free coherent evolution in the singlet-triplet basis [33].
The simultaneous control of two electron spins in an engineered atomic
structure has recently been realized, shedding light on the debated ESR-STM
driving mechanism and showing the potential for multi-spin quantum protocols
on a surface [37].

Quantum sensing with ESR-STM harnesses its unprecedented combination of
Angstrom-scale spatial resolution and tens of nanoelectronvolt (neV) energy
resolution. Unlike traditional STM-based spectroscopy where the energy
resolution is limited to about 0.5 meV at 1 K due to the thermal Fermi-Dirac
broadening of tunneling electrons [38, 39, 40, 41], the energy resolution of
ESR-STM is not limited by electronic thermal broadening because the energy of
the tunneling electrons is not the measured quantity. Instead, the energy in
ESR-STM is benchmarked against the frequency of a supplied radio-frequency
(RF) electromagnetic wave, which drives the spin resonance of a surface spin
and regulates the tunnel current flow through tunneling mangetoresistance. The
energy resolution of ESR-STM is therefore given by the performance of this
“spin regulator”, limited only by the coherence properties of the surface
spin. Using this sensitive ESR-STM measurement scheme, quantum sensors based
on single atomic spins on surfaces have been used to reveal the dipolar
magnetic fields from single-atom magnets at atomic proximity [42, 43, 44] as
well as binding-site-dependent hyperfine interactions [45]. Using single spin
sensors on a surface or a tip, we expect that valuable insight can be provided
into the microscopic magnetic interactions in low-dimensional materials,
strongly correlated systems, and spintronic and magnetic devices.

Quantum simulation with artificial nanomaterials built atom-by-atom serves as
another key application. Various atomic and molecular spins can act as
building blocks for bottom-up spin nanostructures [46, 47]. Local interactions
between spins can be controlled by the atomically precise positioning with STM
and quantified by the neV-resolved ESR-STM spectroscopy. Global interactions
can be provided by substrates, including substrate-mediated exchange
interactions and substrate-induced spin-orbit couplings or superconducting
pairing interactions. The magnetic field of the STM tip provides a convenient
tuning knob that only affects the local spin under the apex of the tip and has
an adjustable field strength that may even exceed the external magnetic field.

This review is organized as follows. In section 2, we introduce the essential
ingredients that enable the coherent control and readout of single spins on
surfaces. Sections 3 to 5 focus on different domains of applications using
spins on surfaces. Section 3 discusses quantum sensing using single atomic
spins on surfaces. Section 4 describes the efforts to improve the quantum
coherence and extend quantum control from single to multiple spins on
surfaces. Section 5 discusses quantum simulation using artificially
constructed 1D and 2D spin arrays on surfaces. The final section 6 provides
concluding remarks.

##  2 Key Ingredients for Coherent Single Spin Experiments on Surfaces

In this section, we review the key ingredients that enable the successful
performance of ESR-STM at the single spin level (for a summary, see Table 1).
After a brief introduction to spins and spin resonance in section 2.1, section
2.2 provides examples of coherently controlled surface spins and their quantum
functionalities, and section 2.3 outlines the experimental setup required for
performing these experiments. Sections 2.4, 2.5, and 2.6 focus on three main
developments that enabled the harnessing of quantum behavior of single surface
spins, namely spin initialization (section 2.4), spin control (section 2.5),
and spin detection (section 2.6). Extending from a single spin to multi-spin
structures, section 2.7 discusses a key advantage of the STM-based approach,
i.e., the atomically precise control of spin-spin interactions via atom
manipulation. Finally, section 2.8 highlights a unique tunability of STM-based
setups, that is, a sizeable, highly local magnetic field from the STM tip.

###  2.1 Spins as Quantum Information Carriers

Spins, as natural physical observables with discrete energy levels, have been
regarded as a promising platform for realizing quantum functionalities since
the dawn of quantum information science [48, 49]. This section explains the
fundamental concept underlying the use of spins as quantum information
carriers. Consider the simplest case, an idealized electron spin-1/2 placed in
a static magnetic field
$\boldsymbol{B}_{\mathrm{ext}}=(0,0,-B_{\mathrm{ext}})$ along the
$-z$-direction. The spin Hamiltonian reads

$H=-\boldsymbol{m}\cdot\boldsymbol{B}_{\mathrm{ext}}=-g\mu_{\mathrm{B}}B_{\mathrm{ext}}S_{z}.$
(1)

Here $\boldsymbol{m}=-g\mu_{\mathrm{B}}\boldsymbol{S}$ is the spin’s magnetic
moment, $g$ is the magnitude of the electron spin’s $g$ factor,
$\mu_{\mathrm{B}}$ is the Bohr magneton, and the spin operators are taken as
unitless (without $\hbar$) throughout this review. The static magnetic field
splits the spin $\ket{\uparrow}$ and $\ket{\downarrow}$ states, or
equivalently, the $\ket{0}$ and $\ket{1}$ states, by the Zeeman energy
$g\mu_{\mathrm{B}}B_{\mathrm{ext}}=\hbar\omega_{0}$, where $\omega_{0}$ is
called the Larmor frequency. A general spin-1/2 state can be visualized as a
vector in a Bloch sphere (Figure 1a, left), parametrized as

$\ket{\psi}=\cos\frac{\theta}{2}\ket{0}+e^{i\phi}\sin\frac{\theta}{2}\ket{1},$
(2)

where $\theta$ and $\phi$ are the spherical coordinates that indicate the spin
direction. The use of spin-1/2 as qubits then relies on the creation of a
well-defined initial state such as $\ket{0}$, the control of the spin to reach
an arbitrary state $\ket{\psi}$, and the detection of the spin state, for
example, by projection onto a certain axis [50].

The control of spin states is typically achieved by an oscillating magnetic
field

$\boldsymbol{B}_{1}=2B_{1}\cos(\omega t+\alpha)\boldsymbol{e}_{X},$ (3)

where the magnetic field has an angular frequency $\omega$ near the Larmor
frequency of the spin ($\omega\approx\omega_{0}$) in the RF range, $\alpha$ is
the phase of the magnetic field, and $t$ is the time. Throughout this review,
we use $\boldsymbol{e}_{X,Y,Z}$ and $\boldsymbol{e}_{x,y,z}$ to indicate the
unit vectors in the lab frame and in the rotating frame, respectively (see
below). Inserting both $\boldsymbol{B}_{\mathrm{ext}}$ and
$\boldsymbol{B}_{1}$ fields into the Hamiltonian in Equation 1 yields the
Schr$\ddot{\mathrm{o}}$dinger equation

$i\hbar\frac{d\ket{\psi(t)}}{dt}=\frac{\hbar}{2}\begin{bmatrix}-\omega_{0}&2\Omega\cos(\omega
t+\alpha)\\\ 2\Omega\cos(\omega
t+\alpha)&\omega_{0}\end{bmatrix}\ket{\psi(t)},$ (4)

where we have converted the $B_{1}$ field strength into an angular frequency
$\Omega=g\mu_{\mathrm{B}}B_{1}/\hbar$. The oscillating field can be decomposed
into a rotating wave and a counter-rotating wave as

$\boldsymbol{B}_{1}=2B_{1}\cos(\omega
t+\alpha)\boldsymbol{e}_{X}=B_{1}[\cos(-\omega
t-\alpha)\boldsymbol{e}_{X}+\sin(-\omega
t-\alpha)\boldsymbol{e}_{Y}]+B_{1}[\cos(\omega
t+\alpha)\boldsymbol{e}_{X}+\sin(\omega t+\alpha)\boldsymbol{e}_{Y}].$ (5)

For a nearly resonant field ($\omega\approx\omega_{0}$), the first, rotating
component of the $B_{1}$ field mostly follows the spin’s Larmor precession and
acts to induce an additional spin rotation about the rotating field axis. This
causes a periodic oscillation of the spin state between $\ket{0}$ and
$\ket{1}$, which is the origin of the spin resonance. The second, counter-
rotating field component in Equation 5 rotates at a very fast rate of
$\omega+\omega_{0}$ relative to the spin’s Larmor precession, and its effect
can often be ignored (known as the rotating wave approximation) [51]. The
Schr$\ddot{\mathrm{o}}$dinger equation under the rotating wave approximation
reads

$i\hbar\frac{d\ket{\psi(t)}}{dt}=\frac{\hbar}{2}\begin{bmatrix}-\omega_{0}&\Omega
e^{i(\omega t+\alpha)}\\\ \Omega e^{-i(\omega
t+\alpha)}&\omega_{0}\end{bmatrix}\ket{\psi(t)}.$ (6)

The physics of spin resonance becomes clearest in a reference frame that
itself rotates at the angular frequency $\omega$ of the oscillating $B_{1}$
field. The spin state in this rotating frame is

$\ket{\tilde{\psi}(t)}=\exp(-i\omega tS_{z})\ket{\psi(t)},$ (7)

whose time evolution is governed by a time-independent Hamiltonian

$i\hbar\frac{d\ket{\tilde{\psi}(t)}}{dt}=\frac{\hbar}{2}\begin{bmatrix}-\omega_{0}+\omega&\Omega
e^{i\alpha}\\\ \Omega
e^{-i\alpha}&\omega_{0}-\omega\end{bmatrix}\ket{\tilde{\psi}(t)}.$ (8)

At resonance ($\omega=\omega_{0}$), the diagonal terms of the Hamiltonian
vanish, and the off-diagonal terms $\Omega e^{\pm i\alpha}$ dominate the time
evolution by periodically flipping the spin between states $\ket{0}$ and
$\ket{1}$. Mathematically, consider an initial state
$\ket{\tilde{\psi}(0)}=\ket{0}$ under a resonant RF wave at
$\omega=\omega_{0}$. The spin state after time $\tau$ becomes

$\ket{\tilde{\psi}(\tau)}=\cos(\frac{1}{2}\Omega\tau)\ket{0}+e^{i(3\pi/2-\alpha)}\sin(\frac{1}{2}\Omega\tau)\ket{1},$
(9)

which corresponds to a spin state at an angle $\theta=\Omega\tau$ and
$\phi=3\pi/2-\alpha$ in the Bloch sphere in the rotating frame. A $B_{1}$
pulse at the resonance frequency thus controls the spin state by rotating it
around an axis in the $xy$ plane (Figure 1a), where the field strength and the
pulse duration jointly determine the rotation angle $\Omega\tau$, and the
field phase $\alpha$ determines the rotation axis. If the spin population in
state $\ket{0}$ is measured at the end of the pulse, a periodic oscillation
will appear

$P_{\ket{0}}(\tau)=\cos^{2}(\frac{1}{2}\Omega\tau)=\frac{1}{2}+\frac{1}{2}\cos(\Omega\tau).$
(10)

The oscillation of populations under (nearly) resonant driving is known as the
Rabi oscillation. At $\Omega\tau=\pi/2$ and $\alpha=0$, for example, the pulse
acts to rotate the spin state by 90∘ around the $x$ axis and turn both
populations of states $\ket{0}$ and $\ket{1}$ to 50%. At off resonance
($\omega\neq\omega_{0}$), the rotation axis is tilted away from the $xy$
plane, and the spin rotation rate, generally known as the Rabi rate, becomes

$\Omega_{\mathrm{R}}=\sqrt{\Omega^{2}+(\omega-\omega_{0})^{2}}.$ (11)

Finally, if we choose to view the spin evolution in the original lab frame
(which does not rotate), the spin motion needs to be combined with a
precession of an angular velocity $-\omega$ around the $z$ axis (see Equation
7).

After spin manipulation, readout can be performed by projecting the spin into
$z$ direction (as in quantum dot experiments [52, 53]) or into the $xy$ plane
(as in ensemble ESR experiments) via spin-to-charge or spin-to-photon
conversions. In an ESR-STM setup, as we shall see in section 2.6, the measured
signals contain both spin-$z$ and $xy$ contributions, which can be
distinguished by their different lineshapes.

In reality, an atomic spin, even when isolated, differs from the above
idealistic case in that (1) both spin and orbital angular momenta are present
and coupled through spin-orbit coupling, (2) multiple electrons in different
orbitals can contribute to the atomic spin, and (3) many nuclei carry a
nuclear spin that couples to the electron spin via the hyperfine interaction.
In addition, if the spin is placed in a molecular or solid-state environment,
its orbital properties can be significantly modified by the surrounding
ligands [54]. Despite these complications, an effective spin Hamiltonian that
approximates orbital effects as spin operators is often sufficient for
describing the spin properties [54]. The effective Hamiltonian for a single
spin can be generally written as

$H=\mu_{\mathrm{B}}\boldsymbol{B}_{\mathrm{ext}}\cdot\mathbf{g}\cdot\boldsymbol{S}+\boldsymbol{S}\cdot\mathbf{D}\cdot\boldsymbol{S}+\mu_{\mathrm{B}}\boldsymbol{B}_{\mathrm{ext}}\cdot\mathbf{g}_{n}\cdot\boldsymbol{I}+\boldsymbol{S}\cdot\mathbf{A}\cdot\boldsymbol{I}+\boldsymbol{I}\cdot\mathbf{P}\cdot\boldsymbol{I},$
(12)

where $\boldsymbol{S}$ is an effective spin operator,
$\boldsymbol{B}_{\mathrm{ext}}$ is the external magnetic field, $\mathbf{g}$
is the $g$-factor tensor, and $\mathbf{D}$ symbolically represents the
magnetic anisotropy that lifts the spin degeneracy due to orbital
contributions. Tensors are denoted in a bold, non-italic font throughout this
review, while vectors are denoted in a bold, italic font. The latter three
nucleus-related terms represent the nuclear Zeeman energy, the hyperfine
interaction, and the electric quadrupole interaction, respectively, and they
will be discussed in more detail in section 3.2. Spin resonance of a realistic
spin can be understood as the resonant transition between two spin eigenstates
where the transition matrix element is non-zero (i.e., containing spin
components allowed by selection rules).

The detailed form of the magnetocrystalline anisotropy term
$\boldsymbol{S}\cdot\mathbf{D}\cdot\boldsymbol{S}$ can be obtained through
group theory analyses, for example, by using the Stevens operators
$\sum_{k,q}B_{k}^{q}O_{k}^{q}(S)$ [54, 55, 56]. The commonly used axial ($D$)
and rhombic ($E$) anisotropy terms are related to the second-order Stevens
coefficients by $D=3B_{2}^{0}$ and $E=B_{2}^{2}$. The corresponding
Hamiltonian terms are defined as

$B_{2}^{0}O_{2}^{0}=D[S_{z}^{2}-\frac{1}{3}S(S+1)],\ \
B_{2}^{2}O_{2}^{2}=E(S_{x}^{2}-S_{y}^{2}),$ (13)

where $x$, $y$, and $z$ indicate the principal axes of the anisotropy tensor,
$\mathbf{D}$. A list of $D$ (and $E$, when applicable) values for spins
mentioned in this review can be found in Table 2. The role of the axial
anisotropy term ($D$) is to split the spin levels based on $|S_{z}|$. For
example, spins with $D<0$ have an easy axis along $z$, i.e., the lowest energy
states have the maximal $|S_{z}|$ ($S_{z}=\pm S$) while higher energy states
are spin states with smaller $|S_{z}|$. This state configuration creates an
effective energy barrier (the “anisotropy barrier”) for spin-reversal between
the two lowest-energy states, hence increasing the spin lifetime [57] (an
anisotropy barrier is depicted above the Fe atom in Figure 1a). On top of the
effect of axial anisotropy, a non-zero rhombic term ($E$) can provide further
splitting of the spin states. In addition, it mixes the spin composition of
different states, allowing for additional spin-reversal mechanisms such as
under-barrier transitions and quantum tunneling of magnetization [57]. Due to
the lack of mirror symmetry at a surface, spins on surfaces experience at
least the axial anisotropy. The magnetocrystalline anisotropy, however, cannot
split conjugated doublets in half-integer spin atoms due to the time-reversal
symmetry [54]. As a result, states in spin-1/2 systems are not split by
anisotropy terms (another way to see this is that for spin-1/2, $S_{i}^{2}$ is
just a constant (with $i=x,y,z$) and may be ignored).

###  2.2 Examples of Coherently Controlled Spins on Surfaces

Single electron spins on surfaces, unlike many other quantum systems, can
exist in a wide range of hosts, including transition metal and rare earth
adatoms, molecules with magnetic atomic centers or radicals, and spin defects.
Commonly used atomic species frequently have isotopes with nonzero nuclear
spins, adding to the diversity of surface spins. Most experiments conducted in
the ESR-STM setting to date, however, have focused on adatoms, partially owing
to surface scientists’ familiarity with them. The preparation of spin-carrying
adatoms on surfaces follows standard surface-science procedures in which
single magnetic atoms and molecules are vacuum-deposited onto a cold substrate
in the STM or a cooling stage. In practice, a substrate temperature of less
than 100 K is usually sufficient to prevent atoms from clustering.

A specific substrate, two-monolayer (2ML) MgO on Ag(100), has been the surface
of choice for ESR-STM for good reason. The thin MgO insulator acts as a
decoupling layer between the spins and the metallic substrate, reducing
interactions and decoherence, while MgO of 2ML to 4ML thickness is still
conductive enough for stable STM operations [58]. MgO/Ag(100) can also be
relatively easily grown into thin crystalline films that allow for atom
manipulation [59]. Although an early report of ESR-STM [29] attributed the ESR
driving to MgO’s crystal fields, subsequent theoretical and experimental
investigations show that the role of crystal fields can be replaced by the
tip’s magnetic field gradient, and the latter likely dominates [60] (a more
complete discussion on the proposed ESR-STM driving mechanisms can be found in
section 2.5). ESR-STM driving should then impose no special requirements on
the substrates.

Other substrates are under active investigation for use in ESR-STM. A suitable
substrate should maintain spin integrity, provide sufficient isolation from
the decoherence sources (such as substrate conduction electrons), allow
sufficient ESR driving, and permit atom manipulation for multi-spin studies.
Among passivated metal surfaces [61], 1ML Cu2N on Cu(100) is a popular
substrate for spin-related studies (see, for example, section 5.1), yet ESR-
STM for adatoms on Cu2N has not been realized. The reason could be due to
Cu2N’s lower decoupling effect compared to MgO, decoherence caused by
substrate nuclear spins (nearly all Cu and N atoms have nonzero nuclear
spins), or simply insufficient experimental trials. The decoupling effects of
Cu2N and MgO can be estimated by considering the scattering of a surface spin
by conduction electrons starting and ending in the metal substrates, which
induces an effective conductance of $\sim 3\,\mu\mathrm{S}$ for Cu-binding-
site Mn on Cu2N [62], two orders of magnitude higher than that of Fe on 2ML
MgO ($\sim 40\,\mathrm{nS}$) [58] (however, we note that the adatoms on N
binding sites of Cu2N can have much longer lifetime than those on Cu binding
sites [63]). NaCl is another frequently used decoupling substrate for
molecules. Although we found that atoms can often be embedded into the NaCl
lattice due to its relatively large lattice constant, we expect NaCl to be a
promising substrate for spin-carrying molecules. New types of substrates that
better isolate the surface spins from low-energy excitations of substrates are
worth future investigation. These substrates may include bulk semiconductors
such as silicon where excellent coherence has been demonstrated [5],
superconductors or correlated insulators where no low-energy excitations exist
below a threshold energy, and novel platforms such as van der Waals
heterostructures where electrical contacts and isolation layers can be
individually engineered. Furthermore, surface spins other than adatoms, such
as spin defects [7] or skyrmions [64, 65], may introduce new spin control and
detection strategies on these new substrates (Table 1).

Figure 1: Coherently controlled spins on 2ML MgO/Ag(100). a) Quantum
functionalities of surface spins. Left: Quantum control of a single atomic
spin-1/2 localized in a hydrogenated Ti atom. Middle left: Quantum sensing of
magnetic fields from a single-atom magnet (Ho) using a spin-2 Fe atom. Middle
right: Control of coupled electron and nuclear spins in a Cu atom. Right:
Spin-1/2 FePc molecule and its potential applications in a molecular quantum
network. O (Mg) atoms of the substrate are depicted in yellow (green). b)
Identification of typical surface spins on MgO using STM d$I$/d$V$ spectra
with spin-averaging tips (i.e., non-spin-polarized tips) at a typical magnetic
field ($\leq$ 1 T) for ESR-STM measurements. Successive d$I$/d$V$ spectra are
shifted by +0.5 a.u. for clarity. c) Same as (b) but with spin-polarized (SP)
tips. The tip’s spin polarization (a critical factor in its performance as an
ESR tip) can be characterized by the heights of the IETS steps, which lie at
around the zero bias for spin-1/2 systems (all except Fe) and at the spin-flip
energies for higher spins (Fe, see text). Whether the IETS conductance is
higher at positive bias or negative bias indicates the relative orientation
between the tip spin and the surface spin, see section 2.6.2 and Ref. [35].
Successive d$I$/d$V$ spectra are shifted by +0.5 a.u. for clarity. d,e)
Examples of STM topograph of isolated Ti atoms, Fe atoms, and FePc molecules
on MgO.

Figure 1a depicts a gallery of single spins on 2ML MgO/Ag(100) (with the spin
properties summarized in Table 2). Single hydrogenated Ti atoms, hereafter
referred to as Ti, host spin-1/2 and provide a two-level quantum system where
continuous-wave (CW) [66] and pulsed spin manipulation [30] were demonstrated
(as sketched on the left side of Figure 1a). In contrast, single Fe atoms on
MgO host spin-2 with a large magnetic moment ($\sim 5.44\mu_{\mathrm{B}}$
[42]) that is fixed perpendicular to the sample plane by a large easy-axis
anisotropy $D=-4.7$ meV (as reflected in its d$I$/d$V$ spectra in Figure 1b,c,
see below) [67]. Despite the anisotropy barrier, an ESR transition of Fe can
be driven between the two lowest energy levels due to intermixing of the spin
eigenstates [29]. Because of these properties, Fe is a sensitive quantum
sensor of local out-of-plane magnetic fields. Fe sensors have been used to
investigate the magnetic fields of nearby single-atom magnets including Ho
[43], Dy [44], or a second Fe atom [42]. Single Cu atoms, on the other hand,
provide a model system with strongly coupled electronic and nuclear spins
where the nuclear spin can be initialized by pumping the electron spin [35].
Finally, as shown on the right side of Figure 1a, single-molecular spin-1/2
localized in FePc has the potential to be used as a molecular qubit [34]. The
aforementioned building blocks have been used to construct a variety of
artificial spin structures, each of which makes use of different quantum
functionalities of the surface spins. These structures are summarized in Table
3 and will be the subjects of sections 3–5.

Different atomic species on 2ML MgO/Ag can be readily identified using STM
topographic imaging and d$I$/d$V$ spectroscopy as shown in Figure 1b–e. Ti
atoms deposited on 2ML MgO/Ag substrate have two preferential binding sites:
the oxygen site (i.e., on top of an oxygen atom of MgO) and the bridge site
(i.e., between two oxygen atoms). Under typical scanning conditions of
$V_{\mathrm{DC}}=0.1$ V and $I_{\mathrm{t}}=$ 20 pA, a bridge-site Ti atom has
a large apparent height (around 1.9 Å), whereas an oxygen-site Ti atom has a
lower local density of states and thus a lower apparent height (around 1.0 Å)
(Figure 1d) (throughout this review, a positive bias voltage means a positive
voltage applied to the sample side). Ti atoms can be laterally manipulated
along the MgO lattice directions by parking the tip 1.5 to 2 lattice constants
ahead of Ti at a setpoint of $V_{\mathrm{DC}}=0.1$ V and $I_{\mathrm{t}}=$ 20
pA and then approaching the tip towards the MgO surface by 0.3 nm at around
$V_{\mathrm{DC}}=0.3$ V to pull Ti towards the tip. In STM d$I$/d$V$
spectroscopy, a bridge-site Ti atom shows a rather featureless d$I$/d$V$
spectrum with a gradual decrease of d$I$/d$V$ at increasing bias voltage,
while an oxygen-site Ti atom exhibits strong inelastic tunneling spectroscopic
(IETS) steps at around $\pm 80$ mV that may be related to the excitation of a
vibrational mode or into a higher-lying orbital state (Figure 1b). IETS steps
generally occur when the STM bias exceeds a threshold voltage corresponding to
a bosonic excitation such as phonons and spin flips, while an additional
inelastic tunneling channel opens up and contributes to the tunnel current
[68, 69]. Under a spin-polarized tip (Figure 1c), Ti atoms at both binding
sites exhibit d$I$/d$V$ IETS steps at around zero bias voltage. These IETS
steps arise from spin flips between the $\ket{\pm 1/2}$ spin states of Ti,
which only require their Zeeman energy difference of $\sim$100 $\mu$eV at a
typical magnetic field for ESR-STM measurements and should thus appear at
$V_{\mathrm{DC}}=\pm 100\ \mu$eV, essentially the zero bias voltage (100
$\mu$eV corresponds to about 24 GHz, or an out-of-plane field of about 0.86 T
on bridge-site Ti). At a higher magnetic field, the two IETS steps of Ti can
be separately resolved, but at lower fields typical for ESR-STM, they merge
into one feature that we loosely refer to as a zero-bias step [70]. It turns
out that the difference of the IETS step heights at positive and negative bias
voltages (or equivalently, the height of the zero-bias step) is a good
indicator of the tip’s spin polarization (see section 2.6.2 and Ref. [71]). A
good ESR-STM tip for Ti requires a strong spin polarization of the tip, which
needs a zero-bias step height of at least 20% of the total d$I$/d$V$ signal
strength as a rule of thumb (as marked in Figure 1c). ESR-STM can drive the
spin resonance of both oxygen- and bridge-site Ti on 2ML MgO. Because oxygen-
site Ti has a lower local density of states than bridge-site Ti, the tip is
typically closer to the former, resulting in a higher Rabi rate but also
stronger tip-induced magnetic fields and decoherence.

Fe atoms are another commonly used atomic species, particularly for quantum
sensing and the preparation of spin-polarized tips. Fe on 2ML MgO/Ag typically
sits on the oxygen sites. In STM topography, Fe’s apparent height (around 1.5
Å) is between the bridge- and oxygen-site Ti, and an Fe atom exhibits a
distinctive “dark halo” around it under a sharp STM tip (Figure 1d). In a
d$I$/d$V$ spectrum, Fe on 2ML MgO has clear IETS steps at $\pm 14$ mV due to
transitions to anisotropy-induced higher-energy spin levels (Figure 1a,b)
[58]. Under a spin-polarized STM tip (Figure 1c), the Fe’s IETS step heights
at 14 mV and $-14$ mV become unequal, and the step height difference reflects
the tip’s spin polarization along Fe’s spin direction (out of the sample
plane), see section 2.6.2 and Ref. [71]. Fe can be picked up onto the tip by
approaching the tip towards the surface by 0.2–0.7 nm at around 0.6 V. Fe can
often be dropped off from the tip by approaching it towards the surface by 0.7
nm or deeper at around $-0.6$ V. This technique is known as vertical
manipulation [72, 46]. Three to ten Fe atoms on the tip are often enough to
create strong spin polarization of the tip and allow ESR-STM measurements.

Cu atoms and FePc molecules are two more examples of spin-1/2 systems. Cu on
2ML MgO/Ag has an apparent height of about 3.2 Å under typical scanning
conditions, whereas FePc molecules are distinguishable by their four-lobe
shape (Figure 1d,e). Despite having a weak zero-bias Kondo-like resonance in
their d$I$/d$V$ spectra (Figure 1b), both systems host spin-1/2 and allow ESR
transitions [35, 34]. Under a spin-polarized tip, Cu and FePc spectra show
IETS steps near zero bias owing to the same mechanism as Ti, which, combined
with the weak Kondo feature, yields interesting spectral shapes (Figure 1c).
Furthermore, 63Cu and 65Cu isotopes contain nuclear spins of 3/2 which are
strongly coupled to Cu’s electron spins (see section 3.2 and Ref. [35]).

Table 1: Key ingredients for coherent single spin control on a surface | Current status | Future directions
---|---|---
| Atoms and molecules evaporated | Various substrates: bulk semiconductors,
Spin preparation | onto a thin insulating film | 2D materials, superconductors;
|  $\to$ Requires preparation chamber | More forms of spins: spin defects, skyrmions
| Thermal; | Spin pumping schemes
Spin initialization | Spin-transfer torque | using other energy levels
|  $\to$ Requires low temperature, magnetic field |
| RF $E$-field applied to a tip or an antenna, | Using $B$-field gradient from single-atom magnets;
Spin control | converted to RF $B$-field in the tip’s $B$-field gradient | Better engineered antenna
|  $\to$ Requires high-frequency cabling |
| Tunnel magnetoresistive readout | Force detection;
Spin detection | using spin-polarized STM | Optical detection;
|  $\to$ Requires spin-polarized STM tip | Single-shot readout

###  2.3 Experimental Apparatus

The requirement for preparing, initializing, controlling, and detecting
individual spins on a surface places demands on the experimental apparatus.
The instrument is typically a low-temperature (typically 1 Kelvin) STM
equipped with a moderate magnet (typically 1 Tesla), GHz-frequency coaxial
cables, and

low-temperature evaporation capabilities, as summarized in Table 1. Some
examples of ESR-STM instruments can be found in Refs. [73, 74, 75, 76]. As
shown in Figure 2a, at the heart of the apparatus is an STM that allows access
to individual spins on a surface. A spin-polarized STM tip is used for
individual spin resonance driving (section 2.5), individual spin readout
(section 2.6), and optionally spin-torque initialization of the spin state
(section 2.4). When studying spins on passivated substrates, a spin-polarized
STM tip can be conveniently prepared by picking up typically three to ten
magnetic atoms (e.g., Fe or Mn) onto the tip as discussed in the previous
section. Alternative forms of spin-polarized STM tips include a tip made of a
ferromagnetic or an antiferromagnetic wire or a metal tip coated with a thin
magnetic layer [22].

A static magnet, built around the STM, establishes the Zeeman energy splitting
of the spin levels. A magnetic field of around 1.5 T is usually sufficient
because most ESR-STM experiments are conducted at resonance frequencies below
40 GHz limited by the cables’ RF transmission and the output range of RF
signal generators (Figure 2b). A single-axis magnetic field, either in or out
of the sample plane, should suffice for many studies, especially on organic
radicals where the angle-dependent $g$-factor anisotropy is small. A multi-
axis vector magnetic field, on the other hand, is convenient for studying
systems such as transition metal atoms on a surface, where the $g$-factor
anisotropy and the magnetocrystalline anisotropy can be significant [54]. In
coupled-spin structures with small magnetocrystalline anisotropy, a vector
magnetic field can be used to modify the dipolar spin-spin interaction by
changing the spin directions (see section 2.7). Another practical benefit of
an adjustable field angle is that it can be used to maximize the ESR signal of
a given spin-polarized tip. This is because the ESR signal intensity can vary
strongly with the field angle due to changes in the driving strength and the
detection sensitivity [77].

A cryostat that houses the STM and the magnet is critical for the stable
operation of the system. Equally importantly, a cryostat provides a low spin
temperature for thermal initialization of the spin states (see section 2.4).
ESR-STM has so far been performed at temperatures ranging from 0.05 K [31] to
5 K [32] and is possible up to at least 10 K (unpublished), a temperature
achievable in most low-temperature STMs.

Figure 2: Experimental apparatus for STM-based coherent spin experiments. a)
Design of an ESR-STM system that includes a ultrahigh vacuum (UHV) chamber for
sample preparation, a liquid helium (LHe) cryostat for thermalization, and a
superconducting magnet. High-frequency coaxial cables composed of different
materials are used between different temperature stages. The STM head is
attached to the coldest stage, which is typically a 1K pot or a mixing chamber
(MC). b) RF transfer functions measured in several ESR-STM systems (1 K
homemade ESR-STM: homemade Joule-Thomson refrigerator, American Magnetics Inc.
cryostat, 7 T single-axis magnet mostly in the sample plane; mK homemade ESR-
STM: Janis JDR-250 dilution refrigerator, Cryomagnetics 9 T + 6 T vector
magnet; 0.4 K commercial STM: Unisoku USM1300, 6 T + 5 T vector magnet). In
the transfer functions, 0 dB corresponds to zero attenuation from the RF
cabling. c,d) Schematics of electronics for continuous-wave (CW) (c) and
pulsed (d) ESR-STM experiments. A bias tee is used to combine DC and RF bias
voltages before applying them to the tunnel junction. The RF voltage is pulse
modulated using a lock-in amplifier in the CW scheme or an arbitrary waveform
generator (AWG) in the pulsed scheme. e–g) Three sweeping schemes of ESR-STM
measurements: RF frequency sweeps (e), external magnetic field sweeps (f), and
tip’s magnetic field sweeps (g) (during the sweep of one parameter, the other
two are fixed). (e) Reproduced with permission.[29] 2015, AAAS. (f) Reproduced
with permission.[74] 2020, APS. (g) Reproduced with permission.[44] 2021,
Springer Nature.

To enable the coherent control of the spin states, coaxial cables with high
transmission in the GHz range are used to connect room-temperature
feedthroughs down to the tip, the sample, or an antenna in the STM head. Here,
we use a homemade ESR-STM (equipped with a dilution refrigerator, Janis
JDR-250) as an example to illustrate the high-frequency cabling used in the
ESR-STM setup:

  * •

From the room-temperature flange to the ‘4K’ stage, heat transfer must be
minimized. We thus use stainless steel semi-rigid cables (Coax Japan,
SC-119/50-SSS-SS) terminated by SMA connectors (Pasternack, PE4116).

  * •

From the ‘4K’ stage to the mixing chamber, we use superconducting NbTi cables
(Coax Japan, SC-160/50-NbTi-NbTi, with a critical temperature of around 10 K)
which provide high RF transmission while maintain a weak heat transfer.

  * •

From the mixing chamber to the STM head, we use semi-rigid copper cables (Coax
Japan, SC-219/50-SC) for high thermal conduction.

  * •

Finally, for the wire going to the tip, we have an additional section of
copper flexible wire (Cooner wire, CW2040-3650P) to allow the free motion of
the STM coarse approach walker. To connect the flexible and semi-rigid copper
cables, a special connector (Rosenberger 19S105-500L5) is used.

The connectors between different sections of the cables are usually rigidly
placed on the corresponding cold stages. It is advantageous to add RF
attenuators at the connections to reduce RF noise coming from the higher-
temperature sides [78].

The transmission of high-frequency cables can be characterized by transfer
function measurements. The details of transfer function measurements using an
STM can be found in Ref. [79]. In essence, one can use the RF broadening of a
sharp d$I$/d$V$ feature to evaluate the RF voltage supplied to the tunnel
junction. By comparing the RF output power of a signal generator and the RF
voltage reaching the tunnel junction at different frequencies, one can obtain
the frequency-dependent loss (i.e., the transfer function) of the RF cabling
system. Using a well-calibrated transfer function, the RF signal generator’s
power output can be adjusted to supply a constant RF voltage to the tunnel
junction over a wide frequency range, which is required for accurate
frequency-sweep ESR-STM measurements (see below).

In Figure 2b, we compare the RF transfer functions of several ESR-STM setups.
The relatively poor transmission in one system (‘Commercial’) is due to an
imperfect design of the connectors and the use of long, lossy flexible cables.
Other homemade cabling systems provide satisfactory transmission over a wide
frequency range. All of the systems in Figure 2b use STM tip cables for RF
transimission, which necessitates a section of flexible coaxial cable for STM
walker motion or for the operation of internal damping system such as springs.
These flexible RF cables cause significant RF loss (limiting the available RF
frequency up to about 30 GHz) and, consequently, heat generation near the STM
junction. An antenna-based design in which the RF power is capacitively
coupled to the tunnel junction partially overcomes these shortcomings [74].
Higher RF voltages in a wider frequency window (up to 100 GHz) can be applied
to the STM junction using an antenna (not shown) [74, 76].

RF components outside of the cryostat are experiment-specific. Figure 2c
depicts the setup for continuous-wave ESR-STM experiments (which also works
for transfer function measurements). Here a bias tee (e.g., SigaTek SB15D2 or
SHF BT45R) is used to combine the DC bias voltage with the RF output of a
signal generator (e.g., Keysight E8267d) before sending the signal to the STM.
A lock-in amplifier is often employed to improve the signal-to-noise ratio by
chopping the RF signal on and off (at a typical frequency of 100 Hz) [29]. The
STM feedback loop can be engaged in these experiments as long as its response
time is set to be longer than the lock-in period. As another example, Figure
2d depicts the setup for pulsed ESR-STM experiments. Here an arbitrary
waveform generator (e.g., Tektronix AWG70000B) is used to trigger the RF
signal generator to produce nanosecond-long RF pulses. The arbitrary waveform
generator can also generate DC bias pulses for the STM readout.

There are three ways to effectively scan across the magnetic resonance
frequency in an ESR-STM measurement, namely the frequency sweeps, the external
magnetic field sweeps, and the tip’s magnetic field sweeps (Figure 2e–g), each
having its own advantages and disadvantages:

  * •

The frequency-sweep method (Figure 2e) takes advantage of the broadband RF
transmission across the tunnel junction. No mechanical motion is involved in a
frequency sweep, resulting in no additional vibrational noise in the tunnel
junction. The frequency-sweep ESR spectra are simple to interpret. The
frequency-sweep method, however, necessitates a calibrated, high-quality
transfer function over a wide frequency range, which may be difficult to
realize for systems with poor transmission properties or employing resonator
designs.

  * •

In an external magnetic field sweep (Figure 2f), the RF wave is fixed at a
specific frequency while the external magnetic field is swept across the
magnetic resonance. This is the method of choice when the RF wave is
sufficiently strong only within a narrow frequency range, such as in
resonator-based setups. In STM, sweeping the external magnetic field requires
extra caution because it may cause mechanical instabilities from the
superconducting magnet or piezoelectric elements. The tip’s magnetization
might also be altered during an external field sweep, which can affect both
the ESR driving and the detection. Despite these challenges, the authors of
Refs. [31, 32, 74] were able to successfully demonstrate the use of external
magnetic field sweeps in ESR-STM.

  * •

A tip’s magnetic field sweep (Figure 2g) is a unique tool in ESR-STM due to
the presence of a spatially varying tip’s magnetic field (see Section 2.8 and
Ref. [70]). The concept is similar to that of an external field sweep, except
that instead of sweeping the external magnet, the spatial separation between
the magnetic tip and the surface spin is swept, which can significantly change
the magnetic field experienced by the spin [70, 80]. This method is typically
used as a quick preparatory step to characterize the tip properties and the
spin resonance frequencies, but it can also be used to collect high-quality
data [44, 81]. A cross-reference to the two preceding methods is required to
convert the tip-sample separation into the magnetic field of the tip. If a
magnetic tip has bistable magnetization, the tip’s magnetic field sweep can
reveal two (rather than one) resonance peaks, slightly complicating the
interpretation (Figure 2g) [44].

Table 2: Examples of well-characterized spins on surfaces. A negative $D$ value indicates easy-axis anisotropy. $\parallel$ and $\perp$ denote that the corresponding quantity’s direction is parallel or perpendicular to the sample surface, respectively. $\parallel^{O}$ denotes MgO’s in-plane O-impurity-O direction (see Figure 1a), $\parallel^{N}$ denotes Cu2N’s in-plane N-impurity-N direction, and $\parallel^{H}$ denotes Cu2N’s other in-plane, hollow-impurity-hollow direction (see Figure 12). Note∗: it remains unclear at the moment whether the 80-meV IETS step of oxygen-site Ti is related to a higher-lying orbital state or a vibrational mode. Note∗∗: the magnetic moment of bridge-site Ti is $0.96\ \mu_{\mathrm{B}}$ along the other in-plane direction [77]. Note∗∗∗: Higher-order anistropy terms mix the Fe spin states, allowing the ESR transition of Fe on MgO to occur [29]. Other anisotropy terms for Ho [82] and Dy [44] on MgO are also known. On Cu2N, the copper binding site lacks four-fold symmetry and induces transverse anisotropy $E$ of 0.007 meV for Mn and 0.31 meV for Fe [83].  Atom | Site | Spin | $\mathinner{\\!\left\lvert\boldsymbol{m}\right\rvert}(\mu_{\mathrm{B}})$ |  $D$ (meV) | ESR | Manipulation |  IETS (meV) | Kondo | Ref.
---|---|---|---|---|---|---|---|---|---
2ML MgO on Ag(100) substrate |  |  |  |  |
Ti | Oxygen | 1/2 | 0.835∥, 0.305⟂ | 0 | $\ket{\pm 1/2}$ | Lateral | 80∗, 0 | No | [31, 66]
Ti | Bridge | 1/2 | 0.83∥O, 0.99${}^{\perp\ **}$ | 0 | $\ket{\pm 1/2}$ | Lateral | 0 | No | [36, 77]
Fe | Oxygen | 2 | 5.44⟂ | $-4.7^{\perp}$ | $\ket{\pm 2}^{***}$ | Vertical | 14 | No | [42, 67]
Cu | Oxygen | 1/2 | 0.99 | 0 | $\ket{\pm 1/2}$ | Difficult | 0 | Weak | [35]
FePc | Oxygen | 1/2 | 1.058 | 0 | $\ket{\pm 1/2}$ | Difficult | 0 | Weak | [34]
Ho | Oxygen | 8 | 10.1⟂ | $-2.5^{\perp}$ | No | Vertical | No | No | [43, 82]
Dy | Oxygen | 15/2 | 9.9⟂ | $-4.5^{\perp}$ | No | Vertical | No | No | [44]
1ML Cu2N on Cu(100) substrate |  |  |  |  |
Mn | Copper | 5/2 | 4.75 | $-0.039^{\perp}$ | N/A | Vertical | 0.2 | No | [83, 84]
Fe | Copper | 2 | 4.22∥N | $-1.55^{\parallel N}$ | N/A | Vertical | 0.2, 3.8, 5.7 | No | [83]
Co | Copper | 3/2 | 3.3 | 2.75∥H | N/A | Vertical | 5.5 | $\ket{\pm 1/2}$ | [85]
Ti | Copper | 1/2 | N/A | 0 | N/A | Vertical | N/A | $\ket{\pm 1/2}$ | [85]
Pt(111) substrate |  |  |  |  |
Fe | hcp | 5/2 | 5 | 0.08⟂ | N/A | Lateral | 0.19 | No | [86]
Fe | fcc | 5/2 | 6 | $-0.19^{\perp}$ | N/A | Lateral | 0.75 | No | [86]
Figure 3: Initialization, detection, and driving schemes of individual spins
using ESR-STM. a) Surface spin initialization schemes using either a low
temperature bath (left) or spin-polarized tunnel current (right). b) Single
spin detection schemes in ESR-STM. Left: STM DC-bias readout of the surface
spin polarization along its quantization axis that produces a symmetric,
Lorentzian lineshape. Right: STM RF-bias homodyne readout of the surface
spin’s precession in a plane perpendicular to the quantization axis that
produces an antisymmetric lineshape. c) Single spin driving schemes in ESR-
STM. A strong, local RF magnetic field (i–iii) can be generated by combining
an RF electrical field across the tunnel junction (applied to the tip or an
antenna) with a spatially-varying static magnetic field (from a magnetic tip
or a nearby single-atom magnet). Alternatively, a global RF magnetic field can
be generated through a microcoil (iv).

###  2.4 Single Spin Initialization

ESR measurements at the single spin level require a strong initial
polarization of the spin states. This section discusses two electron spin
initialization methods demonstrated in ESR-STM: cryogenic cooling and spin-
transfer torque from spin-polarized tunnel current. For the initialization of
nuclear spin states in a strongly coupled electron-nuclear spin system, see
section 4.3.1. Please note that while ESR-STM is used to measure a single
spin, all ESR-STM measurements reported so far have been performed in a time-
ensemble averaged fashion (i.e., by initializing and measuring a single spin
repeatedly, see more in section 2.6.1). As a result, an ensemble description
of the spin states, as used hereafter, is typically sufficient (see, however,
section 2.6.2).

Thermal initialization of electron spins relies on the cryogenic environment
that hosts modern STM systems. In most cases spins on surfaces can be
naturally initialized to nearly the substrate lattice temperature due to
couplings to substrate phonons and conduction electrons. The thermal spin
polarization (of a thermal density matrix) can be expressed using a Boltzmann
ratio. Consider an effective two-level spin system in a magnetic field
$\boldsymbol{B}_{\mathrm{ext}}$ at a temperature $T_{\text{spin}}$. The
Boltzmann population in the lower level is

$P=\frac{1}{\exp[-f_{\text{ESR}}/(20.8366\text{ GHz/K}\times
T_{\text{spin}})]+1}=\frac{1}{\exp(\Delta\boldsymbol{m}\cdot\boldsymbol{B}_{\mathrm{ext}}/k_{B}T_{\text{spin}})+1},$
(14)

where $k_{B}/h=20.8366$ GHz/K is the ratio of the Boltzmann and Planck
constants, and $\Delta\boldsymbol{m}$ is the magnetic moment difference
between the two spin levels involved in the ESR transition of frequency
$f_{\text{ESR}}$. We summarize the measured values of $\Delta\boldsymbol{m}$
for Fe and Ti spins on 2ML MgO and their conversion relations at the end of
this section for the convenience of the readers.

Initialization beyond the thermal polarization requires additional pumping. In
ESR-STM, spin-transfer torque from spin-polarized tunneling electrons offers a
convenient way to achieve this goal [71, 62, 87]. As will be explained in
section 2.6.2, when spin-polarized tunneling electrons pass through a surface
spin, their spins interact and may exchange angular momentum during the
tunneling process, resulting in a spin flip-flop process. Consider a simple
case in which the tip spin is fully polarized to the $\ket{\uparrow}$ state
and a surface spin-1/2 is thermally initialized to be a mixed state with
$70\%$ of the population in the $\ket{\uparrow}$ state and 30$\%$ in the
$\ket{\downarrow}$ state. At positive sample bias, spin-up electrons tunnel
from the tip to the sample, which can cause an inelastic tunneling process
associated with a spin flip-flop. Since the tunneling electron can only be
flipped from $\ket{\uparrow}$ to $\ket{\downarrow}$ due to its assumed initial
polarization in $\ket{\uparrow}$, the surface spin can only go from
$\ket{\downarrow}$ to $\ket{\uparrow}$ (but not the other way around) due to
angular momentum conservation. As a result, the surface spin becomes more
polarized than the thermal state. The ultimate achievable spin polarization
can be predicted using a rate equation that depends on the spin pumping rate
(as determined by the tip’s spin polarization, tunnel current amplitude, the
ratio of inelastic to elastic tunneling events, etc.) and the spin relaxation
rate [62].

Spin transfer torque has been used to initialize several surface spin systems.
A recent experiment on spin-1/2 Ti atoms on 2ML MgO discovered that even a
short current pulse (with less than one tunneling electron per pulse on
average) is efficient at initializing the Ti spin under the tip (see section
2.7 and Ref. [33]). For higher-spin systems such as spin-5/2 Mn atoms on Cu2N,
it was found that a low tunnel current can pump the Mn spin from $\ket{+5/2}$
to $\ket{+3/2}$, while a very high tunnel current (that pumps faster than spin
relaxations) can pull the Mn spin further up in the ladder, even reaching the
highest $\ket{-5/2}$ state [62]. For spin-2 Fe atoms on 2ML MgO, spin-transfer
torque has been used in conjunction with ESR-STM to achieve stronger
initialization and larger ESR signals [88]. We anticipate that future
initialization schemes involving optical or RF pumping through other atomic
levels will provide higher-fidelity spin initialization on surfaces.

Useful relations for calculating thermal populations of Fe and Ti on MgO: In
its ground state, Fe on 2ML MgO hosts magnetic moment $5.44\pm 0.03\
\mu_{\mathrm{B}}$ in the out-of-plane direction [42]. The Fe ESR frequency
corresponding to the Zeeman energy is given by

$f_{\text{Fe}}=\Delta E/h=2\times 5.44\ (\mu_{\mathrm{B}}/h)\times
B_{{\mathrm{ext}}}^{\perp}=152\text{ GHz/T}\times B_{{\mathrm{ext}}}^{\perp},$
(15)

where $\mu_{\mathrm{B}}/h=13.9962$ GHz/T is used, and the resonance frequency
is determined solely by the out-of-plane component of the external magnetic
field, $B_{{\mathrm{ext}}}^{\perp}$. The thermal population of Fe in its
ground state is obtained by combining Equation 14 and 15

$P_{\text{Fe}}=\\{1+\exp[-B_{{\mathrm{ext}}}^{\perp}/(0.137\text{ T/K}\times
T_{\text{spin}})]\\}^{-1}.$ (16)

In an out-of-plane magnetic field $B_{{\mathrm{ext}}}^{\perp}$ of 0.1 T, for
example, the Fe resonance frequency is $15.2$ GHz, and Fe’s ground-state
populations are 54.3%, 67.5%, 86.1%, and 99.9% at $T_{\text{spin}}=$ 4.2 K, 1
K, 0.4 K, and 0.1 K, respectively. This estimation demonstrates the importance
of achieving a low temperature for a reasonable initial spin polarization.

The magnetic moment of spin-1/2 Ti on 2ML MgO depends on the magnetic field
direction and deviates from 1 $\mu_{\mathrm{B}}$ due to orbital contributions,
as shown in Table 2. In an external magnetic field $B_{\mathrm{ext}}$ along
the out-of-plane direction, for example, oxygen- and bridge-site Ti have ESR
frequencies of

$f_{\text{TiO}}=8.54\text{ GHz/T}\times B_{\mathrm{ext}},\ \
f_{\text{TiO}}=27.7\text{ GHz/T}\times B_{\mathrm{ext}},$ (17)

and thermal populations of

$P_{\text{TiO}}=\\{1+\exp[-B_{\mathrm{ext}}/(2.44\text{ T/K}\times
T_{\text{spin}})]\\}^{-1},\ \
P_{\text{TiB}}=\\{1+\exp[-B_{\mathrm{ext}}/(0.75\text{ T/K}\times
T_{\text{spin}})]\\}^{-1}.$ (18)

###  2.5 Single Spin Control in ESR-STM and Its Mechanism

Coherent control of spins is the defining feature of ESR-STM compared to other
STM-based spin measurement schemes such as IETS or magnetization curves (see
section 5.1). The initial idea of ESR-STM was formulated based on the well-
known massive electric field (on the order of 109 V/m) generated by a DC bias
voltage across an STM tunnel junction. It was imagined that a moderate RF
voltage supplied to the tunnel junction can similarly generate a very large RF
electric field, allowing single spins to be addressed locally while producing
little heat (unlike when using a magnetic coil). Although an oscillating
magnetic field is required to directly induce ESR transitions with $\Delta
m=\pm 1$ (see section 2.1), it was previously demonstrated in other solid-
state spin systems such as semiconductor quantum wells and quantum dots that
an oscillating electric field can indirectly drive $\Delta m=\pm 1$
transitions by modulating spin Hamiltonian parameters such as $g$-factors [89]
or mechanically oscillating the electrons’ spin density in the presence of a
magnetic field gradient [90, 91].

The ESR-STM driving scheme proved to be effective, and it has now been
performed on a variety of spins, including Ti, Fe, Cu, and FePc on MgO (Table
2). In addition to the original scheme of applying an RF wave directly to the
tip, RF driving through an antenna (that capacitively couples to the tip) has
also been reported (Figure 3c) [74]. Recently, a “remote” driving scheme was
demonstrated, in which ESR driving was shown for a remote spin placed near the
tip but not directly in the tunnel junction (see Figure 3c, section 4.3.2, and
Ref. [37]). It was discovered that remote driving works only when an single-
atom magnet, in this case Fe, is positioned close to the remote spin (and
their separation sensitively affects the driving strength) [37].

These experimental observations lead us to the conclusion that a large
magnetic field gradient, either from a magnetic tip or a nearby single-atom
magnet, is important for ESR driving in STM. As a result, the most likely
driving mechanism of ESR-STM, in our opinion, is that an RF voltage
mechanically oscillates the spin-carrying electron in the presence of a static
magnetic field gradient [60], as in the case of quantum dots [90, 91]. This
oscillation produces an effective RF magnetic field that can then drive the
$\Delta m=\pm 1$ ESR transitions. This conclusion is consistent with a careful
examination of ESR-STM spectra obtained over a wide parameter range [74].

There are numerous other ESR-STM mechanisms that have been proposed (for a
recent review, see Ref. [92]). Direct driving from oscillating magnetic fields
in the tunnel junction was estimated to be negligible compared to the observed
Rabi rates [74, 70], although a careful RF simulation of the tunnel junction
has yet to be performed. In an early report of ESR-STM of Fe on MgO, the
driving mechanism was proposed to be related to MgO’s crystal field and spin-
orbit coupling (combined with the mechanical oscillation of atomic spin
density) [29]. Further experimental and theoretical studies, however, show
that this mechanism is typically weaker than the aforementioned driving
through the tip’s magnetic field gradient [60] and cannot explain experimental
results such as ESR driving of spin-1/2 atoms [36, 35]. In an interesting
proposal of a cotunneling driving mechanism [93], the RF electric field
periodically modifies the hopping amplitude between the surface atom and a
metallic reservoir, resulting in an oscillating driving term after tracing out
the reservoir’s degrees of freedom. Other proposals rely on spin-transfer
torque induced by spin-polarized tunnel current [94], RF modulation of the
tunnel barrier [95], spin-phonon interactions [96, 97], or modulations of
$g$-factor anisotropy [98]. Nevertheless, these proposed mechanisms do not
address the observed necessity of a magnetic field gradient, only work for a
spin in the tunnel junction, or do not obey other experimental observations
[74], and thus are unlikely to provide the dominant driving force.

###  2.6 Single Spin Detection

The ability to detect a single spin state is at the heart of the emerging
spin-based quantum technology. Single spin detection has been achieved in
optical spectroscopy, RF force microscopy, scanning magnetometry, and
electrical measurements. Optically detected magnetic resonance of single spins
was first demonstrated in single molecules [99, 100] and then extended to
defects in solids such as nitrogen-vacancy centers in diamond [101]. For
single spin detection, the change of an optical fluorescence signal is
monitored while the frequency of an RF wave is swept across the magnetic
resonance. In force microscopy, single spin detection was achieved by
recording the change in the vibration frequency of an oscillating cantilever
upon spin resonance, which varies due to the magnetic exchange interaction
between a magnetic tip and the spin [102, 25]. Scanning magnetometers with
tips based on either a superconducting quantum interference device [27] or a
single nitrogen-vacancy center [28] have also reached single electron-spin
sensitivity, where the tips, tens of nanometers away from the single spin,
sense its stray magnetic field.

Unlike the aforementioned techniques, electrical readout of single spin states
hinges on the spin-to-charge conversion. In semiconductor quantum dots and
donor atoms, for example, the spin-to-charge conversion typically makes use of
a neighboring charge reservoir, whose chemical potential is tuned between the
spin-up and spin-down states. This results in a spin-dependent tunneling
event, during which the change in charge can be capacitively sensed by a
second quantum dot nearby [52, 53]. In single-molecule magnets, electron and
nuclear spins can be read out from the conductance of an effective quantum dot
in a break junction configuration [19, 103, 104]. In the STM setup, early
efforts to detect spin resonance used a non-magnetic tip to read an RF tunnel
current at the Larmor frequency of an electron spin, where the spin
sensitivity could be due to exchange coupling between the local spin and the
tunneling electrons [105, 106].

Most later studies of ESR-STM, however, rely on the use of spin-polarized STM
tips. Owing to the spin sensitivity of the tip, traditional DC tunnel current
detection is sufficient to directly probe the spin states of the atom in the
tunnel junction, resulting in a simpler circuit design and higher data
quality. Below we discuss two complementary descriptions of the spin-polarized
tunneling readout.

####  2.6.1 Ensemble Description of Readout

So far, ESR-STM measurements have been carried out in a time-ensemble averaged
fashion, which means that the spin initialization, control, and measurement
sequences are performed repeatedly, and an averaged tunnel current readout
reflects the ensemble average of the spin state being measured. As shown in
Figure 2c,d, a lock-in amplifier is typically used to improve the signal-to-
noise ratio, where a driven state (that one would like to measure) placed in
the lock-in A cycle is contrasted with a reference state (typically the
thermal state) placed in the lock-in B cycle. The lock-in output signal then
yields the difference between the driven and the reference states. Similar
schemes are used in pulsed measurements, but the total pulse sequence is
typically much shorter than the length of each lock-in cycle and is thus
repeated within each lock-in cycle. Typical measurement parameters for the
readout of one spin state (i.e., one data point during a sweep) are a total
data acquisition time of 1$\sim$5 seconds, a lock-in frequency of 100$\sim$500
Hz, and for pulsed measurements, a total pulse sequence length of 0.5$\sim$2
$\mu$s (which needs to be longer than the spin’s $T_{1}$ time for re-
initialization).

The ensemble readout signal of a spin state by a spin-polarized tip is
described by tunneling magnetoresistance (TMR), where the tunnel conductance
depends on the relative alignment between the tip spin and the surface spin as
[22]

$G=G_{\mathrm{j}}[1+a\braket{\boldsymbol{S}_{\mathrm{tip}}}\cdot\braket{\boldsymbol{S}}].$
(19)

Here, $G_{\mathrm{j}}$ is the spin-averaged junction conductance, $a$ is a
prefactor that describes the TMR contribution to the total conductance, and
$\braket{\boldsymbol{S}_{\mathrm{tip}}}$ and $\braket{\boldsymbol{S}}$ are the
expectation values of the tip and surface spins (taken in the same reference
frame). The ESR-STM readout can then be understood as a modification of
$\braket{\boldsymbol{S}}$ upon a near-resonant RF wave, which results in a
detectable change in $G$. To detect this change in $G$ and to drive the ESR
transition, a combination of a DC bias voltage ($V_{\mathrm{DC}}$) and an RF
bias voltage ($V_{\mathrm{RF}}$) is applied to the STM junction, yielding the
total bias voltage

$V=V_{\mathrm{DC}}+V_{\mathrm{RF}}\cos(\omega t+\phi),$ (20)

where $\omega$ and $\phi$ denote the angular frequency and the phase of the RF
wave, respectively. The tunnel current is given by the product of the
tunneling conductance and the bias voltage

$I(t)=GV=G_{\mathrm{j}}[1+a\braket{\boldsymbol{S}_{\mathrm{tip}}}\cdot\braket{\boldsymbol{S}}][V_{\mathrm{DC}}+V_{\mathrm{RF}}\cos(\omega
t+\phi)].$ (21)

The current signal is then read out using a preamplifier with a bandwith of
around 1 kHz, resulting in a time-averaged current $\overline{I(t)}$ that
contains no RF components.

The two contributions to the detected signal, DC and homodyne (Figure 3b), are
most simply illustrated in a CW measurement, which we will discuss below (in a
pulsed measurement, these two contributions carry similar forms but depend on
more parameters). A CW measurement uses a prolonged data acquisition time
wherein the initial Rabi spin oscillations (see section 2.1) have already
decayed, and a steady spin state is instead reached through the balance of
driving and dissipations. The steady-state solution of a spin-1/2 system is
historically derived from the Bloch equation (and the same result can be
obtained from modeling an open quantum spin-1/2 system [107, 37]). The Bloch
equation in the rotating frame reads [108, 109]

$\frac{d\braket{\boldsymbol{S}}}{dt}=\braket{\boldsymbol{S}}\times[-(\omega-\omega_{0})\boldsymbol{e}_{z}-\Omega\boldsymbol{e}_{x}]-\frac{\braket{S_{z}}-\braket{S_{z}^{0}}}{T_{1}}\boldsymbol{e}_{z}-\frac{\braket{S_{x}}\boldsymbol{e}_{x}+\braket{S_{y}}\boldsymbol{e}_{y}}{T_{2}},$
(22)

where $T_{1}$ is the longitudinal relaxation time that tends to bring the
system to the thermal-equilibrium spin polarization $\braket{S_{z}^{0}}$, and
$T_{2}$ is the transverse relaxation time that tends to diminish the
transverse magnetization. The Larmor frequency is
$\omega_{0}=g\mu_{\mathrm{B}}B_{\mathrm{ext}}/\hbar$ (with the static field
$B_{\mathrm{ext}}$ applied in the $-z$ direction), and the driving field
strength is $\Omega=g\mu_{\mathrm{B}}B_{1}/\hbar$ (with $B_{1}$ applied in the
$x$ direction in the rotating frame). By setting all time derivatives to zero,
the steady-state solution of the surface spin in the rotating frame is
obtained as [108, 109]

$\displaystyle\braket{S_{x}}=\frac{\Omega(\omega-\omega_{0})T_{2}^{2}}{1+(\omega-\omega_{0})^{2}T_{2}^{2}+\Omega^{2}T_{1}T_{2}}\braket{S_{z}^{0}},\
\ \braket{S_{y}}=\frac{-\Omega
T_{2}}{1+(\omega-\omega_{0})^{2}T_{2}^{2}+\Omega^{2}T_{1}T_{2}}\braket{S_{z}^{0}},$
$\displaystyle\braket{S_{z}}=\frac{1+(\omega-\omega_{0})^{2}T_{2}^{2}}{1+(\omega-\omega_{0})^{2}T_{2}^{2}+\Omega^{2}T_{1}T_{2}}\braket{S_{z}^{0}}.$
(23)

While the tip spin $\braket{\boldsymbol{S}_{\mathrm{tip}}}$ is stationary in
the lab frame, it rotates at the angular frequency $+\omega$ around the $z$
axis in the rotating frame (of the surface spin) as

$\braket{\boldsymbol{S}_{\mathrm{tip}}}=\braket{S^{\mathrm{tip}}_{z}}\boldsymbol{e}_{z}+\braket{S^{\mathrm{tip}}_{xy}}[\cos{(\omega
t+\psi)}\boldsymbol{e}_{x}+\sin{(\omega t+\psi)}\boldsymbol{e}_{y}],$ (24)

where $\psi$ denotes the initial phase of the tip spin with respect to the $x$
axis. Inserting Equation 23 and 24 into Equation 21 and taking a time average
yields the detected tunnel current

$\overline{I(t)}=G_{\mathrm{j}}V_{\mathrm{DC}}[1+a\braket{S^{\mathrm{tip}}_{z}}\braket{S_{z}}]+\frac{1}{2}G_{\mathrm{j}}aV_{\mathrm{RF}}\braket{S^{\mathrm{tip}}_{xy}}[\cos{(\psi-\phi)}\braket{S_{x}}+\sin{(\psi-\phi)}\braket{S_{y}}].$
(25)

Here the low bandwidth of the current preamplifier allows us to ignore the RF
components of the tunnel current. Nonetheless, a $V_{\mathrm{RF}}$-dependent
term (the second term in Equation 25) remains. This term, derived from the
component
$G_{\mathrm{j}}a(\braket{\boldsymbol{S}_{\mathrm{tip}}}\cdot\braket{\boldsymbol{S}})V_{\mathrm{RF}}\cos(\omega
t+\phi)$ in Equation 21, arises because the relative rotation of the tip and
surface spins acts as an RF rectifier, allowing the RF voltage to drive
current only at specific times during an RF cycle, resulting in an average DC
current.

As previously mentioned, in typical CW ESR-STM measurements a lock-in
detection scheme is used for higher signal-to-noise ratio, which contrasts
$\overline{I(t)}$ of a driven state ($V_{\mathrm{RF}}$, $\Omega\neq 0$) with a
thermal state ($V_{\mathrm{RF}}=\Omega=0$). The final ESR signal measured by a
lock-in amplifier can then be expressed as

$I_{\text{ESR}}=\Delta I_{\text{DC}}+\Delta
I_{\text{Homodyne}}=G_{\mathrm{j}}aV_{\mathrm{DC}}\braket{S^{\mathrm{tip}}_{z}}[\braket{S_{z}}-\braket{S_{z}^{0}}]+\frac{1}{2}G_{\mathrm{j}}aV_{\mathrm{RF}}\braket{S^{\mathrm{tip}}_{xy}}[\cos{(\psi-\phi)}\braket{S_{x}}+\sin{(\psi-\phi)}\braket{S_{y}}].$
(26)

The measured ESR signal thus contains two contributions, labeled as DC and
homodyne in the equation above. As sketched in Figure 3b left, the DC ESR
signal, $\Delta I_{\text{DC}}$, originates from a population change of the
surface spin under RF driving (i.e., $\braket{S_{z}}-\braket{S_{z}^{0}}$) and
is sensed by the tip magnetization along the surface spin’s quantization axis
$z$ ($\braket{S^{\mathrm{tip}}_{z}}$). In contrast, as shown in Figure 3b
right, the homodyne ESR signal, $\Delta I_{\text{Homodyne}}$, detects the
transverse magnetization of the surface spin ($\braket{S_{x}}$ and
$\braket{S_{y}}$). As explained above, the transverse magnetization is sensed
by a combination of the rotating transverse tip magnetization
($\braket{S^{\mathrm{tip}}_{xy}}$) and an oscillating RF voltage at the
driving frequency, hence the name, homodyne detection. The DC ESR signal
follows $\braket{S_{z}}$ and has a symmetric Lorentizian-peak lineshape, while
the homodyne ESR signal follows $\braket{S_{x}}$ and $\braket{S_{y}}$ and can
have an asymmetric lineshape (Figure 3b) [108, 109].

Finally, in pulsed ESR measurements, the homodyne ESR signal is present during
the RF driving (unless for remote driving of a spin not in the tunnel
junction, see section 4.3.2). The DC readout, if turned on after the driving,
measures the spin population along the $z$ axis following spin manipulation.
Some relaxation effects are included due to the finite duration of the DC
readout pulse.

####  2.6.2 Single-Event Description of Readout

The future possibility of using surface spins as qubits necessitates non-
averaging, single-event spin detection, which can be made possible by
employing existing electrical or optical detection schemes developed in other
quantum systems [110, 111]. To properly describe single readout events and the
post-measurement spin states, one needs to go beyond the ensemble description
in the previous section and understand the microscopic interactions that occur
between individual tunneling electrons and surface spins during the readout
process. In addition, this description enables us to understand the
initialization mechanism via spin-transfer torque (section 2.4) and why a
tip’s spin polarization is reflected in the spin-flip IETS step heights
(Figure 1c).

Here we follow a microscopic model presented in Refs. [71, 62], which was
originally developed to describe the IETS d$I$/d$V$ spectra for Mn and Fe on
Cu2N. This treatment is consistent with theoretical studies in Refs. [87, 112,
113, 114, 115]. A more complete model that includes higher-order tunneling
events can be found in Ref. [116]. Consider an initial state
$\ket{\sigma_{i}m_{i}}$ composed of a tunneling electron in the spin state
$\ket{\sigma_{i}}$ along the quantization axis $z$ and a local surface spin in
spin state $\ket{m_{i}}$ along the quantization axis $z^{\prime}$. During
tunneling events, the tunneling electron scatters with the surface spin. The
tunneling probabilities are argued to take the following form

$Y(m_{f},\sigma_{f},m_{i},\sigma_{i})=\frac{1}{Y_{0}}|\braket{\sigma_{f}m_{f}}{\boldsymbol{\sigma}\cdot\boldsymbol{S}+u}{\sigma_{i}m_{i}}|^{2}=\frac{1}{Y_{0}}|\braket{\sigma_{f}m_{f}}{(\sigma_{z}S_{z}+u)+\frac{1}{2}(\sigma_{-}S_{+}+\sigma_{+}S_{-})}{\sigma_{i}m_{i}}|^{2},$
(27)

where $u$ represents spin-independent potential scattering between the
tunneling electron and the surface spin,
$\boldsymbol{\sigma}\cdot\boldsymbol{S}$ represents their spin-dependent
Kondo-like scattering, and $Y_{0}$ is a normalization factor. In the
following, we consider the simple situation where the tunneling electron and
the surface spin are polarized in the same direction ($z\parallel
z^{\prime}$). In this case, elastic tunneling, mediated by
$\sigma_{z}S_{z}+u$, maintains the tip and surface spin states before and
after tunneling (i.e., $\ket{\sigma_{f}m_{f}}=\ket{\sigma_{i}m_{i}}$).
Inelastic tunneling is created by the flip-flop terms
$\sigma_{-}S_{+}+\sigma_{+}S_{-}$, resulting in a final state
$\ket{\sigma_{f}m_{f}}=\ket{\sigma_{i}+1,m_{i}-1}$ or
$\ket{\sigma_{i}-1,m_{i}+1}$.

Using this microscopic model, post-measurement spin states can be described.
Consider a surface spin originally in the state $\ket{m_{i}}$, a tip spin
fully polarized in the $+1/2$ state, and positive sample bias. If the next
tunneling event is elastic, the spin will stay in the spin-$m_{i}$ state. If
inelastic tunneling instead happens, the surface spin will be flipped to
$m_{i}+1$ because the tunneling electron spin must begin in the $+1/2$ state
in the tip and can only flip to $-1/2$ after flip-flop scattering. Without
detailed knowledge of this tunneling event, the surface spin is described by a
density matrix of a mixed state of $\ket{m_{i}}$ and $\ket{m_{i}+1}$, whose
mixing ratio is determined by the elastic vs. inelastic tunneling rates.
Considered this way, the tunneling measurement appears to be not simply a
projection. The post-measurement state for a general spin state measured with
a partially polarized tip should carry a similar form but be dependent on more
parameters. Future theoretical and experimental investigations are needed to
formulate a fully fledged quantum measurement theory for spin-polarized
tunneling readout in ESR-STM.

There are two other interesting observations one can make from this model.
First, both the elastic and the inelastic tunnel currents can be dependent on
the surface spin state and hence be used as a spin readout method. Second, the
difference of the IETS step heights at positive and negative bias voltages is
directly related to the tip’s spin polarization. The former statement can be
seen by considering the elastic tunnel probability, which typically depend on
the surface spin state $m_{i}$ because in general

$\frac{1}{Y_{0}}|\braket{\sigma_{i}m_{i}}{(S_{z}\sigma_{z}+u)}{\sigma_{i}m_{i}}|^{2}\neq\frac{1}{Y_{0}}|\braket{\sigma_{i}m_{j}}{(S_{z}\sigma_{z}+u)}{\sigma_{i}m_{j}}|^{2},$
(28)

for $m_{i}\neq m_{j}$, due to the interplay of the two interaction terms,
$S_{z}\sigma_{z}$ and $u$ [71]. Similarly, the inelastic tunneling probability
as governed by the spin flip-flop terms also in general depends on the surface
spin state [71]. As a result, both elastic and inelastic tunnel currents can
be used to read out the spin state. It is generally believed that the
inelastic contribution dominates the spin readout signals of Ti on MgO, as
evidenced by Ti’s short $T_{1}$ time under a tunnel current, whereas the
elastic contribution dominates for Fe on MgO because of Fe’s long $T_{1}$ time
despite tunnel current effects (partially due to its anisotropy barrier) [58].
A careful study of the elastic and inelastic contributions to the tunnel
current, as done for Mn and Fe on Cu2N [71, 62], is still lacking for spins on
MgO.

In order to make the latter claim, we need to relate the inelastic tunnel
current to the tip’s spin polarization, which can be defined as

$\eta_{\text{tip}}=\frac{\text{DOS}(+1/2)-\text{DOS}(-1/2)}{\text{DOS}(+1/2)+\text{DOS}(-1/2)},$
(29)

where $\text{DOS}(+1/2)$ and $\text{DOS}(-1/2)$ are the tip’s spin +1/2 and
$-$1/2 density of states near the Fermi level, respectively. We can also
define an effective “sample” spin polarization [71]

$\eta_{\text{s}}=\frac{\sum_{\sigma_{f}}[Y(m_{f},\sigma_{f},m_{i},+1/2)-Y(m_{f},\sigma_{f},m_{i},-1/2)]}{\sum_{\sigma_{f}}[Y(m_{f},\sigma_{f},m_{i},+1/2)+Y(m_{f},\sigma_{f},m_{i},-1/2)]},$
(30)

which ranges between $-$1 and +1 and describes the “spin-filtering” effect
during tunneling. When $\eta_{\text{s}}=1$, for example,
$Y(m_{f},\sigma_{f},m_{i},+1/2)$ is much greater than
$Y(m_{f},\sigma_{f},m_{i},-1/2)$, and so only spin-$+1/2$ tunneling electrons
can tunnel through the surface spin. Using these definitions of spin
polarizations, the inelastic tunnel conductance from the tip to the sample and
from the sample to the tip can be loosely written as [71]

$\displaystyle G_{\text{inelastic,
t}\to\text{s}}=G/4\times[(1+\eta_{\text{tip}})(1+\eta_{\text{s}})+(1-\eta_{\text{tip}})(1-\eta_{\text{s}})],$
(31) $\displaystyle G_{\text{inelastic,
s}\to\text{t}}=G/4\times[(1+\eta_{\text{tip}})(1-\eta_{\text{s}})+(1-\eta_{\text{tip}})(1+\eta_{\text{s}})],$
(32)

The difference between the IETS steps at positive and negative sample bias
voltages then yields the tip’s spin polarization

$P=\frac{G_{\text{Inelastic, t}\to\text{s}}-G_{\text{Inelastic,
s}\to\text{t}}}{G_{\text{Inelastic, t}\to\text{s}}+G_{\text{Inelastic,
s}\to\text{t}}}=\eta_{\text{tip}}\cdot\eta_{\text{s}}.$ (33)

This result is analogous to Equation 19 in conventional TMR formulation and
indicates that the IETS step asymmetry generally characterizes the tip’s spin
polarization $\eta_{\text{tip}}$ (as stated in section 2.2 and Figure 1). When
the quantization axes of the tunneling electron and the surface spin are not
aligned, the interpretation becomes less transparent and requires a careful
simulation [62].

###  2.7 Spin-Spin Interactions and Coupled Spin Systems

To go beyond single spin quantum control, it is necessary to magnetically
couple multiple spins in a controllable way. Using vertical and lateral atom
manipulation techniques of STM, artificial spin structures with well-defined
interactions can be built on a surface [46, 72]. These multi-spin systems
constructed atom-by-atom can then be employed for quantum control (section 3),
sensing (section 4), and simulation (section 5).

To realize these functionalities, a good understanding of spin-spin
interactions and their influences on spin states is required. In this section,
we summarize experimental findings by focusing on two coupled spin-1/2 atoms,
such as Ti on 2ML MgO (Figure 4a) [33, 36, 66]. Spins on a thin insulating
layer are mostly coupled via dipolar and exchange interactions. The spin
Hamiltonian for two coupled spins, $\boldsymbol{S}_{1}$ and
$\boldsymbol{S}_{2}$, is given by

$H=\mu_{\mathrm{B}}(\boldsymbol{B}_{\mathrm{ext}}+\boldsymbol{B}_{\mathrm{tip}})\cdot\mathbf{g}_{1}\cdot\boldsymbol{S}_{1}+\mu_{\mathrm{B}}\boldsymbol{B}_{\mathrm{ext}}\cdot\mathbf{g}_{2}\cdot\boldsymbol{S}_{2}+\boldsymbol{S}_{1}\cdot\mathbf{J}_{\text{e}}\cdot\boldsymbol{S}_{2}+\boldsymbol{S}_{1}\cdot\mathbf{J}_{\text{d}}\cdot\boldsymbol{S}_{2}.$
(34)

Here, we assume that the tip acts on the first spin ($\boldsymbol{S}_{1}$)
with an effective magnetic field ($\boldsymbol{B}_{\mathrm{tip}}$) (see
section 2.8), and $\mathbf{J}_{\text{e}}$ and $\mathbf{J}_{\text{d}}$
represent the exchange and dipolar couplings, respectively. In typical ESR-STM
measurement conditions, $\boldsymbol{B}_{\mathrm{ext}}$ is greater than
$\boldsymbol{B}_{\mathrm{tip}}$ and determines the quantization axis (taken as
the $z$ axis below). The exchange coupling term is typically well approximated
by $J_{\mathrm{e}}\boldsymbol{S}_{1}\cdot\boldsymbol{S}_{2}$, where the
exchange coupling tensor is taken to be symmetric and described by a coupling
strength $J_{e}$. In most cases, the dipolar coupling is small compared to the
Zeeman energy (determined by $\boldsymbol{B}_{\mathrm{ext}}$), so we can
further adopt the secular approximation of the dipolar coupling [109]. Under
these approximations the Hamiltonian becomes

$\displaystyle
H=\mu_{\mathrm{B}}g_{1}(-{B}_{\mathrm{ext}}+{B}_{\mathrm{tip}})S_{1z}-\mu_{\mathrm{B}}g_{2}{B}_{\mathrm{ext}}S_{2z}+J_{\mathrm{e}}\boldsymbol{S}_{1}\cdot\boldsymbol{S}_{2}+J_{\mathrm{d}}(3S_{1z}S_{2z}-\boldsymbol{S}_{1}\cdot\boldsymbol{S}_{2})$
$\displaystyle=\mu_{\mathrm{B}}g_{1}(-{B}_{\mathrm{ext}}+{B}_{\mathrm{tip}})S_{1z}-\mu_{\mathrm{B}}g_{2}{B}_{\mathrm{ext}}S_{2z}+(J_{\mathrm{e}}+2J_{\mathrm{d}})S_{1z}S_{2z}+(J_{\mathrm{e}}-J_{\mathrm{d}})(S_{1x}S_{2x}+S_{1y}S_{2y}),$
(35)

where $g_{1}$ and $g_{2}$ are the g-values projected along the external
magnetic field direction, and

$J_{\mathrm{d}}=\mu_{0}\mu_{\mathrm{B}}^{2}g_{1}g_{2}(1-3\cos^{2}\theta)/2\pi
r^{3}$ is the dipolar coupling constant with $\theta$ being the angle between
the quantization axis ($z$) and the connecting vector of two spins
($\boldsymbol{r}$). The last two terms show the effects of exchange and
dipolar couplings on the spin states, i.e., an energy splitting by
$J_{\mathrm{e}}+2J_{\mathrm{d}}$ and a mixing of the spin states by
$J_{\mathrm{e}}-J_{\mathrm{d}}$ because the last term can be written as spin
flip-flop terms
$(J_{\mathrm{e}}-J_{\mathrm{d}})/2\times(S_{1}^{+}S_{2}^{-}+S_{1}^{-}S_{2}^{+})$.

It is also known that the exchange and dipolar couplings of spins have
distinct dependence on distance and direction. The exchange interaction is
typically spatially isotropic (i.e., direction independent) but it decays
exponentially over distance according to
$J_{\mathrm{e}}=J_{\mathrm{0}}\exp{[-(r-r_{0})/d_{\mathrm{e}}]}$, where
$d_{\mathrm{e}}$ is the decay constant and $J_{\mathrm{0}}$ is the exchange
coupling strength at $r=r_{0}$. The dipolar interaction, on the other hand,
decays more slowly and is direction-dependent, as evidenced by the
$\theta$-dependence in its coupling constant $J_{\mathrm{d}}$. These two
magnetic interactions can be disentangled by measuring the ESR spectra for a
series of spin pairs at different separations and orientations. Roughly
speaking, for pairs at very close distance, the isotropic exchange interaction
dominates and can be determined at any orientation, while farther-spaced pairs
need to be constructed with different orientations in order to determine the
anisotropic dipolar interactions. Using this procedure, the interactions
between bridge-bridge Ti pairs [36], bridge-oxygen Ti pairs [36], oxygen-
oxygen Ti pairs [66], and the dipolar coupling of Fe-Fe pairs [42] have been
obtained.

Figure 4: Tuning eigenstates of two coupled spins using a magnetic tip. a) STM
image of two Ti atoms on MgO. b) Eigenstates and eigenenergies as a function
of the tip’s magnetic field. An avoided level crossing appears at the tuning
point, where the tip field compensates for the $g$-factor difference. c)
Eigenstates of two coupled spins represented by Bloch spheres in the singlet-
triplet subspace. When $\varepsilon$ $\sim$ 0, $\ket{S}$ and $\ket{T_{0}}$
states are the eigenstates. With increasing
$\mathinner{\\!\left\lvert\varepsilon\right\rvert}$, the spin eigenstates
become closer to the Zeeman product states, as characterized by the mixing
parameter $\xi$. d) Pump-probe measurements of free coherent evolution
observed at a “tuning point” when singlet-triplet states are formed (28.5 pA).
The tuning point corresponds to $\xi$ $\sim$ $\pi/2$ in (c), and the free
coherent evolution is a Larmor precession in the singlet-triplet subspace with
frequency $(J_{\mathrm{e}}-J_{\mathrm{d}})/h$. (a,b,d) Reproduced with
permission.[33] 2021, AAAS.

Spin-spin interactions in a spin pair can change the spin eigenstates and
result in the creation of singlet-triplet states. In a basis composed of
Zeeman product states ($\ket{00},\ket{01},\ket{10},\ket{11}$), the eigenstates
of the coupled-spin Hamiltonian (Equation 35) are

$\begin{gathered}\ket{T_{-1}}=\ket{00},\ket{S(\xi)}=\cos\frac{\xi}{2}\ket{01}-\sin\frac{\xi}{2}\ket{10}\\\
\ket{T_{0}(\xi)}=\sin\frac{\xi}{2}\ket{01}+\cos\frac{\xi}{2}\ket{10},\ket{T_{+1}}=\ket{11}.\end{gathered}$
(36)

Here, $\xi$ is a mixing parameter (Figure 4c) determined by the ratio of the
spin flip-flop coupling energy ($J_{\mathrm{e}}-J_{\mathrm{d}}$) to the Zeeman
energy difference of the two spins ($\varepsilon$) as

$\tan\xi=\frac{J_{\mathrm{e}}-J_{\mathrm{d}}}{\varepsilon}=\frac{J_{\mathrm{e}}-J_{\mathrm{d}}}{\mu_{\mathrm{B}}[(g_{2}-g_{1})B_{\mathrm{ext}}+g_{1}B_{\mathrm{tip}}]}.$
(37)

When the Zeeman energy difference, $\varepsilon$, is small compared to the
coupling energy ($\varepsilon\ll J_{\mathrm{e}}-J_{\mathrm{d}}$), the mixing
parameter $\xi\sim\pi/2$, and the eigenstates correspond to the singlet and
triplet states, $\ket{S}=(\ket{01}-\ket{10})/\sqrt{2}$ and
$\ket{T_{0}}=(\ket{01}+\ket{10})/\sqrt{2}$. In contrast, when the Zeeman
energy difference is large ($\varepsilon\gg J_{\mathrm{e}}-J_{\mathrm{d}}$),
the mixing paramter $\xi\sim 0$, and the eigenstates are Zeeman product states
$\ket{01}$ and $\ket{10}$ (Figure 4b,c). Note that the tip’s magnetic field
enters the Zeeman energy difference of the spins (Equation 37) and can thus
affect the spin eigenstates. Two strongly coupled spins
($J_{\mathrm{e}}-J_{\mathrm{d}}\gg\varepsilon$), for example, are robust
against the local magnetic field fluctuations and can be used to enhance spin
coherence (see section 4.1 and Ref. [36]).

On the other hand, if two spins are weakly coupled, the tip’s magnetic field
can sensitively affect the formation of the singlet-triplet states (Figure
4b). The authors of Ref. [33] recently demonstrated this by first creating the
singlet-triplet states using a carefully tuned tip’s magnetic field and then
observing a “Larmor precession” in the singlet-triplet subspace. As shown in
Figure 4a, two Ti atoms with slightly different Zeeman energies ($\sim$1 GHz)
were used, with a magnetic tip located over one of the atoms to provide a
local, tunable magnetic field. ESR-STM was then used to identify a specific
tip-atom separation, dubbed the “tuning point”, where the tip’s magnetic field
is such that the two spins have the same Zeeman energies ($\varepsilon=0$).
This results in a perfect mixing of $\xi=\pi/2$ and the creation of the
singlet-triplet states.

To show the existence of the singlet-triplet states at the tuning point,
electrical pump-probe experiments were performed, where a DC pump pulse is
used to initialize the spin state followed by a DC probe pulse that measures
the evolved spin state after some time delay. In the weakly-coupled Ti spin
pair, the Zeeman energy is the dominating energy scale, and so the ground
state is $\ket{00}$. A DC pump pulse from a spin polarized tip is utilized to
locally excite the spin underneath the tip apex (say, the first spin), which,
to some fidelity, flips it and creates the spin state $\ket{10}$ (this spin
flipping is due to spin-transfer torque, see section 2.4). At the tuning
point, the state $\ket{10}$ can be regarded as a superposition of the singlet
state $\ket{S}$ and the triplet state $\ket{T_{0}}$, and a free evolution then
occurs in the $xy$ plane of the singlet-triplet subspace (green trajectory in
upper panel of Figure 4c and the 28.5 pA curve of Figure 4d). This precession
can be detected using a DC probe pulse that measures along the
$\ket{10}$-$\ket{01}$ axis in the singlet-triplet subspace because the first
spin in the $\ket{10}$ and $\ket{01}$ states has a different spin orientation
during the Lamor precession. In contrast, away from the tuning point (lower
panel of Figure 4c), the excited spin state $\ket{10}$ is essentially an
excited eigenstate, with only an exponential decay towards the ground state
$\ket{00}$ but no coherent evolution (26 pA/30.5 pA of Figure 4d), in some
sense similar to the relaxation of a single-spin excited state. This work
showcases the power of combined high energy and temporal resolutions in STM-
based ESR and electrical pump-probe measurements. In principle, pulsed DC and
RF measurements can be extended to larger spin structures or magnetic
materials where local spin flips may induce interesting dynamics.

###  2.8 Creating a Tunable Local Magnetic Field Using the STM Tip

Figure 5: Magnetic resonance imaging (MRI) in STM. a) Constant-current STM
image of an oxygen-site Ti on MgO. b) An MRI scan in the same area with a
spin-polarized tip showing a spatial pattern of resonance signals. The RF
frequency is fixed at $f$= 25 GHz, and the spatial variations originate from
magnetic interactions between the tip spin and the surface spin. c) More MRI
scans at different tip-atom distances. Reproduced with permission. [117] 2019,
Springer Nature.

A unique feature of ESR-STM is the presence of a nearby magnetic tip. The
tip’s magnetic moment is largely classical because of its proximity to the
metallic tip body and hence fast decoherence. While magnetic tips with only
one Fe atom on the apex were found to be paramagnetic [71], ESR tips commonly
used (with three to ten Fe atoms) often have large enough anisotropy to keep
the tip spin fixed along a certain direction (typically at a 15–60∘ angle to
the external field applied during the preparation of the spin-polarized tip
[77]). Similar to two coupled surface spins as discussed in section 2.7, the
tip spin $\braket{\boldsymbol{S}_{\mathrm{tip}}}$ and the surface spin
$\boldsymbol{S}$ are also coupled via dipolar and exchange interactions as

$H=g\mu_{\mathrm{B}}\boldsymbol{B}_{\mathrm{tip}}\cdot\boldsymbol{S}=g\mu_{\mathrm{B}}(\boldsymbol{B}_{\mathrm{tip-
dip}}+\boldsymbol{B}_{\mathrm{tip-
ex}})\cdot\boldsymbol{S}=\frac{g_{\mathrm{tip}}g\mu_{0}\mu_{\mathrm{B}}^{2}}{4\pi
r^{3}}[\braket{\boldsymbol{S}_{\mathrm{tip}}}-3(\braket{\boldsymbol{S}_{\mathrm{tip}}}\cdot\hat{\boldsymbol{r}})\hat{\boldsymbol{r}}]\cdot\boldsymbol{S}+J_{0}e^{-(r-r_{0})/d_{\mathrm{e}}}\braket{\boldsymbol{S}_{\mathrm{tip}}}\cdot\boldsymbol{S},$
(38)

where $g_{\mathrm{tip}}$ and $g$ are the $g$-factors of the tip spin and the
surface spin, respectively, $\boldsymbol{r}$ and $\hat{\boldsymbol{r}}$ are
the connecting vector from the tip spin to the surface spin and the
corresponding unit vertor, and $J_{0}e^{-(r-r_{0})/d_{e}}$ characterizes the
exponential decay of the exchange coupling strength. The tip’s magnetic field
(from the magnetic interactions between the tip and the surface spins) was
found to be continuously tunable from 1 mT to 10 T, covering 4 orders of
magnitude [70].

The dipolar and exchange contributions in Equation 38 can be disentangled by
performing ESR-STM measurements over a wide range of tip-atom separations
[32]. Another interesting way to visualize the tip’s magnetic interactions
with a surface magnetic atom is by obtaining ESR-STM signals while scanning
the tip laterally over the atom. An example of this so-called nano-MRI
(magnetic resonance imaging) method is shown in Figure 5. MRI scans contain
unique signatures of the magnetic tips used for scanning, which, upon a
careful analysis, can reveal the dipolar and exchange couplings between the
tip and the surface spins [117].

Table 3: Summary of ESR-STM experiments performed on MgO/Ag substrates System studied | Short description | Section | Year | Ref.
---|---|---|---|---
Isolated Fe | First demonstration of ESR-STM | 2 | 2015 | [29]
Isolated Fe | Measurement of $T_{1}$ time at high $B$-field (pump-probe, no ESR) | 2 | 2017 | [58]
Isolated Fe | Measurement of $T_{2}$ time; spin-transfer torque initialization | 4 | 2018 | [88]
Isolated Fe | Optimization of Fe ESR in a vector $B$-field; tip $B$-field sweep | 2 | 2019 | [81]
Isolated Ti, Ti-Ti | Demonstration of ESR-STM on spin-1/2; Ti-Ti interaction | 2,3 | 2017 | [66]
Isolated Ti | Measuring the exchange interaction between tip and surface spins | 2 | 2019 | [70]
Isolated Ti |  $g$-factor anisotropy in a vector magnetic field; external $B$-field sweep | 2 | 2021 | [31]
Isolated Ti | Tip spin direction; $g$-factor anisotropy in a vector magnetic field | 2,3 | 2021 | [77]
Isolated Ti, Fe | Magnetic resonance imaging of tip spin-surface spin magnetic interactions | 2 | 2019 | [117]
Isolated Ti, Fe | On ESR-STM driving mechanism; external $B$-field sweep | 2 | 2020 | [32]
Isolated Ti, Fe | Measurement and binding-site control of hyperfine couplings | 3 | 2018 | [45]
Isolated Cu | Manipulation of nuclear spin states through hyperfine coupling | 3,4 | 2018 | [35]
Fe-Fe pair | Sensing of Fe’s effective magnetic field | 2,3 | 2017 | [42]
Fe-Ho pair | Sensing of Ho’s effective magnetic field; Ho initialization | 2,3 | 2017 | [43]
Fe-Dy pair | Sensing of Dy’s effective magnetic field; tip $B$-field sweep | 2,3 | 2021 | [44]
Ti-Ti pair | Enhanced coherence in singlet-triplet states | 2,4 | 2018 | [36]
Ti-Ti pair | Spin-flip initialization; coherent evolution in the singlet-triplet basis | 2 | 2021 | [33]
Ti-Ti-Fe | Simultaneous and individual driving of two Ti spins using one tip | 2,4 | 2021 | [37]
Ti-Ti-Ti-Ti | Quantum simulation of a resonating valence-bond state | 5 | 2021 | [118]
FePc, FePc-FePc | ESR-STM study of molecular spins and their interactions | 2,3 | 2021 | [34]

##  3 Quantum Sensing Using Individual Atomic Spins on Surfaces

Quantum sensing, in broad terms, encompasses all measurement approaches using
quantum systems as sensors and harnessing the high sensitivity of quantum
states to external perturbations [119]. Ideal quantum sensors respond
exclusively to the desired external signals, where their sensitivity is
determined by the coupling strength to external signals as well as the quantum
sensors’ coherence qualities. Different quantum systems excel at sensing
different physical quantities. Trapped ions [120] and Rydberg atoms [121], for
example, are good electric-field sensors, while some superconducting circuits
[122, 123] and spin-based systems such as semiconductor quantum dots [52] and
color centers in diamond [124, 125, 126, 127] respond sensitively to magnetic
fields.

The detection of small signals with high spatial resolution is a primary goal
of quantum sensing. In magnetic field sensing, a challenging objective is to
achieve the sensing of single spins and their interactions at the atomic scale
[128, 129, 130]. Traditional STM-based spectroscopy, such as IETS, attains
atomic resolution but has an energy resolution limited by the thermal Fermi-
Dirac broadening of tunneling electrons [38, 39, 40, 41]. The energy
resolution of ESR-STM, on the other hand, is not limited by electronic thermal
broadening because the energy of the tunneling electrons is not the measured
quantity. Instead, the energy in ESR-STM is measured against the frequency of
a supplied RF wave, which resonantly drives a surface spin state and regulates
the tunnel current flow through the TMR effect (see section 2.6.1). The energy
resolution of ESR-STM is thus limited only by the performance of this TMR-
based “spin regulator”, given by a combination of $T_{1}$ and $T_{2}$ of the
surface spin depending on the specific detection mechanism (for CW ESR
measurement, the energy resolution is related to the linewidth specified in
Equation 44). Due to their distinct energy resolutions, traditional IETS
spectra can be used to quantify strong exchange coupling or Ruderman-Kittel-
Kasuya-Yosida (RKKY) interaction (see section 5.1), while ESR-STM can detect
considerably weaker electronic dipole-dipole interactions and hyperfine
couplings.

In this section, we mainly describe the use of single magnetic atoms on
surfaces as the smallest quantum sensors. Spins on surfaces satisfy the basic
criteria for quantum sensing, such as the existence of discrete energy levels,
the ability to initialize, manipulate, and read out these levels, and the
interaction with external signals [119]. Surface spins are mostly used to
sense atomic-scale local magnetic fields, either from a nearby magnetic atom
(section 3.1) or a nuclear spin (section 3.2). The outcome is a shift in ESR
frequencies with an energy resolution of about 40 neV under current CW ESR-STM
measurement conditions. In section 3.3, we briefly discuss recent developments
of tip-based sensing that might soon enter the quantum regime.

###  3.1 Single Spin Sensing of Dipolar Fields at the Atomic Scale

Fe atoms on MgO serve as a good sensor for local out-of-plane magnetic fields,
as their magnetic moments are fixed perpendicular to the surface plane by a
large magnetic anisotropy (Table 2) [29, 67]. As shown in Figure 6a, using
vertical atom manipulation, an Fe atom (sensor) was positioned close to
another magnetic atom (target, which is also Fe in this case) to probe the
magnetic field emanating from it. During the lock-in measurement time (about
1–10 ms, see section 2.6.1), due to perturbations, the target Fe atom
occasionally flips its spin and hence the magnetic field emanating from it,
resulting in two ESR peaks observed on the sensor atom (Figure 6b,c). The ESR
peak height ratio thus corresponds to the Boltzmann occupations of the two
lowest spin states of the target Fe atom at this temperature, while the ESR
peak splitting directly yields the sensor-target magnetic interaction [42].
The ESR peak splitting was found to increase at reduced separations ($r$)
between the two Fe atoms (see Figure 6d) with a trend that follows an
$r^{-3}$-power law, which is characteristic of the magnetic dipole-dipole
interaction (section 2.7). Fitting to the measured splitting (Figure 6d)
reveals the magnetic moment of Fe atom as $5.44\pm 0.03\ \mu_{\mathrm{B}}$
[42], a remarkable precision for an atomic-scale measurement.

The accurate determination of the Fe’s magnetic moment on MgO allows its use
as a sensor of magnetic fields from other single-atom magnets. This technique
has been used to determine the magnetic moments of Co [42], Ho [43], Dy [44],
and some unknown magnetic species (Figure 6d,e), all with 1% or better
accuracy. For lanthanide atoms (Ho and Dy), Fe sensing was proven to be non-
invasive and does not perturb the magnetic stability [43, 44].

Figure 6: Atomic-scale sensing of dipolar fields from single-atom magnets. a)
Schematic showing the sensing of dipole fields from a target atom using a
sensor spin. b) ESR spectrum obtained from an Fe atom (sensor) separated by
2.46 nm from another Fe atom (target). c) Schematic energy diagram of the
sensor atom under the influence of the dipole fields of the target atom. Two
ESR peaks in (b) correspond to the two transitions marked by grey arrows in
(c). The Zeeman energy of the target spin is not included in the energy level
diagram. d) ESR splitting as a function of atomic separations for pairs of Fe-
Fe, Fe-Co, and Fe-‘tall’ Fe. The observed $r^{-3}$-distance dependence
indicates that dipole-dipole interactions dominate at these separations, while
deviations in the grey area (with $r$ less than 1 nm) result from additional
exchange interactions. The magnetic moments extracted from the fits are
$5.44\pm 0.03\ \mu_{\mathrm{B}}$ for Fe, $5.88\pm 0.06\ \mu_{\mathrm{B}}$ for
Co, and $4.35\pm 0.08\ \mu_{\mathrm{B}}$ for ‘tall’ Fe. e) An STM image of
five Fe atoms used in “nano-GPS”. Through trilateration using 3 sensor atoms,
the location and the magnetic moment of a target, unknown atom can be
determined. Reproduced with permission.[42] 2017, Springer Nature.

An interesting use of Fe sensors is the trilateration measurement of the
position of a target spin, dubbed nano-GPS (Global Positioning System). Figure
6e illustrates the concept. Using three Fe sensors, a unique location (better
than 0.1 nm) and magnetic moment (better than 0.1 $\mu_{\mathrm{B}}$) of an
unknown spin can be determined.

Other sensors such as spin-1/2 Ti and FePc have also been utilized. Ti sensors
have been used to determine the Ti-Ti interactions at different binding sites
[66, 36] and the magnetization direction of a magnetic tip [77]. FePc
molecules have been used to determine FePc-FePc and FePc-Ti couplings [34].
The scheme described in this section is in principle applicable to the sensing
of any magnetic objects that can stay on or near a surface. We expect that
with its high spatial and energy resolutions, ESR-STM will enable the sensing
of complex magnetic molecules, magnetic nanostructures, and various magnetic
materials.

###  3.2 Single Spin Sensing of Hyperfine Interactions on a Surface

Nuclear spins have been widely exploited as probes to the chemical environment
and electronic characteristics of atoms, molecules, and crystals due to their
well-characterized interactions with electron spins. While single nuclear
spins have been detected and controlled using other approaches [131, 132, 103,
133], directly correlating hyperfine spectra with atomic-scale surroundings of
the nucleus had not been demonstrated. Ref. [45] achieved this goal by using
ESR-STM to resolve hyperfine interactions and modify them by changing the
adatom’s binding site.

Before discussing the experimental results, we first introduce some
theoretical concepts related to hyperfine interactions. In the presence of a
nuclear spin $\boldsymbol{I}$, the spin Hamiltonian becomes

$H=H_{\mathrm{EZ}}+H_{\mathrm{NZ}}+H_{\mathrm{HF}}+H_{\mathrm{NQ}}=\mu_{\mathrm{B}}\boldsymbol{B}_{\mathrm{ext}}\cdot\mathbf{g}\cdot\boldsymbol{S}+\mu_{\mathrm{n}}\boldsymbol{B}_{\mathrm{ext}}\cdot\mathbf{g}_{n}\cdot\boldsymbol{I}+\boldsymbol{S}\cdot\mathbf{A}\cdot\boldsymbol{I}+\boldsymbol{I}\cdot\mathbf{P}\cdot\boldsymbol{I},$
(39)

where $\boldsymbol{S}$ and $\boldsymbol{I}$ represent the electron and nuclear
spin operators, respectively, and $\mu_{\mathrm{B}}$ and $\mu_{\mathrm{n}}$
are the Bohr magneton and nuclear magneton, respectively. The $g$-factors of
electron and nuclear spin ($\mathbf{g}$ and $\mathbf{g}_{n}$), the hyperfine
constant ($\mathbf{A}$), and the electric quadrupole constant ($\mathbf{P}$)
are written in the general tensor forms. The first two terms in the
Hamiltonian are the Zeeman energies of electron and nuclear spins, the third
term represents the hyperfine interaction, and the last term corresponds to
the nuclear electric quadrupole interaction. Note that the magnetocrystalline
anisotropy is not included in Equation 39 for simplicity (which is valid in
the cases discussed below). In a typical ESR-STM environment, the electronic
Zeeman interaction is on the order of $\sim$0.1 meV, the magnetic hyperfine
interaction is $\sim$1 $\mu$eV, and the nuclear Zeeman energy is $\sim$10 neV.
As a result, the nuclear Zeeman energy can often be ignored, as we shall do
later, which means that the quantization axis of the nuclear spin is
predominantly determined by the electron spin through the hyperfine coupling
rather than the external magnetic field.

Since the nuclear and electron spins share similar chemical environment, the
$\mathbf{g}$, $\mathbf{A}$, and $\mathbf{P}$ tensors often have collinear
principal axes and can be diagonalized simultaneously. The diagonal form of
the hyperfine interaction tensor reads [54]:

$H_{\mathrm{HF}}=A_{x}S_{x}I_{x}+A_{y}S_{y}I_{y}+A_{z}S_{z}I_{z}=\sum_{i=x,y,z}A_{i}S_{i}I_{i}.$
(40)

The hyperfine interaction, $\mathbf{A}$, is well-known to have two
contributions: the nucleus can couple to the electron spin cloud outside the
nucleus via the dipolar interaction (the dipolar hyperfine term) or it can
directly interact with the finite electron spin density at the nuclear site
(the contact hyperfine term). The nuclear electric quadrupole ($\mathbf{P}$)
interaction, on the other hand, originates from the electric field gradient
present at the site of the nucleus, and its diagonal form is given by:

$H_{\mathrm{NQ}}=P_{x}I_{x}^{2}+P_{y}I_{y}^{2}+P_{z}I_{z}^{2}=\sum_{i=x,y,z}P_{i}I_{i}^{2}.$
(41)

Figure 7: Hyperfine interactions of individual atoms on a surface. a) ESR
spectra measured on 56Fe with zero nuclear spin (blue) and 57Fe with $I$=1/2
(orange). The bottom panels show STM images of 56Fe and 57Fe. b) ESR spectra
measured on bridge-site Ti atoms with different nuclear spins. A single ESR
peak was observed on 48Ti that carries no nuclear spin, while 6 and 8 ESR
peaks were seen for 47Ti ($I$=5/2) and 49Ti ($I$=7/2), respectively. c)
Binding-site-dependent ESR spectra of the same 47Ti atom moved by lateral atom
manipulation. Right: STM images showing the Ti binding sites where the spectra
on the left are taken. The Ti atom was changed from (1) a bridge site to (2)
an oxygen site then back to (3) a bridge site. The interception points of the
grid correspond to oxygen sites of MgO. d) ESR spectrum of 65Cu ($I$=3/2) on
MgO. Lower panel: Schematic energy diagram of Cu with $S$=1/2 and $I$=3/2.
(a–c) Reproduced with permission.[45] 2018, AAAS. (d) Reproduced with
permission.[35] 2018, Springer Nature.

We are now ready to discuss the experimental ESR-STM studies of the hyperfine
and nuclear electric quadrupole interactions in Fe, Ti, and Cu atoms on MgO,
as summarized in Figure 7. Fe atoms on MgO are typically seen to host only one
ESR peak, which corresponds to 56Fe and 54Fe isotopes (both with $I=0$) that
have a combined abundance of 98%. In a small fraction of Fe atoms, however,
two split ESR peaks were observed with a splitting $\Delta f=231\pm 5$ MHz
(Figure 7a). This splitting originates from the nuclear spin of 57Fe ($I=1/2$
with $\sim$2% natural abundance). The two observed ESR peaks correspond to two
nuclear spin states, and their almost equal ESR peak heights indicate that the
thermal populations of the two nuclear spins are nearly identical. The
simultaneous appearance of two ESR peaks in one spectrum implies that the
nuclear spin relaxes faster than the measurement duration of $\sim$1 ms within
a lock-in cycle (see section 2.6.1). To relate the ESR splitting to the
hyperfine coupling strength, first note that the large electron spin
anisotropy of Fe on MgO sets its electron spin perpendicular to the surface.
Through the hyperfine interaction, the nuclear spin is also aligned to this
direction. The Hamiltonian of 57Fe on MgO can then be written as

$H=g_{z}\mu_{\mathrm{B}}B_{z}S_{z}+A_{z}I_{z}S_{z},$ (42)

where the $z$-axis is the out-of-plane direction, and the nuclear electric
quadrupole interaction is disregarded. Since we are only concerned with the
two lowest electron spin states of Fe (which are approximately the $m_{S}=\pm
2$ states) [29], here we treat Fe as an effective two-level system by ignoring
the other three spin states (which justifies the absence of the magnetic
anisotropy term in the Hamiltonian in Equation 39). Because the Fe hyperfine
spectrum involves $m_{S}=\pm 2$ and $m_{I}=\pm 1/2$ states, the ESR splitting
in Figure 7a corresponds to $\Delta f=\Delta m_{S}\Delta m_{I}A_{z}=4A_{z}$,
which allows the determination of the $z$-component of the hyperfine coupling
constant as $A_{z}=58$ MHz.

The hyperfine spectra of Ti on MgO have more diversity than Fe in that Ti’s
hyperfine interactions are nuclei- and site-dependent. Figure 7b shows three
different ESR spectra obtained for bridge-site Ti. Because one ESR peak splits
into $2I+1$ sub-peaks in the presence of a nuclear spin $I$, the observed
one-, six-, and eight-peak ESR spectra correspond to 48Ti ($I=0$), 47Ti
($I=5/2$) and 49Ti ($I=7/2$), respectively. The ESR peak splittings were found
to be equal ($\sim$47 MHz) for 47Ti and 49Ti, indicating their
indistinguishable nuclear gyromagnetic ratios.

Interestingly, STM atom manipulation can be used to alter Ti’s hyperfine
interaction by modifying its binding-site-dependent local environment. Figure
7c shows that the ESR peak splitting was significantly reduced when 47Ti was
moved from a bridge site to an oxygen site. The splitting was restored when
the atom was relocated to another bridge site. In ref. [45], a higher-
resolution ESR spectrum shows that the broad ESR peak for oxygen-site Ti is
composed of several not-fully-resolved peaks with irregular intervals and
nonuniform intensities. To elucidate why this is the case, consider the full
Hamiltonian in Equation 39

$H=H_{\mathrm{EZ}}+H_{\mathrm{HF}}+H_{\mathrm{NQ}}=\mu_{\mathrm{B}}\sum_{i=x,y,z}g_{i}B_{i}S_{i}+\sum_{i=x,y,z}A_{i}S_{i}I_{i}+\sum_{i=x,y,z}P_{i}I_{i}^{2}.$
(43)

For oxygen-site Ti, the smaller ESR peak splittings reflect a reduction of the
hyperfine constant, $A_{i}$, making it comparable to the nuclear quadrupole
interaction strength, $P_{i}$. Therefore, ill-defined eigenstates in the
electron and nuclear spin space result in irregularly spaced ESR peaks seen on
oxygen-site Ti.

A more careful, orbital-based analysis reveals the origin of the very
different hyperfine coupling strengths between bridge- and oxygen-site Ti.
Recall that the hyperfine interaction contains two contributions: an isotropic
contact term and an anisotropic dipolar term. The contact contribution for
bridge-site Ti ($+50$ MHz) was found to be already larger than oxygen-site Ti
($+19$ MHz). The dipolar contribution additionally enhances the hyperfine
coupling of bridge-site Ti but cancels it for oxygen-site Ti due to their
different orbital occupations [45].

Individual Cu atoms on MgO [35] exhibit hyperfine interactions that are two
orders of magnitude greater than those of Ti. Cu atoms have two stable
isotopes, 63Cu and 65Cu, with natural abundances of 69% and 31%, respectively.
On MgO, both isotopes have electron spins of 1/2 and nuclear spins of 3/2, and
the eight spin eigenstates are labeled as $\ket{i}$ ($i=$1 to 8) in Figure 7d.
The ESR spectra of 65Cu show four hyperfine-split ESR peaks. The hyperfine
constant, $A$, extracted from the four ESR frequencies [35] is about 2.86 GHz
for 63Cu and about 3.05 GHz for 65Cu, which highlights how ESR-STM can be used
to distinguish different isotopes atom-by-atom. The large hyperfine constants
of Cu on MgO, compared to Cu in other environments or other atoms on MgO, stem
from the significant $s$-electron contribution ($\sim$60% occupation of the
4$s$ orbital) to the electron spin, which produces a large contact hyperfine
coupling. Due to the strong hyperfine interaction of Cu on MgO, considerable
mixing of electron and nuclear spin states occurs and creates uneven spacing
of the four ESR peaks. The different ESR frequencies of the four peaks also
cause a thermally-induced nuclear polarization along its quantization axis $z$
($\braket{I_{z}}/I\sim$ 1.7% at 1.2 K and 0.65 T), which is reflected by the
different ESR amplitudes of the four peaks in Figure 7d.

As shown in this section, ESR-detected hyperfine interactions are sensitive to
the atomic surroundings. This type of hyperfine spectroscopy, combined with
STM’s imaging and atom manipulation capabilities, can be utilized to extract
information from the local chemical and electronic environments of a single
spin.

###  3.3 Functionalized-Tip-Based Magnetic-Field Sensing

Aside from using spin sensors on a surface as described in the previous
sections, attaching a quantum spin to the STM tip enables scanning
magnetometry at the atomic scale. The difficulty here is to identify a
suitable spin carrier that can preserve its magnetic properties at the tip and
allow a way to detect the magnetic signal. Spin-1 nickelocene molecules were
found to be one such spin carrier, of which the spin states and magnetic
anisotropy are robust in a variety of metallic environments including at a
metallic tip apex, likely related to the protection by the cyclic $\pi^{*}$
orbital of the Cp rings that sandwiches the Ni atom [134]. Given the
resolvable spin states of nickelocene at the tip apex, a local magnetic field
can alter the spin state energies and be detected through IETS or ESR
spectroscopy. This is a major advantage compared to conventional spin-
polarized STM, where the magnetic structure and excitations of the magnetic
tip apex are mostly unknown. IETS-based magnetic-field imaging with
nickelocene on an STM tip was recently accomplished in pioneering works in
Refs. [135, 136], where the magnetic exchange interactions from individual
magnetic atoms or molecules on surfaces were spatially mapped by tracking the
variations in IETS spectra of the nickelocene molecule on the tip [135, 136].
In a similar measurement scheme, a spin-1/2 cobaltocene molecule attached to
an STM tip was found to exhibit a Kondo resonance, and the Kondo resonance
splitting was used to extract the exchange field of an Fe atom on the surface
[137].

Albeit impressive realizations of atomic-scale magnetic-field imaging,
mappings performed using conventional STM spectroscopy are limited to the sub-
meV energy resolution due to thermal broadening effects (see the introduction
in section 3). We expect that future investigations using an ESR-active
molecule attached to an STM tip can allow for the combination of high energy
and spatial resolutions, thus enabling sensitive scanning quantum sensing in
three spatial dimensions at the atomic scale.

##  4 Quantum Control

Over the last two decades, global efforts have been made to promote the
coherence of solid-state qubits to a level suitable for quantum information
processing. Compared with ions or atoms trapped in a vacuum, solid-state spins
are easier to scale up thanks to modern fabrication techniques, but they
suffer from additional decoherence sources in the solid-state environment. One
way to reduce decoherence is by using nuclear spins which can possess
coherence times as long as hours [138] due to their insensitivity to magnetic
noise, but this also means that nuclear spins are more difficult to manipulate
via magnetic fields.

Quantum control of well-characterized single spins and coupled spin systems on
surfaces has recently been demonstrated using ESR-STM [30]. Spins on surfaces
have the potential to form bottom-up quantum nanodevices and shed light on
decoherence mechanisms in solid-state environments. In this section, we first
discuss the sources of decoherence of surface spins. We then discuss recent
developments on the coherent control of a single spin and the extensions to
multi-spin quantum control.

###  4.1 Quantum Coherence of Spins on Surfaces

Decoherence, the loss of a quantum system’s phase information, occurs from its
interactions with the surrounding environment. Ensuring a long coherence time
is of practical importance to achieve high-fidelity quantum-information
processing, high-sensitivity quantum sensing, and high-efficiency quantum
communications. Improving coherence relies on a good understanding of the
decoherence sources.

While the energy relaxation time ($T_{1}$) of surface spins has been
intensively studied using STM [43, 24, 80, 58] and was found to exceed 10 ms
for Fe on MgO in a high magnetic field [58], the coherence time ($T_{2}$) has
barely been explored until recently. The development of ESR-STM has prompted
interest in identifying the decoherence sources of surface spins both
experimentally [88, 36] and theoretically [139, 140, 141, 142, 143, 144, 145].
Section 4.1.1 discusses this experimental endeavor. Section 4.1.2 discusses
the use of singlet-triplet states for the enhancement of quantum coherence.

####  4.1.1 Decoherence of Single Spins on Surfaces

The first systematic experimental study of the coherence time ($T_{2}$) of a
surface spin was performed on isolated Fe atoms on 2ML MgO (Figure 8a) [88].
Under typical ESR-STM measurement conditions, two dominant decoherence sources
were identified: the tunneling electrons and the local magnetic field
fluctuations from the magnetic tip.

Figure 8: Spin coherence and decoherence in ESR-STM. a) Experimental setup for
ESR measurements and the energy-level diagram of the five lowest energy levels
of an Fe atom on MgO/Ag(100). The external magnetic field was applied mostly
along the in-plane direction. The two lowest levels, $\ket{0}$ and $\ket{1}$,
were split by $f_{0}$ $\approx$ 21 GHz (87 $\mu$eV) due to the small out-of-
plane component of the external magnetic field ($B_{z}$). b) ESR linewidth as
a function of the renormalized drive amplitude (see the $x$ axis label). Each
curve corresponds to a different tunnel current (ranging from 1 to 30 pA). c)
Extracted spin coherence time from (b) plotted against the tunnel current.
Reproduced with permission.[88] 2018, AAAS.

The influence of tunneling electrons on the relaxation and decoherence of an
Fe spin can be shown by measuring the ESR peak intensity and broadening as a
function of RF power ($V_{\mathrm{RF}}$) at different current setpoints
($I_{\mathrm{t}}$) (Figure 8b). The ESR peak intensity ($I_{\mathrm{peak}}$)
and the linewidth ($\Gamma$) due to DC readout can be described by the steady-
state solution to the Bloch equations (Equation 23) as

$\Delta
I_{\mathrm{DC}}=\frac{I_{\mathrm{peak}}}{1+(f-f_{0})^{2}/(\Gamma/2)^{2}}\propto(\braket{S_{z}}-\braket{S_{z}^{0}}),\
\
I_{\mathrm{peak}}=I_{\mathrm{sat}}\cdot\frac{\Omega^{2}T_{1}T_{2}}{\Omega^{2}T_{1}T_{2}+1},\
\ \Gamma=\frac{1}{\pi T_{2}}\sqrt{1+\Omega^{2}T_{1}T_{2}},$ (44)

where $f-f_{0}$ is the frequency detuning, and $\Omega$ is the on-resonance
Rabi rate, which is typically proportional to $V_{\mathrm{RF}}$. When the two
spin states are equally populated due to strong RF driving, the ESR peak
intensity detected by $V_{\mathrm{DC}}$ is saturated to $I_{\mathrm{sat}}$
[88]. By fitting $I_{\mathrm{peak}}$ and $\Gamma$ to Equation 44, we can
extract $T_{2}$ under different tunnel conditions from the slope of the curves
shown in Figure 8b. As plotted in Figure 8c, $T_{2}$ rapidly increases with
decreasing tunnel current $I_{\mathrm{t}}$ in a roughly inversely proportional
fashion, showing that tunnel current has a considerable influence on $T_{2}$.
A careful analysis reveals that the decoherence rate has both tunnel-current-
dependent and tunnel-current-independent terms:

$T_{2}^{-1}=\frac{P_{T_{2}}}{e}\cdot I_{\mathrm{t}}+T_{2}^{-1}(0).$ (45)

The average probability of a tunnel electron that causes decoherence,
quantified by $P_{T_{2}}$, was found to be as high as 64% [88]. In contrast,
only a very small fraction of tunneling electrons (0.5%) causes the energy
relaxation ($T_{1}$ process) of the Fe spin due to its large anisotropy
barrier [58]. By extrapolating $T_{2}$ to the zero-current limit, one can
obtain the current-independent decoherence rate, $T_{2}^{-1}(0)$, that
corresponds $T_{2}(0)=50\pm 6\,\mathrm{ns}$ as shown in Figure 8c. The value
of $T_{2}^{-1}(0)$ and its temperature dependence were found to vary widely
from one magnetic tip to the other, indicating that the current-independent
decoherence is caused by thermally induced changes of the tip’s magnetic
moment. The tunnel current and tip’s magnetic fluctuations are thus
detrimental to spin coherence, which can be partially mitigated by using a
remote driving and sensing scheme as discussed in section 4.3.2.

####  4.1.2 Enhanced Quantum Coherence in Singlet-Triplet States

The decoherence in ESR-STM can be significantly suppressed by using specific
two-level transitions that are designed to be insensitive to magnetic field
fluctuations [146, 147, 148, 149]. These so-called “atomic clock transitions”
can be constructed with spins on surfaces by using singlet and triplet states
that carry zero magnetic quantum numbers, as we shall discuss in this section.

Figure 9: Enhanced spin coherence using singlet-triplet transitions. a,b) ESR
spectrum for Ti-Ti pairs with atomic separations of (a) $r$ = 0.92 nm and (b)
0.72 nm in a nearly in-plane magnetic field of 0.9 T. Each pair composes of an
oxygen-site Ti (TiO) and a bridge-site Ti (TiB). For the pair in (a) at $r$ =
0.92 nm, four ESR peaks were observed (the splitting corresponding to the
coupling energy $J$ is labelled). For the closer pair in (b) at $r$ = 0.72 nm,
three ESR peaks were observed where the magenta one arises from the ST0
transition. Insets: STM images of TiO-TiB pairs. Grid line intersections
represent the positions of oxygen atoms in the MgO lattice. The “x” marks
represent the tip positions while obtaining the ESR spectra. c,d) Schematic
energy level diagrams for Ti pairs in (a,b). In (c), the Zeeman energy is the
dominant energy scale, and so the triplet state T– is the ground state. In
(d), the coupling energy $J$ exceeds the Zeeman energy, and so the singlet
state S becomes the ground state. The corresponding resonance peaks in (a,b)
and ESR transitions in (c,d) are marked by the same colors. e) DC bias voltage
dependence of ESR signals of the T0T– (top) and the ST0 (bottom) transitions
measured for the Ti pair in (b). The tip-atom separation was maintained at a
constant value during this measurement. The ST0 transition is measurable at
zero $V_{\mathrm{DC}}$ due to the homodyne detection. f) The spin coherence
time $T_{2}^{*}$ as a function of the DC bias voltage $V_{\mathrm{DC}}$ at a
constant tip-atom separation. Red curves are fits to $1/V_{\mathrm{DC}}$.
$\sharp 1$ and $\sharp 2$ label the results from two different TiO-TiB pairs
(having the same separations but measured with two different tips). Reproduced
with permission.[36] 2018, AAAS.

In Ref. [36], two strongly coupled spin-1/2 Ti atoms on MgO were used to
create singlet and triplet states with zero magnetic quantum numbers ($m=0$)
(Figure 9). In general, two coupled spin-1/2 atoms yield four eigenstates
denoted as $\ket{\text{S}(\xi)}$, $\ket{\text{T}_{0}(\xi)}$,
$\ket{\text{T}_{-}}$, and $\ket{\text{T}_{+}}$ (see section 2.7).
$\ket{\text{S}(\xi)}$ and $\ket{\text{T}_{0}(\xi)}$ become the singlet and
triplet states (S and T0) at the perfect mixing angle $\xi=\pi/2$. This occurs
when the spin-spin interaction substantially exceeds the Zeeman energy
difference between them (see section 2.7). Using atom manipulation, the
spacing between two Ti atoms can be precisely controlled. At atomic separation
below 1.3 nm, the magnetic interaction between Ti atoms in the pair
exponentially increases due to their exchange coupling, and the resulting spin
states become closer to the S and T0 states.

The ESR spectra for two different Ti-pair configurations are shown in Figure
9a,b. When the Ti pair is separated relatively far by 0.92 nm, four ESR
transitions of $\Delta m=\pm 1$ appear in the measurable frequency range
(Figure 9a,c). The four peaks can be divided into two groups, each split by
the coupling energy $J\sim$ 0.8 GHz. Because in this pair the Ti-Ti coupling
energy does not exceed their Zeeman energy difference at 0.9 T, the singlet
(S) and triplet (T0) states are not well developed. When two Ti atoms are
placed closer (0.72 nm, Figure 9b,d), the Ti-Ti magnetic exchange energy
increases to $\sim$30 GHz and dominates over their Zeeman energy difference at
0.9 T. As a result, nearly perfect S and T0 states appear, with the singlet
state S being the ground state. In the frequency range of 5–32 GHz, three ESR
transitions were observed for the 0.72 nm pair, two of which are the triplet-
triplet transitions of $\Delta m=\pm 1$ and the other is from the desired
singlet-triplet transition of $\Delta m=0$ (magenta curve in Figure 9b).

Interestingly, the ESR driving of the singlet-triplet ST0 transition differs
from that of other transitions. First, we should note that unlike traditional
ensemble ESR where the RF magnetic fields are designed to be maximized along a
transverse axis $x$ (perpendicular to the quantization axis, $z$), the RF
magnetic field in ESR-STM reflects the direction of the magnetic field
gradient generated by the STM tip (see section 2.5) and typically contains
components both perpendicular and parallel to the quantization axis. We can
include the RF driving magnetic fields into Equation 35. Assuming that the
tip’s magnetic field acts only on the first spin, the Hamiltonian becomes

$\displaystyle
H=-\hbar\omega_{\mathrm{0,1}}S_{1z}\otimes\mathcal{I}-\hbar\omega_{\mathrm{0,2}}\mathcal{I}\otimes
S_{2z}+(J_{\mathrm{e}}+2J_{\mathrm{d}})S_{1z}\otimes
S_{2z}+(J_{\mathrm{e}}-J_{\mathrm{d}})\times(S_{1x}\otimes
S_{2x}+S_{1y}\otimes S_{2y})$ $\displaystyle+2\hbar\Omega_{1x}\cos(\omega
t+\phi)S_{1x}\otimes\mathcal{I}+2\hbar\Omega_{1z}\cos(\omega
t+\phi)S_{1z}\otimes\mathcal{I}$
$\displaystyle=\frac{1}{2}\Bigg{[}\begin{smallmatrix}-\hbar\omega_{\mathrm{0,1}}-\hbar\omega_{\mathrm{0,2}}+\frac{J_{\mathrm{e}}+2J_{\mathrm{d}}}{2}+\hbar\tilde{\Omega}_{1z}&0&\hbar\tilde{\Omega}_{1x}&0\\\
0&-\hbar\omega_{\mathrm{0,1}}+\hbar\omega_{\mathrm{0,2}}-\frac{J_{\mathrm{e}}+2J_{\mathrm{d}}}{2}+\hbar\tilde{\Omega}_{1z}&J_{\mathrm{e}}-J_{\mathrm{d}}&\hbar\tilde{\Omega}_{1x}\\\
\hbar\tilde{\Omega}_{1x}&J_{\mathrm{e}}-J_{\mathrm{d}}&\hbar\omega_{\mathrm{0,1}}-\hbar\omega_{\mathrm{0,2}}-\frac{J_{\mathrm{e}}+2J_{\mathrm{d}}}{2}-\hbar\tilde{\Omega}_{1z}&0\\\
0&\hbar\tilde{\Omega}_{1x}&0&\hbar\omega_{\mathrm{0,1}}+\hbar\omega_{\mathrm{0,2}}+\frac{J_{\mathrm{e}}+2J_{\mathrm{d}}}{2}-\hbar\tilde{\Omega}_{1z}\end{smallmatrix}\Bigg{]}.$
(46)

Here $\omega_{\mathrm{0,1}}$ and $\omega_{\mathrm{0,2}}$ are the Larmor
frequencies of spins 1 and 2, respectively, $\omega$ is the applied RF
frequency, and $\tilde{\Omega}_{1x,z}=2\Omega_{1x,z}\cos(\omega t+\phi)$ are
the shorthand expressions for the driving terms, where $\Omega_{1x}$ and
$\Omega_{1z}$ are the Rabi rates that characterize the RF field strengths for
spin 1 along the $x$ and $z$ directions in the lab frame.

When the two spins have the same Zeeman splitting,
$\omega_{\mathrm{0,1}}=\omega_{\mathrm{0,2}}$, the eigenstates
$\ket{\text{S}(\xi=\pi/2)}$, $\ket{\text{T}_{0}(\xi=\pi/2)}$,
$\ket{\text{T}_{-}}$, and $\ket{\text{T}_{+}}$ can be connected to the
original Zeeman state basis by a unitary transformation $U$

$\begin{bmatrix}\ket{00}\\\ \frac{1}{\sqrt{2}}(\ket{01}-\ket{10})\\\
\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})\\\
\ket{11}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\\
0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\ 0&0&0&1\\\
\end{bmatrix}\begin{bmatrix}\ket{00}\\\ \ket{01}\\\ \ket{10}\\\
\ket{11}\end{bmatrix}=U\begin{bmatrix}\ket{00}\\\ \ket{01}\\\ \ket{10}\\\
\ket{11}\end{bmatrix}.$ (47)

In the new singlet-triplet basis, the Hamiltonian becomes

$UHU^{\dagger}=\frac{1}{2}\Bigg{[}\begin{smallmatrix}-\hbar\omega_{\mathrm{0,1}}-\hbar\omega_{\mathrm{0,2}}+\frac{J_{\mathrm{e}}+2J_{\mathrm{d}}}{2}+\hbar\tilde{\Omega}_{1z}&\frac{-\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}&\frac{\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}&0\\\
\frac{-\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}&-\frac{3J_{\mathrm{e}}}{2}&-\hbar\omega_{\mathrm{0,1}}+\hbar\omega_{\mathrm{0,2}}+\hbar\tilde{\Omega}_{1z}&\frac{\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}\\\
\frac{\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}&-\hbar\omega_{\mathrm{0,1}}+\hbar\omega_{\mathrm{0,2}}+\hbar\tilde{\Omega}_{1z}&\frac{J_{\mathrm{e}}}{2}-2J_{\mathrm{d}}&\frac{\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}\\\
0&\frac{\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}&\frac{\hbar\tilde{\Omega}_{1x}}{\sqrt{2}}&\hbar\omega_{\mathrm{0,1}}+\hbar\omega_{\mathrm{0,2}}+\frac{J_{\mathrm{e}}+2J_{\mathrm{d}}}{2}-\hbar\tilde{\Omega}_{1z}\end{smallmatrix}\Bigg{]}.$
(48)

Focusing on the central $2\times 2$ block of Equation 48 that corresponds to
the S and T0 states, we can see that the energy levels for the ST0 transition
(diagonal terms) are determined only by the coupling energies
($J_{\mathrm{e}}$ and $J_{\mathrm{d}}$), and so the S and T0 states are
insensitive to magnetic field fluctuations. The off-diagonal terms of the
central $2\times 2$ ST0 matrix represents the Zeeman energy difference between
the two spins. As a result, in order to resonantly drive the ST0 transition, a
local RF field ($\tilde{\Omega}_{1z}$) needs to be supplied to only one of the
two spins along its quantization axis, $z$ (to supply a Zeeman energy
difference). Other transitions with $\Delta m=\pm 1$, in contrast, have energy
splittings that are sensitive to magnetic field fluctuations and can be driven
by both global and local transverse RF magnetic fields (as represented by the
$\tilde{\Omega}_{1x}$ terms in Equation 48).

The noise immunity of the singlet-triplet transition was found to greatly
enhance its coherence time. Under typical ESR conditions of
$V_{\mathrm{DC}}=40\,\mathrm{mV}$ and $I_{\mathrm{t}}=10\,\mathrm{pA}$, the
spin coherence time for the singlet-triplet (ST0) transition was found to be
$T_{2}^{*}\sim 100\,\mathrm{ns}$. This value is $\sim$8 times longer than the
$T_{2}^{*}$ for the T0T– transition (where $\Delta m\neq 0$) in the same