E*KL*,*kl*=C*l*2+(*k*+1/2)*ħω*s+(2*K*+|*L*|+1) *ħω*b, (1) where the rotational constant C≈0.037, the phonon for the stretching vibration has energy 1.75ω0 and the phonon for the bending vibration coincides with that of the c.m. motion, i.e. ωb= ω0. Note that the bending vibration can itself carry an angular momentum *ħL* and thus rotational angular momentum *ħl* does not necessary coincide with the total angular momentum *ħ*(*L*+*l*). The calculations have shown that the molecule preserves its structure at *rs* =3 (*ħω*0~1 meV for GaAs), i.e. below the "theoretical" Fermi liquid to WM transition. Here the rotational sequence shows nearly equal spacing having value ω0/2 but the stretching

We used an accurate configuration-interaction (CI) method employing a numerical meanfield basis set to study the excitation spectrum of a six-electron WM (Blundell & Chacko, 2011). The CI method (Szabo&Ostlund, 1996; Blundell&Joshi, 2010) is more suitable for the systematic study of excited states than other methods such as Hartree-Fock-based methods and variational Quantum Monte-Carlo (which can treat only the lowest-energy state of a given symmetry). Dots with N ≥ 6 electrons in general have more than one classical isomer (see, for example, the [1,5] and [0,6] isomers in Fig. 1) and therefore isomeric states should form an important part of the phenomenology of excited states. Now, for a circularly

Fig. 2. Electron spatial pair correlation functions for *rs*=2.75, 4, 8 and 20 (left) and excitation energies of the six-electron dot versus Wigner-Seitz radius *r*s (right). The quantities ∆Erot, ∆Evib,

and ∆Eiso are approximate rotational, vibrational, and isomeric excitation energies, respectively, inferred from a classical model (see text); ∆E(3S) and ∆E(3P) are excitation energies to the lowest 3S and 3P states calculated by CI. All excitation energies are scaled by *rs*.

exhibiting collective rotations, as well as stretching and bending vibrations:

vibration does not change.

**3.2 Excited states of six-electron Wigner molecules**

**3.2.1 Configuration-interaction calculations of spin states**

symmetric external potential and a state of definite *Lz*, the electronic density (in the "laboratory" frame) in 2D must also be circularly symmetric (Hirose&Wingreen, 1999) and in the Wigner limit the density therefore becomes a series of concentric rings (see, for example, Ghosal, et al., 2006). To reveal the Wigner localization, we therefore consider the internal many-body correlations by means of the electronic (charge-charge) paircorrelation functions (PCFs) *g*(**r**; **r**0) (Maksym, 1996; Reimann et al., 2000; Yannouleas&Landman**,** 2000). The quantity *g*(**r**; **r**0) is proportional to the conditional probability of finding an electron at the position **r** given that another (reference) electron is present at **r**0. Calculated *g*(**r**; **r**0) functions are presented for *rs* = 2.75, 4, 8, and 20 in Fig. 2, from which it is seen that a partially correlated state is observed even at *rs* = 2.75. Recall that *rs* is expressed in units of the effective Bohr radius *aB\**, where *aB\** ≈ 8.7 nm for the InP/GaInP dots in our experiments.

Our calculations have shown that the evolution of the excitation energy of the lowest 3*S*, 5*S*, 7*S*, and 3*P* states relative to the 1*S* ground state versus *rs* yields approximately parallel straight lines on a logarithmic plot 6 ≤ *rs* ≤ 10, the excitation energy ∆*E* of these states being well fit by an expression of the form ∆*E* = *c* exp(–*m rs*), with *c*(3*P*) = 0.020 Ha\*, *c*(5*S*) = 0.028 Ha\*, *c*(3*S*) = 0.048 Ha\*, *c*(7*S*) = 0.054 Ha\*, and *m* ≈ 0.30 (*aB\**)–1. The energy units Ha\* here are effective Hartrees, with 1 Ha\* ≈ 13.2 meV for the InP/GaInP dots used in our experiments.

### **3.2.2 Low-lying excitations in the classical limit**

At large *rs* the quantum excitation energy of a quasi-2D Wigner molecule may be written approximately in a way analogous to that for a planar molecule

$$E(\mathbf{P}) = E\_{\rm cl}(\mathbf{P}) + L\_z^2 / (2I\_{\rm P}) + \Sigma\_a \left[ \Omega\_b(\mathbf{P})(n\_a + 1/2) \right] + E\_{\rm spin} \tag{2}$$

where *E*cl(P) is the classical electrostatic energy of isomer P, *IP* is its moment of inertia, and *na* is the number of vibrational quanta in a normal mode with frequency Ω*a*(P). The energy *E*spin is the spin-spin interaction energy of the spins of the localized electrons. As an example of a typical rotational excitation energy, we note that the ground-state [1,5] isomer has a moment of inertia *IP* = 8.9 *rs 2N1/3*, and it then follows using Eq. (2) that the *S*- to *P*-wave excitation energy is given in the classical limit by (in effective a.u.) ∆*E*rot = 0.0309 *rs* –2. Similarly, noting that the frequency of the first classical normal mode of the [1,5] isomer is Ω1 = 0.650ω0 (see below), we find that the excitation energy of one vibrational quantum in this mode is (in effective a.u.) ∆*E*vib = 0.415 *rs –*3/2. For isomeric excitations we found (Blundell &Chacko, 2011) ∆*E*iso = *E*cl (0, 6) – *E*cl (1, 5) = 0.0714 *rs –*1. To clarify the role of rotational excitations, we show in Fig. 2 the classical estimate of the *S*- to *P*-wave excitation energy ∆*E*rot as a function of *rs*, together with the excitation energy calculated by CI (from the 1*S* ground state to the lowest 3*P* state, which is the lowest-lying *P*-wave state).

We also show a typical spin excitation energy ∆*E*(3*S*), defined as the energy of the lowest 3*S* state relative to the 1*S* ground state, as calculated by CI. From Fig. 2, one sees that the smallest rotational excitation for *rs* <10 is in fact somewhat larger than the classical estimate ∆*E*rot. This is simply because a spin excitation is also involved. One also sees in Fig. 2 that for *rs* < 6, the "spin" excitation energy (due to atomic-like exchange and correlation effects) is nominally comparable to the isomeric and vibrational energies. Thus, although at these values of *rs* it is possible to find partial Wigner localization in a recognizable geometry, it is

Molecular States of Electrons: Emission of

**4.1 Structural properties**

layer thickness 5, 20, 40 and 60 nm.

Single Molecules in Self-Organized InP/GaInP Quantum Dots 131

at intermediate *rs* is an isomer rather than a vibrational excitation of the ground state. We thus expect low-lying isomeric states to be a generic feature of the excitation spectrum of dots with *N* = 6 and *N* ≥ 9 electrons at intermediate *rs* values. Also, the rotational excitation energies ∆*E*rot

Our InP/GaInP QD samples were grown by Metal-Organic Chemical Vapor Deposition (MOCVD) in a horizontal AIX200/4 reactor under pressure of 100 mbar. Trimethylgallium (TMGa), trimethylaluminium (TMAl) and trimethylindium (TMIn) metalorganic compounds were used as the group III element sources. Arsine (AsH3) and phosphine (PH3) were used as the group V element sources. (100) GaAs substrates misoriented by 2o towards the [110] direction were used. Initially a 50 nm-thick GaAs layer was deposited on the wafer. Then 50 nm thick Ga0.52In0.48P (GaInP) lattice-matched with GaAs layer was grown. The QDs were grown at 725o C by depositing 7 monolayers (ML) of InP (Vinokurov et al,. 1999; Chu et al., 2009,). We studied the structures with uncapped dots and structures having GaInP cap

The dot density (~2\*109 cm-2) and their sizes (base ~10-200 nm, height ~5-60 nm) were measured using atomic force microscopy (AFM) for the uncapped samples and transmission electron microscopy (TEM) for the capped samples. These data are presented in Fig.4a-d. A clear bimodal size distribution is seen from AFM images in Fig.4a consisting of large dots having size 100-200 nm and density 0.6x109 cm-2 and the small dots having sizes 10-70 nm and density 1.2x109 cm-2. From TEM measurements a lens shape of the large dots was revealed (see Fig.4c and d). Due to a residual n-type doping of the GaInP layer in the MOCVD growth process (n~1016 cm-3) the modulation doping of InP QDs occurs and they can contain up to 20 electrons (Hessman et al., 2001). Thus these QDs can represent natural WM and for dot sized 150-200 nm the *rs* ~4 can be achieved for few electron dots. The electron density can be varied from 1010 up to 5x1011 cm-2 and thus these InP QDs offer much more flexibility in varying of WM parameters then "conventional" GaAs/AlGaAs QDs used in SETs (Singha et al, 2010).

Fig. 4. Atomic force (a and b, size 2.5x2.5 *μ*m2) and transmission electron (c – plan-view, d –

We estimated that our InP QDs can contain up to 10% Ga. Pure InP QDs can be grown by depositing nominally 0.5 ML of InP at growth temperatures 580 C and using in-situ annealing/growth interruption. Such growth conditions suppress Ga and In intermixing

cross section) microscopy images of InP/GaInP QDs structures.

are generally found to be small compared to ∆*E*iso and ∆*E*vib, similar to Fig. 2 for *N* = 6.

**4. InP/GaInP quantum dots as natural electronic molecules**

not generally possible to discuss isomeric and vibrational excitations separately from spin excitations at these densities.

#### **3.2.3 Vibrational modes**

The classical normal modes for the [1,5] ground-state isomer are shown in Fig. 3. There are 2N – 1 = 11 normal modes grouped into five doubly degenerate modes and one nondegenerate mode, which is a breathing mode at high frequency. The lowest frequency mode can be thought of as a dipolar oscillation of the central electron accompanied by a distortion of the outer ring. The third and fourth modes, at Ω=1.223ω0 and Ω=1.314ω0, correspond to quadrupole and octupole distortions, respectively, of the outer ring, with the central electron remaining fixed. The second mode is a collective dipolar oscillation of the c.m. of the system at frequency Ω=ω0 (exactly), in which the whole structure remains undistorted during the oscillation. The existence of such a classical mode can be shown to be a general result for a system in a harmonic confining potential having an interaction depending only on the relative coordinates of the particles. The quantum mechanical analog of this result is the Kohn theorem, according to which under the same circumstances the c.m. motion decouples exactly from the "relative coordinates," and one can describe the system by a wave function in relative coordinates combined with oscillations of the c.m. in the harmonic confining potential. Note that the breathing mode of the six electron molecule has frequency Ω=1.732ω0, which is nearly the same as the frequency of the breathing mode of the two electron molecule considered above.

Fig. 3. Normal modes of the ground-state isomer of the classical six-electron dot. The normal mode frequency Ω is indicated as a multiple of the frequency of the parabolic potential ω0.

#### **3.3 Generalization to larger size N**

We can use the classical arguments that work well for *N* = 6 to show that aspects of the same excited-state phenomenology apply to larger *N* as well. The classical model yields more than one isomer for *N* = 6 and for *N* ≥ 9. We have used the "basin hopping" algorithm to generate and study the classical isomers in the size range up to *N* = 20. As *N* increases, the energy separation of isomers tends to become smaller. Thus the excitation energy ∆*E*iso(*N*) for *N* electrons satisfies ∆*E*iso(6) = 0.10ω0 2/3, ∆*E*iso(9) = 0.044ω0 2/3 and ∆*E*iso(19) = 0.013ω0 2/3 (in effective a.u.). It is then generally the case that the first excited level (spin multiplet) for fixed *Lz*

at intermediate *rs* is an isomer rather than a vibrational excitation of the ground state. We thus expect low-lying isomeric states to be a generic feature of the excitation spectrum of dots with *N* = 6 and *N* ≥ 9 electrons at intermediate *rs* values. Also, the rotational excitation energies ∆*E*rot are generally found to be small compared to ∆*E*iso and ∆*E*vib, similar to Fig. 2 for *N* = 6.

### **4. InP/GaInP quantum dots as natural electronic molecules**

### **4.1 Structural properties**

130 Fingerprints in the Optical and Transport Properties of Quantum Dots

not generally possible to discuss isomeric and vibrational excitations separately from spin

The classical normal modes for the [1,5] ground-state isomer are shown in Fig. 3. There are 2N – 1 = 11 normal modes grouped into five doubly degenerate modes and one nondegenerate mode, which is a breathing mode at high frequency. The lowest frequency mode can be thought of as a dipolar oscillation of the central electron accompanied by a distortion of the outer ring. The third and fourth modes, at Ω=1.223ω0 and Ω=1.314ω0, correspond to quadrupole and octupole distortions, respectively, of the outer ring, with the central electron remaining fixed. The second mode is a collective dipolar oscillation of the c.m. of the system at frequency Ω=ω0 (exactly), in which the whole structure remains undistorted during the oscillation. The existence of such a classical mode can be shown to be a general result for a system in a harmonic confining potential having an interaction depending only on the relative coordinates of the particles. The quantum mechanical analog of this result is the Kohn theorem, according to which under the same circumstances the c.m. motion decouples exactly from the "relative coordinates," and one can describe the system by a wave function in relative coordinates combined with oscillations of the c.m. in the harmonic confining potential. Note that the breathing mode of the six electron molecule has frequency Ω=1.732ω0, which is nearly the same as the frequency of the breathing mode

Fig. 3. Normal modes of the ground-state isomer of the classical six-electron dot. The normal mode frequency Ω is indicated as a multiple of the frequency of the parabolic potential ω0.

We can use the classical arguments that work well for *N* = 6 to show that aspects of the same excited-state phenomenology apply to larger *N* as well. The classical model yields more than one isomer for *N* = 6 and for *N* ≥ 9. We have used the "basin hopping" algorithm to generate and study the classical isomers in the size range up to *N* = 20. As *N* increases, the energy separation of isomers tends to become smaller. Thus the excitation energy ∆*E*iso(*N*) for *N*

2/3, ∆*E*iso(9) = 0.044ω0

effective a.u.). It is then generally the case that the first excited level (spin multiplet) for fixed *Lz*

2/3 and ∆*E*iso(19) = 0.013ω0

2/3 (in

excitations at these densities.

of the two electron molecule considered above.

**3.3 Generalization to larger size N**

electrons satisfies ∆*E*iso(6) = 0.10ω0

**3.2.3 Vibrational modes**

Our InP/GaInP QD samples were grown by Metal-Organic Chemical Vapor Deposition (MOCVD) in a horizontal AIX200/4 reactor under pressure of 100 mbar. Trimethylgallium (TMGa), trimethylaluminium (TMAl) and trimethylindium (TMIn) metalorganic compounds were used as the group III element sources. Arsine (AsH3) and phosphine (PH3) were used as the group V element sources. (100) GaAs substrates misoriented by 2o towards the [110] direction were used. Initially a 50 nm-thick GaAs layer was deposited on the wafer. Then 50 nm thick Ga0.52In0.48P (GaInP) lattice-matched with GaAs layer was grown. The QDs were grown at 725o C by depositing 7 monolayers (ML) of InP (Vinokurov et al,. 1999; Chu et al., 2009,). We studied the structures with uncapped dots and structures having GaInP cap layer thickness 5, 20, 40 and 60 nm.

The dot density (~2\*109 cm-2) and their sizes (base ~10-200 nm, height ~5-60 nm) were measured using atomic force microscopy (AFM) for the uncapped samples and transmission electron microscopy (TEM) for the capped samples. These data are presented in Fig.4a-d. A clear bimodal size distribution is seen from AFM images in Fig.4a consisting of large dots having size 100-200 nm and density 0.6x109 cm-2 and the small dots having sizes 10-70 nm and density 1.2x109 cm-2. From TEM measurements a lens shape of the large dots was revealed (see Fig.4c and d). Due to a residual n-type doping of the GaInP layer in the MOCVD growth process (n~1016 cm-3) the modulation doping of InP QDs occurs and they can contain up to 20 electrons (Hessman et al., 2001). Thus these QDs can represent natural WM and for dot sized 150-200 nm the *rs* ~4 can be achieved for few electron dots. The electron density can be varied from 1010 up to 5x1011 cm-2 and thus these InP QDs offer much more flexibility in varying of WM parameters then "conventional" GaAs/AlGaAs QDs used in SETs (Singha et al, 2010).

Fig. 4. Atomic force (a and b, size 2.5x2.5 *μ*m2) and transmission electron (c – plan-view, d – cross section) microscopy images of InP/GaInP QDs structures.

We estimated that our InP QDs can contain up to 10% Ga. Pure InP QDs can be grown by depositing nominally 0.5 ML of InP at growth temperatures 580 C and using in-situ annealing/growth interruption. Such growth conditions suppress Ga and In intermixing

Molecular States of Electrons: Emission of

photon emission. Their classical coordinate *d*f

vibronic structure of conventional molecules.

**5.1 Optical spectroscopy of quantum dots**

classical distance *d*i

*d*i

*d*i

Single Molecules in Self-Organized InP/GaInP Quantum Dots 133

In Fig.5c we show classical positions of the trion and the electrons in the 2e-WM before (upper cartoon) and after (lower cartoon) the radiative recombination. It can be shown (Govorov et al., 2006) using the minimization of the classical energy of the electrons that the

e of the electron to the dot center is a factor *β*= mtrω0tr2/meω0

e in the ground state obeys relation *d*i

tr, the corresponding distance for the center of mass of the trion. This is because the trion confinement potential and corresponding harmonic frequency ω0tr, is stronger than the electron confinement potential and corresponding harmonic frequency ω0. In a similar way, we can determine the classical configuration of the two electrons left in the final state after

tr as seen in Fig.5c. The wave functions of the initial and final states are peaked at the classical coordinates and decay exponentially along the radial axis away from their peak positions (see for example Fig.2). Since *β* ≠ 1, the classical coordinates for the initial and final states are different. This gives a coupling of the radiative recombination transition with vibrational modes of the WM which is the origin of the shake-up process in the photoluminescence of WM, to be discussed in the experimental part. Such "electron-

From symmetry considerations one can see from Fig.5c that the annihilation of the trion induces stretching and center-of-mass distortion along the 2e-WM axis, thus generating the stretching and translation modes discussed above. Thus the emission spectra of a 2e-WM are expected to have contributions from the "zero-phonon" line (ZPL) and two sets of Stokes phonon replicas having energies n*ħω*0 and n*ħ*1.7*ω*0, where n=1, 2, … . This is similar to the

Similar symmetry considerations based on classical electron arrangements shown in Fig.1 predict weak phonon Stokes emission for WM having a central electron, i.e. for N=6-9, and strong phonon Stokes emission for the two electrons at the center, i.e. N=2 and N=10-14.

For study of the effects of the electron localization in single InP/GaInP QDs we used the near-field scanning optical microscopy (NSOM) technique (Betzig&Trautman, 1993) in combination with magneto-PL spectroscopy. The basic principle of NSOM (Synge, 1928), providing a way to overcome the diffraction limit of light of conventional optics, is to use the coupling of the evanescent electromagnetic field and the radiative electromagnetic waves in the vicinity of a nano-probe placed near the boundary between two media. This principle was realized nearly three decades ago using nano-apertures (Lewis et al., 1984, Pohl et al., 1984), metallic (Fischer and Pohl ,1989) and dielectric (Coutjon et al 1989, Reddick at al., 1989) nano-tips, allowing nanometer-scale spatial resolution in optical experiments with a wide range of applications including experiments with single semiconductor QDs (Flack et al., 1996, Toda et al., 1996). The high spatial resolution, scanning ability and nondestructive character of the experiment in combination with a high magnetic field and timeresolved techniques, allows the use of NSOM to study structural parameters (Mintairov et al., 2001), the spin structure of exciton states (Ortner et al., 2003, Toda et al., 1998), the temporal coherence of the wave functions (Toda et al., 2000), and the mechanisms of carrier migration (K. Matsuda et al., 2000) and relaxation (Toda et al., 1999) in semiconductor QDs.

phonon" coupling can be expressed via Huang-Rhys factors (Huang&Rhys, 1950).

**5. The near-field scanning optical microscopy (NSOM) technique**

2 larger than

e *> d*f e *>*

and result in the pyramidal dot shape having base ~60 nm and height ~15 nm (Georgsson et al., 1995). Even smaller InP QDs having base 30 nm and height 7 nm were grown using 4ML InP deposition at 550 C (Ren et al., 1999).

In Fig.5a we present a cartoon showing the formation of a WM in our InP/GaInP QDs. Nine electrons come from the adjacent donor atoms located in the GaInP. The classical arrangement of a 9e-WM consists of eight electrons surrounding one electron at the center. The corresponding tentative band diagram of this QD is shown in Fig.5b. The barrier is formed by GaInP having band gap energy 1.97 eV (Janssens et al., 2003; Pryor et al., 1996). The "bandgap" of the QD material includes vertical confinement energy (~150 meV) and bandgap increase (~50 mev) due to Ga/In intermixing. We estimated the QD material bandgap to be ~1.7 eV, which is nearly 200 meV higher than the bandgap of InP. Due to strain effects and Ga/In intermixing we expect type-I band alignment between the QD and barrier material (Janssens et al., 2003; Pryor et al., 1997).

### **4.2 Photoluminescence of Wigner molecules**

Under photexcitation an electron-hole pair is generated in the dot and it forms a trion with the central electron (see in Fig.5a). The formation of trions in a dilute electron gas is well established (Finkelstein et al., 1995) and thus we can assume that a radiative recombination of the trion also forms the photoluminescence (PL) spectra of the WM. The selection rules for radiative transitions of the trion involving the electron having spin ± 1/2 and the hole having spin ± 3/2 do not change the electron spin and thus only charge excitations are expected to dominate the emission spectra of the WM.

Fig. 5. Arrangement of electrons (classical positions) (a) and corresponding band diagram (b) of photo-excited nine-electron InP QD (base ~150 nm); classical positions of the trion and electrons in two electron WM before (upper) and after (lower) radiative recombination (c). Trion and adjacent ionized donors are also shown in a. Thick(thin) curves in b are the band diagrams for filled(empty) dot.

and result in the pyramidal dot shape having base ~60 nm and height ~15 nm (Georgsson et al., 1995). Even smaller InP QDs having base 30 nm and height 7 nm were grown using 4ML

In Fig.5a we present a cartoon showing the formation of a WM in our InP/GaInP QDs. Nine electrons come from the adjacent donor atoms located in the GaInP. The classical arrangement of a 9e-WM consists of eight electrons surrounding one electron at the center. The corresponding tentative band diagram of this QD is shown in Fig.5b. The barrier is formed by GaInP having band gap energy 1.97 eV (Janssens et al., 2003; Pryor et al., 1996). The "bandgap" of the QD material includes vertical confinement energy (~150 meV) and bandgap increase (~50 mev) due to Ga/In intermixing. We estimated the QD material bandgap to be ~1.7 eV, which is nearly 200 meV higher than the bandgap of InP. Due to strain effects and Ga/In intermixing we expect type-I band alignment between the QD and

Under photexcitation an electron-hole pair is generated in the dot and it forms a trion with the central electron (see in Fig.5a). The formation of trions in a dilute electron gas is well established (Finkelstein et al., 1995) and thus we can assume that a radiative recombination of the trion also forms the photoluminescence (PL) spectra of the WM. The selection rules for radiative transitions of the trion involving the electron having spin ± 1/2 and the hole having spin ± 3/2 do not change the electron spin and thus only charge excitations are

Fig. 5. Arrangement of electrons (classical positions) (a) and corresponding band diagram (b) of photo-excited nine-electron InP QD (base ~150 nm); classical positions of the trion and electrons in two electron WM before (upper) and after (lower) radiative recombination (c). Trion and adjacent ionized donors are also shown in a. Thick(thin) curves in b are the band

InP deposition at 550 C (Ren et al., 1999).

barrier material (Janssens et al., 2003; Pryor et al., 1997).

expected to dominate the emission spectra of the WM.

diagrams for filled(empty) dot.

**4.2 Photoluminescence of Wigner molecules**

In Fig.5c we show classical positions of the trion and the electrons in the 2e-WM before (upper cartoon) and after (lower cartoon) the radiative recombination. It can be shown (Govorov et al., 2006) using the minimization of the classical energy of the electrons that the classical distance *d*i e of the electron to the dot center is a factor *β*= mtrω0tr2/meω0 2 larger than *d*i tr, the corresponding distance for the center of mass of the trion. This is because the trion confinement potential and corresponding harmonic frequency ω0tr, is stronger than the electron confinement potential and corresponding harmonic frequency ω0. In a similar way, we can determine the classical configuration of the two electrons left in the final state after photon emission. Their classical coordinate *d*f e in the ground state obeys relation *d*i e *> d*f e *> d*i tr as seen in Fig.5c. The wave functions of the initial and final states are peaked at the classical coordinates and decay exponentially along the radial axis away from their peak positions (see for example Fig.2). Since *β* ≠ 1, the classical coordinates for the initial and final states are different. This gives a coupling of the radiative recombination transition with vibrational modes of the WM which is the origin of the shake-up process in the photoluminescence of WM, to be discussed in the experimental part. Such "electronphonon" coupling can be expressed via Huang-Rhys factors (Huang&Rhys, 1950).

From symmetry considerations one can see from Fig.5c that the annihilation of the trion induces stretching and center-of-mass distortion along the 2e-WM axis, thus generating the stretching and translation modes discussed above. Thus the emission spectra of a 2e-WM are expected to have contributions from the "zero-phonon" line (ZPL) and two sets of Stokes phonon replicas having energies n*ħω*0 and n*ħ*1.7*ω*0, where n=1, 2, … . This is similar to the vibronic structure of conventional molecules.

Similar symmetry considerations based on classical electron arrangements shown in Fig.1 predict weak phonon Stokes emission for WM having a central electron, i.e. for N=6-9, and strong phonon Stokes emission for the two electrons at the center, i.e. N=2 and N=10-14.

### **5. The near-field scanning optical microscopy (NSOM) technique**

### **5.1 Optical spectroscopy of quantum dots**

For study of the effects of the electron localization in single InP/GaInP QDs we used the near-field scanning optical microscopy (NSOM) technique (Betzig&Trautman, 1993) in combination with magneto-PL spectroscopy. The basic principle of NSOM (Synge, 1928), providing a way to overcome the diffraction limit of light of conventional optics, is to use the coupling of the evanescent electromagnetic field and the radiative electromagnetic waves in the vicinity of a nano-probe placed near the boundary between two media. This principle was realized nearly three decades ago using nano-apertures (Lewis et al., 1984, Pohl et al., 1984), metallic (Fischer and Pohl ,1989) and dielectric (Coutjon et al 1989, Reddick at al., 1989) nano-tips, allowing nanometer-scale spatial resolution in optical experiments with a wide range of applications including experiments with single semiconductor QDs (Flack et al., 1996, Toda et al., 1996). The high spatial resolution, scanning ability and nondestructive character of the experiment in combination with a high magnetic field and timeresolved techniques, allows the use of NSOM to study structural parameters (Mintairov et al., 2001), the spin structure of exciton states (Ortner et al., 2003, Toda et al., 1998), the temporal coherence of the wave functions (Toda et al., 2000), and the mechanisms of carrier migration (K. Matsuda et al., 2000) and relaxation (Toda et al., 1999) in semiconductor QDs.

Molecular States of Electrons: Emission of

and thus can be as high as coated ones.

etching (d-f) and FIB trimming (f).

**5.3 NSOM imaging**

Single Molecules in Self-Organized InP/GaInP Quantum Dots 135

4-10-2. Uncoated fiber tips provide an order of magnitude higher PL signals but in general they have poorer spatial resolution. Below, however, we will demonstrate an apertureless mode of uncoated tip in which spatial resolution is determined just by the apex diameter

Fig. 6. SEM images of the near-field optical fiber probes: side view of the two types of tapers used (a, b); 45o tilt images of apertures having diameters (nm) and coating: c - 280 and Au, d - 270 and Al, e - 60 and Al/Ti and f- 120 and Al. The apertures were prepared by pulling (c),

Fig.7a-d presents our results of NSOM imaging of ~ 2 *μ*m long section of a CdSe nanowire (NW) having diameter 40 nm taken with an uncoated fiber tip (Mintairov et al., 2010). It shows topographic (a) and monochromatic NSOM (b) images together with a set of twelve spectra measured along a 400 nm size central section of the wire (c) and a false-color wavelengthposition plot of these spectra (d). Note that the topographic image is taken with the same fiber as the NSOM image. For further discussion we will denote the plot in Fig.7d as linear scan spectra (LSS) plot (image). Since we use accumulation time up to few seconds for each point of LSS image it can be considered as "static" image. The monochromatic NSOM image (see

In the topographic image (Fig.7a) the NW is seen as a vertical stripe having width ~100 nm. It has some distortions arising from the noise of the tuning-fork feedback control. We can estimate the width (*W*) and height (*H*) of the wire in the topography image to be *W*~100 and *H*~50 nm, respectively, and accounting for the fact that topographical height is equal to the

In the NSOM image (Fig.7b) the wire has a much larger width (~250 nm). It also reveals ~500 and ~100 nm intensity modulation along the wire. The fine scale modulation ~100 nm has the same scale as the topography. The nearly three times larger width of the NW in the NSOM image compared to the topography indicates an optical coupling between the wire and the tip before their physical contact, and reflects the "apertureless" nature of imaging using an uncoated tip. In such an apertureless regime the spatial resolution along the NW depends on the tip-wire distance and it is equal to the tip apex size (~50 nm for our tip)

Fig.7b), taken at 20 ms per pixel, thus can be considered as a "dynamic" image.

wire diameter we can get value of the tip apex size to be *TA*=*W-H*~50 nm.

Achieving spatial resolution as high as 30 nm allows optical mapping of exciton wave functions in a single QD (Matsuda et al., 2003). Such spatial resolution is of the order of the electron separation in the Wigner localization regime and thus the NSOM technique opens the possibility to probe the position of the individual electrons in the electronic molecules. However, present GaAs/AlGaAs SET structures require relatively thick AlGaAs cap layers (~70 nm), which provides vertical modulation doping but does not allow achievement of spatial resolution below 200 nm (Mintairov et al., 2003). In contrast InP/GaInP QDs, having lateral modulation doping (see Fig.5b), can form a WM for cap layer thickness down to zero nm (see below). Thus the tip-QD distance separation can be zero and spatial resolution as high as 10 nm can be possible, as was demonstrated for single molecules (Hosaka& Saiki, 2001).

### **5.2 Experimental details**

### **5.2.1 NSOM set up**

For our low-temperature magneto-PL measurements we used an Oxford Instruments CryoSXP cryogenic scanner together with a liquid helium cryostat with a superconducting magnet providing magnetic fields up to 12 T. Shear-force tip-sample distance control and scanning are governed by a Vecco AFM controller from Digital Instruments, providing a scan range over an area of 3x3 *μ*m2 at 4.2K. For characterization of InP/GaInP structures at room temperature we used an NSOM-2000 system from NANONICs Inc. having a scan range 90x90 *μ*m2. The near-field photoluminescence (PL) spectra were taken in collectionillumination mode, i.e. laser excitation and PL emission collection using the same NSOM fiber probe. Emission was excited by the 488 nm line of an Ar ion laser and dispersed using a 270 mm focal length monochromator. The spectra were measured using a nitrogen-cooled CCD detector and 1200 gr/mm grating. Monochromatic NSOM images were measured using a 300 gr/mm grating and a GaAs photomultiplier working in the photon-counting regime with an accumulation time of 20 ms per pixel. Excitation power density was ~20-100 and 0.1-1 W/cm-2 for 300 and 10 K, respectively. The spectral resolution of the system is 0.2- 0.4 meV. We measured spectra in a magnetic field up to 10 T. A quarter wave-plate and linear polarizer were used for the separation of right- and left-hand circular polarization.

### **5.2.2 NSOM probes**

As near-field probes we used tapered single mode fiber tips. We used both metal coated and uncoated tips. Coated tips were prepared by electron beam deposition of metal (Al or Au in combination with Ti) having thickness ~50-200 nm. The physical structure of the aperture and its optical quality were controlled using scanning electron microscopy (SEM) imaging, visual observation of the light coming from the tip under a microscope with x100 objective, and by measuring its far-field transmission for 632.8 nm wavelength (HeNe laser). SEM images of typical near-field optical fiber probes are presented in Fig. 6a-f. The taper was prepared by using a pulling technique ((Lewis et al., 1984, Pohl et al., 1984). We used two pulling regimes in which the taper angle increases towards the tip end (see Fig.6a and b). Both regimes give tips (apertures) with diameters 200 nm and aperture angles 30-40o (see Fig. 6c) but within a few micrometers from the aperture they have different taper angles. To make smaller apertures we used further chemical etching in a hydrofluoric acid solution (Otsu, 1998). We also use a focused ion beam milling technique to flatten the apertures (Fig.6f). The transmission of our metal coated fiber probes was 104-10-2. Uncoated fiber tips provide an order of magnitude higher PL signals but in general they have poorer spatial resolution. Below, however, we will demonstrate an apertureless mode of uncoated tip in which spatial resolution is determined just by the apex diameter and thus can be as high as coated ones.

Fig. 6. SEM images of the near-field optical fiber probes: side view of the two types of tapers used (a, b); 45o tilt images of apertures having diameters (nm) and coating: c - 280 and Au, d - 270 and Al, e - 60 and Al/Ti and f- 120 and Al. The apertures were prepared by pulling (c), etching (d-f) and FIB trimming (f).

### **5.3 NSOM imaging**

134 Fingerprints in the Optical and Transport Properties of Quantum Dots

Achieving spatial resolution as high as 30 nm allows optical mapping of exciton wave functions in a single QD (Matsuda et al., 2003). Such spatial resolution is of the order of the electron separation in the Wigner localization regime and thus the NSOM technique opens the possibility to probe the position of the individual electrons in the electronic molecules. However, present GaAs/AlGaAs SET structures require relatively thick AlGaAs cap layers (~70 nm), which provides vertical modulation doping but does not allow achievement of spatial resolution below 200 nm (Mintairov et al., 2003). In contrast InP/GaInP QDs, having lateral modulation doping (see Fig.5b), can form a WM for cap layer thickness down to zero nm (see below). Thus the tip-QD distance separation can be zero and spatial resolution as high as 10 nm can be possible, as was demonstrated for single molecules (Hosaka& Saiki, 2001).

For our low-temperature magneto-PL measurements we used an Oxford Instruments CryoSXP cryogenic scanner together with a liquid helium cryostat with a superconducting magnet providing magnetic fields up to 12 T. Shear-force tip-sample distance control and scanning are governed by a Vecco AFM controller from Digital Instruments, providing a scan range over an area of 3x3 *μ*m2 at 4.2K. For characterization of InP/GaInP structures at room temperature we used an NSOM-2000 system from NANONICs Inc. having a scan range 90x90 *μ*m2. The near-field photoluminescence (PL) spectra were taken in collectionillumination mode, i.e. laser excitation and PL emission collection using the same NSOM fiber probe. Emission was excited by the 488 nm line of an Ar ion laser and dispersed using a 270 mm focal length monochromator. The spectra were measured using a nitrogen-cooled CCD detector and 1200 gr/mm grating. Monochromatic NSOM images were measured using a 300 gr/mm grating and a GaAs photomultiplier working in the photon-counting regime with an accumulation time of 20 ms per pixel. Excitation power density was ~20-100 and 0.1-1 W/cm-2 for 300 and 10 K, respectively. The spectral resolution of the system is 0.2- 0.4 meV. We measured spectra in a magnetic field up to 10 T. A quarter wave-plate and linear polarizer were used for the separation of right- and left-hand circular polarization.

As near-field probes we used tapered single mode fiber tips. We used both metal coated and uncoated tips. Coated tips were prepared by electron beam deposition of metal (Al or Au in combination with Ti) having thickness ~50-200 nm. The physical structure of the aperture and its optical quality were controlled using scanning electron microscopy (SEM) imaging, visual observation of the light coming from the tip under a microscope with x100 objective, and by measuring its far-field transmission for 632.8 nm wavelength (HeNe laser). SEM images of typical near-field optical fiber probes are presented in Fig. 6a-f. The taper was prepared by using a pulling technique ((Lewis et al., 1984, Pohl et al., 1984). We used two pulling regimes in which the taper angle increases towards the tip end (see Fig.6a and b). Both regimes give tips (apertures) with diameters 200 nm and aperture angles 30-40o (see Fig. 6c) but within a few micrometers from the aperture they have different taper angles. To make smaller apertures we used further chemical etching in a hydrofluoric acid solution (Otsu, 1998). We also use a focused ion beam milling technique to flatten the apertures (Fig.6f). The transmission of our metal coated fiber probes was 10-

**5.2 Experimental details**

**5.2.1 NSOM set up**

**5.2.2 NSOM probes**

Fig.7a-d presents our results of NSOM imaging of ~ 2 *μ*m long section of a CdSe nanowire (NW) having diameter 40 nm taken with an uncoated fiber tip (Mintairov et al., 2010). It shows topographic (a) and monochromatic NSOM (b) images together with a set of twelve spectra measured along a 400 nm size central section of the wire (c) and a false-color wavelengthposition plot of these spectra (d). Note that the topographic image is taken with the same fiber as the NSOM image. For further discussion we will denote the plot in Fig.7d as linear scan spectra (LSS) plot (image). Since we use accumulation time up to few seconds for each point of LSS image it can be considered as "static" image. The monochromatic NSOM image (see Fig.7b), taken at 20 ms per pixel, thus can be considered as a "dynamic" image.

In the topographic image (Fig.7a) the NW is seen as a vertical stripe having width ~100 nm. It has some distortions arising from the noise of the tuning-fork feedback control. We can estimate the width (*W*) and height (*H*) of the wire in the topography image to be *W*~100 and *H*~50 nm, respectively, and accounting for the fact that topographical height is equal to the wire diameter we can get value of the tip apex size to be *TA*=*W-H*~50 nm.

In the NSOM image (Fig.7b) the wire has a much larger width (~250 nm). It also reveals ~500 and ~100 nm intensity modulation along the wire. The fine scale modulation ~100 nm has the same scale as the topography. The nearly three times larger width of the NW in the NSOM image compared to the topography indicates an optical coupling between the wire and the tip before their physical contact, and reflects the "apertureless" nature of imaging using an uncoated tip. In such an apertureless regime the spatial resolution along the NW depends on the tip-wire distance and it is equal to the tip apex size (~50 nm for our tip)

Molecular States of Electrons: Emission of

~60 nm (~150 meV).

in InP/GaInP QDs.

**6.2 Low-temperature emission spectra**

Single Molecules in Self-Organized InP/GaInP Quantum Dots 137

nm. For a 60 nm cap the image contrast strongly decreases, which demonstrates the expected reduction of the spatial resolution (down to 150 nm) due to increased tip-dot separation. By positioning the tip on the bright spots we measured the spectra of individual InP/GaInP QDs at room temperature. Three such spectra for the uncapped sample presented in Fig.8a show that the spectra of single InP QDs at room temperature consists of a single band having wavelength in the range of 710-790 nm (1.75-1.56 eV) and halfwidth *γ*

Fig. 8. Room temperature NSOM spectra (a) together with NSOM (b-d) and AFM (e-g) images (size 2.5x2.5 *μ*m2) of InP/GaInP QD structures having cap layer thickness (nm): 0 (b and e), 5 (c and f) and 20 (d and g). The ~500 nm diameter dashed circles in b-g outline the

In Fig.8b-g we present simultaneously measured NSOM (b, c and d) and AFM (e, f and g) images for d=0, 5 and 20 nm. The InP QDs having height 20-40 nm are clearly seen in the AFM image of the uncapped sample (Fig.8e). However the QD base observed is slightly (~50 nm) larger for the AFM image than in the NSOM image (see the encircled 1 dots in Fig.8b and e), which demonstrates the effect of electron localization. In the capped samples the QD location corresponds to the "valley" of the AFM images; this is seen by comparing the AFM and NSOM images (see the encircled dots for both the 5 and 20 nm cap) and the QD height decreases to 20-10 nm. Thus we observed the growth of the GaInP matrix material at the edges of the QDs. This together with the height reduction demonstrates additional growth control of the dot shape and thus confinement potential

We should also note that the use of the topographic images for the uncapped samples (see Fig.8b and e) allows the experimental determination of the size of the specific QD, which is

Fig.9 shows low-temperature (T=10 K) NSOM emission spectra of an InP/GaInP structure taken with spatial resolution ~200 nm at two fiber tip positions centred at two different QDs (QD2e and QD3m) separated by 400 nm. (This notation will be clarified below). In the range

one of the key parameters determining the electron correlation regime.

same area in the NSOM and AFM images. Inserts in 8a show NSOM images of the

structures having cap layer thickness 0, 5, 20 and 60 nm.

when the tip and the wire are in contact. Thus the 100 nm size modulation of the emission intensity seen in the contact part of the NSOM image corresponds to an intrinsic uniformity of the NW emission of ~50 nm.

Fig. 7. Topography (a) and NSOM (b) images of NW1 at 50 K together with the set of twelve spectra taken during linear scan having length 400 nm (c) and their false-color wavelengthtip-position plot (d). Fiber tip positions for linear scan are marked by dots in (b). Image size in (a) and (b) is 0.8x2 *μ*m2. Detection energy in (b) is 1.8 eV

Comparison of Fig.7b and Fig.7c shows that variations of the intensity of the spectra follow the intensity variations in the monochromatic NSOM image. For example, the strong (weak) NSOM intensity in Fig.7b at points 9-11 (1-6) corresponds to strong (weak) spectra in Fig.7c taken at corresponding points. Some intensity variations seen in the spectra (like the intensity increase at point 8 and 4) are not observed in the NSOM image due to photon counting noise arising from the "dynamic" character of the image. Analysis of Fig.7c and d shows that intensity fluctuations are accompanied by changes of the spectral position and the width of the NW emission peak. For neighboring points separated by only 36 nm the changes of peak position and width by a few meV can be detected.

### **6. NSOM characterization and emission spectra of InP/GaInP structures**

### **6.1 Room temperature NSOM imaging**

Fig.8a-d shows the results of room temperature NSOM imaging of InP/GaInP QD structures having cap layer thickness d=0, 5, 20 and 60 nm. The images were taken at detection wavelength 750 nm. NSOM images of our structures (see inserts in Fig.8a and Fig.8b-d) show a set of bright spots having density ~20 *μ*m-2 and size ~150-250 nm related to single QDs. Clearly-resolved single QD images are observed for 5 and 20 nm capped QDs as well as for the uncapped QDs. It is important to note the strong emission intensity for the uncapped QDs at room temperature. We believe that this is the first observation of such strong emission from uncapped QDs allowing ultra-high spatial-resolution. Accounting for the ~100 nm base of the QDs we estimated the tip apex size (spatial resolution) to be 25-75

when the tip and the wire are in contact. Thus the 100 nm size modulation of the emission intensity seen in the contact part of the NSOM image corresponds to an intrinsic uniformity

Fig. 7. Topography (a) and NSOM (b) images of NW1 at 50 K together with the set of twelve spectra taken during linear scan having length 400 nm (c) and their false-color wavelengthtip-position plot (d). Fiber tip positions for linear scan are marked by dots in (b). Image size

Comparison of Fig.7b and Fig.7c shows that variations of the intensity of the spectra follow the intensity variations in the monochromatic NSOM image. For example, the strong (weak) NSOM intensity in Fig.7b at points 9-11 (1-6) corresponds to strong (weak) spectra in Fig.7c taken at corresponding points. Some intensity variations seen in the spectra (like the intensity increase at point 8 and 4) are not observed in the NSOM image due to photon counting noise arising from the "dynamic" character of the image. Analysis of Fig.7c and d shows that intensity fluctuations are accompanied by changes of the spectral position and the width of the NW emission peak. For neighboring points separated by only 36 nm the

**6. NSOM characterization and emission spectra of InP/GaInP structures**

Fig.8a-d shows the results of room temperature NSOM imaging of InP/GaInP QD structures having cap layer thickness d=0, 5, 20 and 60 nm. The images were taken at detection wavelength 750 nm. NSOM images of our structures (see inserts in Fig.8a and Fig.8b-d) show a set of bright spots having density ~20 *μ*m-2 and size ~150-250 nm related to single QDs. Clearly-resolved single QD images are observed for 5 and 20 nm capped QDs as well as for the uncapped QDs. It is important to note the strong emission intensity for the uncapped QDs at room temperature. We believe that this is the first observation of such strong emission from uncapped QDs allowing ultra-high spatial-resolution. Accounting for the ~100 nm base of the QDs we estimated the tip apex size (spatial resolution) to be 25-75

in (a) and (b) is 0.8x2 *μ*m2. Detection energy in (b) is 1.8 eV

changes of peak position and width by a few meV can be detected.

**6.1 Room temperature NSOM imaging**

of the NW emission of ~50 nm.

nm. For a 60 nm cap the image contrast strongly decreases, which demonstrates the expected reduction of the spatial resolution (down to 150 nm) due to increased tip-dot separation. By positioning the tip on the bright spots we measured the spectra of individual InP/GaInP QDs at room temperature. Three such spectra for the uncapped sample presented in Fig.8a show that the spectra of single InP QDs at room temperature consists of a single band having wavelength in the range of 710-790 nm (1.75-1.56 eV) and halfwidth *γ* ~60 nm (~150 meV).

Fig. 8. Room temperature NSOM spectra (a) together with NSOM (b-d) and AFM (e-g) images (size 2.5x2.5 *μ*m2) of InP/GaInP QD structures having cap layer thickness (nm): 0 (b and e), 5 (c and f) and 20 (d and g). The ~500 nm diameter dashed circles in b-g outline the same area in the NSOM and AFM images. Inserts in 8a show NSOM images of the structures having cap layer thickness 0, 5, 20 and 60 nm.

In Fig.8b-g we present simultaneously measured NSOM (b, c and d) and AFM (e, f and g) images for d=0, 5 and 20 nm. The InP QDs having height 20-40 nm are clearly seen in the AFM image of the uncapped sample (Fig.8e). However the QD base observed is slightly (~50 nm) larger for the AFM image than in the NSOM image (see the encircled 1 dots in Fig.8b and e), which demonstrates the effect of electron localization. In the capped samples the QD location corresponds to the "valley" of the AFM images; this is seen by comparing the AFM and NSOM images (see the encircled dots for both the 5 and 20 nm cap) and the QD height decreases to 20-10 nm. Thus we observed the growth of the GaInP matrix material at the edges of the QDs. This together with the height reduction demonstrates additional growth control of the dot shape and thus confinement potential in InP/GaInP QDs.

We should also note that the use of the topographic images for the uncapped samples (see Fig.8b and e) allows the experimental determination of the size of the specific QD, which is one of the key parameters determining the electron correlation regime.

### **6.2 Low-temperature emission spectra**

Fig.9 shows low-temperature (T=10 K) NSOM emission spectra of an InP/GaInP structure taken with spatial resolution ~200 nm at two fiber tip positions centred at two different QDs (QD2e and QD3m) separated by 400 nm. (This notation will be clarified below). In the range

Molecular States of Electrons: Emission of

InP/GaInP QDs having base 60 nm.

QD5m, 1.7662 for QD1i, and 1.7073 for QD2e.

energy Fermi edge state (Hawrylak, 1992; Kukushkin et al., 1989).

Single Molecules in Self-Organized InP/GaInP Quantum Dots 139

exciton is observed (Sugisaki et al., 2002). The manifold structure was also observed for ∆E values as large as 10 meV (Blome et al., 2000; Hessman et al., 2001) for pyramidal

Fig. 10. Low-temperature (10K) NSOM LSS plot of InP/GaInP QD structure (a) and spectra of individual dots (b-e) The s-peak energy (in eV) in (b-e) is 1.9241 for QD1s, 1.6994 for

Our measurements of metallic QDs using a magnetic field discussed below have shown that the observed multi-peak structure of the emission spectra of InP/GaInP in Fig.10, c-e results from the filling of several electron shells. Here the spectral line shape is created by the radiative recombination of a photo-excited hole localized inside QD with electrons resulting from the "metallic character" of the dot; this is similar to the acceptor-related emission of a two-dimensional electron gas (2DEG) (Hawrylak, 1992). However in our QDs, i.e. a confined 2DEG, the spectral shape is determined by zero-dimensional confinement selection rules, and has maxima at the lowest s-state energy due to the s-state hole involved in the recombination process (see also the band diagram in Fig.5b), while in the 2DEG the spectral shape is determined by the density of the electronic states and is dominated by the high

The observed number of shells and the shell spacing allows us to estimate the number of electrons and the "effective" dot size, assuming a parabolic confinement potential (Jacak et al., 1998). The dots QD5m and QD3e in Fig.10c and e have spacing 3.5 and 2.5 meV, giving dot sizes ~90 and ~100 nm, respectively. They have three occupied s, p and d shells giving the possible number of electrons to be 7-12. Thus the electron density in these QD is ~5- 10x1010 cm-2, which is similar to the density used in 2DEG GaAs/AlGaAs structures. Using the values of the dot size and the number of electrons we estimate *rs* values of the InP QDs to be 1.5-2.5. Our observation of the narrowing of shell peaks and their fine structure for

1.67-2.0 eV (740-620 nm) the spectra are dominated by the bands of the "central" dots, i.e. by QD2e in position 1 and by QD3m in position 2, having emission energy ~1.70 eV. Weaker bands of neighbouring dots located close to the tip edge are also seen at ~20 meV higher energy. The band shape of the large QDs reveals multipeak (manifold) structures, which will be discussed in detail below. The spectra also contain sharp lines observed in the range 1.85-1.95 eV, which are related to small InP QDs, and a broader band at 1.97 eV which is related to the GaInP matrix. The emission energy of our large InP QD is ~50 meV higher than the emission energy of the pyramidal InP QDs having base 60 nm, observed by other authors (Blome et al., 2000; Hessman et al., 2001), which indicates Ga/In intermixing. Using this energy difference we can estimate the value of the Ga composition of our InP QDs to be ~10%. Such intermixing resulting in significant increase of the dot size is favourable for the WM formation. Using low-temperature NSOM we measured the emission spectra of ~50 single InP QDs, allowing us to observe the effects of Wigner localization, which will be analysed below.

Fig. 9. 10K–NSOM spectra of InP/GaInP QD structures taken with spatial resolution 200 nm

### **7. Experimental study of Wigner molecules**

#### **7.1 Fermi liquid to Wigner molecule transition**

Fig.10a shows an LSS plot, taken over a linear scan of 1.6 *μ*m, for a 60 nm capped sample measured at T=10 K using a coated fiber having a 100 nm aperture. Four QDs denoted by QD1m, QD2e, QD3m and QD4e are observed in Fig.10a in the spectral range 1.72-1.68 eV. The spectra at tip positions 1 and 2 centered at QD2e and QD3m were shown in Fig.9 over a wider spectra range. Fig.10b-e compares the spectra of three large dots QD5m, QD1i, QD2e and a small lone dot QD1s. The LSS image in Fig.10a demonstrates a drastic difference in the fine structure of the emission manifold of the dots having subscript m and e. This difference is also clear from the spectra of dots QD5m and QD2e in Fig.10c and e, respectively. In the m-type dots (further referred to as metallic) up to three components (s, p and d) of the emission manifold are observed and they have an energy splitting (∆E) of ~4 meV and halfwidth (*γ*) of 3-5 meV. For the e-type dots (further referred to as excitonic) a fine structure of the s and p components consisting of several ultranarrow lines is clearly seen in Fig.10a, having *γ*<0.2 meV and ∆E as small as 1-2 meV. In Fig.10d we present the spectrum of a dot having a mixed structure consisting of few sharp lines and wider peaks. In contrast, for a small dot (Fig.10b) having ∆E >15 meV a single line related to a neutral

1.67-2.0 eV (740-620 nm) the spectra are dominated by the bands of the "central" dots, i.e. by QD2e in position 1 and by QD3m in position 2, having emission energy ~1.70 eV. Weaker bands of neighbouring dots located close to the tip edge are also seen at ~20 meV higher energy. The band shape of the large QDs reveals multipeak (manifold) structures, which will be discussed in detail below. The spectra also contain sharp lines observed in the range 1.85-1.95 eV, which are related to small InP QDs, and a broader band at 1.97 eV which is related to the GaInP matrix. The emission energy of our large InP QD is ~50 meV higher than the emission energy of the pyramidal InP QDs having base 60 nm, observed by other authors (Blome et al., 2000; Hessman et al., 2001), which indicates Ga/In intermixing. Using this energy difference we can estimate the value of the Ga composition of our InP QDs to be ~10%. Such intermixing resulting in significant increase of the dot size is favourable for the WM formation. Using low-temperature NSOM we measured the emission spectra of ~50 single InP QDs, allowing us to observe the effects of Wigner

Fig. 9. 10K–NSOM spectra of InP/GaInP QD structures taken with spatial resolution 200 nm

Fig.10a shows an LSS plot, taken over a linear scan of 1.6 *μ*m, for a 60 nm capped sample measured at T=10 K using a coated fiber having a 100 nm aperture. Four QDs denoted by QD1m, QD2e, QD3m and QD4e are observed in Fig.10a in the spectral range 1.72-1.68 eV. The spectra at tip positions 1 and 2 centered at QD2e and QD3m were shown in Fig.9 over a wider spectra range. Fig.10b-e compares the spectra of three large dots QD5m, QD1i, QD2e and a small lone dot QD1s. The LSS image in Fig.10a demonstrates a drastic difference in the fine structure of the emission manifold of the dots having subscript m and e. This difference is also clear from the spectra of dots QD5m and QD2e in Fig.10c and e, respectively. In the m-type dots (further referred to as metallic) up to three components (s, p and d) of the emission manifold are observed and they have an energy splitting (∆E) of ~4 meV and halfwidth (*γ*) of 3-5 meV. For the e-type dots (further referred to as excitonic) a fine structure of the s and p components consisting of several ultranarrow lines is clearly seen in Fig.10a, having *γ*<0.2 meV and ∆E as small as 1-2 meV. In Fig.10d we present the spectrum of a dot having a mixed structure consisting of few sharp lines and wider peaks. In contrast, for a small dot (Fig.10b) having ∆E >15 meV a single line related to a neutral

localization, which will be analysed below.

**7. Experimental study of Wigner molecules 7.1 Fermi liquid to Wigner molecule transition**

exciton is observed (Sugisaki et al., 2002). The manifold structure was also observed for ∆E values as large as 10 meV (Blome et al., 2000; Hessman et al., 2001) for pyramidal InP/GaInP QDs having base 60 nm.

Fig. 10. Low-temperature (10K) NSOM LSS plot of InP/GaInP QD structure (a) and spectra of individual dots (b-e) The s-peak energy (in eV) in (b-e) is 1.9241 for QD1s, 1.6994 for QD5m, 1.7662 for QD1i, and 1.7073 for QD2e.

Our measurements of metallic QDs using a magnetic field discussed below have shown that the observed multi-peak structure of the emission spectra of InP/GaInP in Fig.10, c-e results from the filling of several electron shells. Here the spectral line shape is created by the radiative recombination of a photo-excited hole localized inside QD with electrons resulting from the "metallic character" of the dot; this is similar to the acceptor-related emission of a two-dimensional electron gas (2DEG) (Hawrylak, 1992). However in our QDs, i.e. a confined 2DEG, the spectral shape is determined by zero-dimensional confinement selection rules, and has maxima at the lowest s-state energy due to the s-state hole involved in the recombination process (see also the band diagram in Fig.5b), while in the 2DEG the spectral shape is determined by the density of the electronic states and is dominated by the high energy Fermi edge state (Hawrylak, 1992; Kukushkin et al., 1989).

The observed number of shells and the shell spacing allows us to estimate the number of electrons and the "effective" dot size, assuming a parabolic confinement potential (Jacak et al., 1998). The dots QD5m and QD3e in Fig.10c and e have spacing 3.5 and 2.5 meV, giving dot sizes ~90 and ~100 nm, respectively. They have three occupied s, p and d shells giving the possible number of electrons to be 7-12. Thus the electron density in these QD is ~5- 10x1010 cm-2, which is similar to the density used in 2DEG GaAs/AlGaAs structures. Using the values of the dot size and the number of electrons we estimate *rs* values of the InP QDs to be 1.5-2.5. Our observation of the narrowing of shell peaks and their fine structure for

Molecular States of Electrons: Emission of

the SU emission, as was discussed in section 4.2.

**7.3.1 Optically induced intra-dot magnetic field**

magnetic field B=0, 1, 2 … 10T.

**7.3 Emission spectra of metallic dots in a magnetic field**

versus magnetic field. Dashed lines are drawn for clarity.

Single Molecules in Self-Organized InP/GaInP Quantum Dots 141

levels and related magneto-plasmons (Butov et al., 1992; Hawrylak&Potemski, 1997; Nash et al., 1993). SU emission from the excited states was also observed in an InAs QD ensemble (Paskov et al., 2000). In the Wigner localization regime one can expect a vibronic structure of

To probe Wigner localization in the metallic dots we used a magnetic field which effectively increases *rs* by squeezing the electron motion. With increase of the magnetic field the WM is formed above the molecular-droplet transition at magnetic field >4T ((Maksym et al., 2000;

Fig. 12a and b presents circular polarized emission spectra of QD11 and QD8 measured at

The s-shell band in Fig. 12a (QD11) at zero field is not polarized. With increasing magnetic field it becomes circularly polarized. The dominant emission of this band is σ+-polarized for B=1-3 and 9-10T and σ--polarized for B=5-7T. Zeeman splitting varies from +0.3 meV for 4T

Fig. 12. Circularly polarized components (σ- - solid, σ+ - dotted) of emission spectra of QD11 and QD8 in magnetic field (0, 1, 2, …10 T). Inserts show position of the energy of the s-peak

The s-shell band in Fig. 12b (QD8) shows different magnetic field behavior. First, unlike QD11, it has strong circular polarization and Zeeman splitting at zero magnetic field. Both circular polarization and Zeeman splitting disappear at 2T. Second, the band has a strong *low energy shift* (slope~0.8 meV/T) for magnetic fields 0-2T. For higher fields it has a high energy shift with a slope of 0.25 meV/T. Our observations of the circular polarization at zero magnetic field and negative magnetic field shift at B=0-2 T for QD8 reflect a strong

Reimann&Manninen, 2002; Szafran et al., 2003). Details of this phenomenon follow.

to -0.3 meV for 8T. The band shifts to higher energies with magnetic field.

excitonic dots indicates a formation of the excitonic/trion emission at a critical electron density at or below ~5x1010 cm-2. Such a transition (from a broad Fermi sea emission to exciton and trion narrow lines) was observed in a 2DEG at higher densities (Finkelstein et al., 1995). Since the formation of the excitonic/trion emission in a many-electron system implies exciton/trion *localization* we can conclude that the multi-shell structure in these QDs reflects the formation of a Wigner molecule (WM). We believe this to be the first observation of a WM in a semiconductor QD.

### **7.2 Electron shell filling in the Fermi liquid regime**

In Fig.11a and b we show the spectra of five metallic dots having varying ∆E and shell fillings. The QD7, QD8 and QD9 in Fig.11a have emission from three shells (~10 electrons) and show a progressive decrease of ∆E from 5.9 to 4.6 and to 3.5 meV, corresponding to sizes of the confining potential changing from ~65 to ~80 and to ~90 nm. On the other hand, QD10, QD7, and QD11 in Fig.11b show a progressive increase of the number of the shells from two for QD10 to three for QD7 and to four for QD11 (see spectra in a magnetic field below), thus demonstrating changes of the number of electrons from ~6 to ~20. The increase of electron numbers is accompanied by a decrease of ∆E, as expected.

Fig. 11. Low-temperature (10K) NSOM spectra of five metallic single InP/GaInP QDs . The s-peak energy ES is 1.7064, 1.6992, 1.6994 and 1.7041, 1.6981 eV for QD7, QD8, QD9 and QD10, QD11, respectively. Peaks \* are contributions of neighboring QDs.

In addition to the shell peaks the spectra of single InP QDs display emission features related to so-called shake up or Auger processes denoted in Fig11a and b as SU. These features appear as low energy tails of the s-shell peak and for some dots they resolve into a separate band shifted by ~∆E/2 (see QD9 and QD10). The scaling of the SU emission energy with ∆E is seen in Fig.11b. The SU emission occurs in many-electron systems when a recombining electron-hole pair excites surrounding electrons via the Coulomb interaction. In a 2DEG the SU emission appears at high magnetic fields via excitation of electrons into higher Landau

excitonic dots indicates a formation of the excitonic/trion emission at a critical electron density at or below ~5x1010 cm-2. Such a transition (from a broad Fermi sea emission to exciton and trion narrow lines) was observed in a 2DEG at higher densities (Finkelstein et al., 1995). Since the formation of the excitonic/trion emission in a many-electron system implies exciton/trion *localization* we can conclude that the multi-shell structure in these QDs reflects the formation of a Wigner molecule (WM). We believe this to be the first observation

In Fig.11a and b we show the spectra of five metallic dots having varying ∆E and shell fillings. The QD7, QD8 and QD9 in Fig.11a have emission from three shells (~10 electrons) and show a progressive decrease of ∆E from 5.9 to 4.6 and to 3.5 meV, corresponding to sizes of the confining potential changing from ~65 to ~80 and to ~90 nm. On the other hand, QD10, QD7, and QD11 in Fig.11b show a progressive increase of the number of the shells from two for QD10 to three for QD7 and to four for QD11 (see spectra in a magnetic field below), thus demonstrating changes of the number of electrons from ~6 to ~20. The increase

Fig. 11. Low-temperature (10K) NSOM spectra of five metallic single InP/GaInP QDs . The s-peak energy ES is 1.7064, 1.6992, 1.6994 and 1.7041, 1.6981 eV for QD7, QD8, QD9 and

In addition to the shell peaks the spectra of single InP QDs display emission features related to so-called shake up or Auger processes denoted in Fig11a and b as SU. These features appear as low energy tails of the s-shell peak and for some dots they resolve into a separate band shifted by ~∆E/2 (see QD9 and QD10). The scaling of the SU emission energy with ∆E is seen in Fig.11b. The SU emission occurs in many-electron systems when a recombining electron-hole pair excites surrounding electrons via the Coulomb interaction. In a 2DEG the SU emission appears at high magnetic fields via excitation of electrons into higher Landau

QD10, QD11, respectively. Peaks \* are contributions of neighboring QDs.

of a WM in a semiconductor QD.

**7.2 Electron shell filling in the Fermi liquid regime**

of electron numbers is accompanied by a decrease of ∆E, as expected.

levels and related magneto-plasmons (Butov et al., 1992; Hawrylak&Potemski, 1997; Nash et al., 1993). SU emission from the excited states was also observed in an InAs QD ensemble (Paskov et al., 2000). In the Wigner localization regime one can expect a vibronic structure of the SU emission, as was discussed in section 4.2.

To probe Wigner localization in the metallic dots we used a magnetic field which effectively increases *rs* by squeezing the electron motion. With increase of the magnetic field the WM is formed above the molecular-droplet transition at magnetic field >4T ((Maksym et al., 2000; Reimann&Manninen, 2002; Szafran et al., 2003). Details of this phenomenon follow.

### **7.3 Emission spectra of metallic dots in a magnetic field**

### **7.3.1 Optically induced intra-dot magnetic field**

Fig. 12a and b presents circular polarized emission spectra of QD11 and QD8 measured at magnetic field B=0, 1, 2 … 10T.

The s-shell band in Fig. 12a (QD11) at zero field is not polarized. With increasing magnetic field it becomes circularly polarized. The dominant emission of this band is σ+-polarized for B=1-3 and 9-10T and σ--polarized for B=5-7T. Zeeman splitting varies from +0.3 meV for 4T to -0.3 meV for 8T. The band shifts to higher energies with magnetic field.

Fig. 12. Circularly polarized components (σ- - solid, σ+ - dotted) of emission spectra of QD11 and QD8 in magnetic field (0, 1, 2, …10 T). Inserts show position of the energy of the s-peak versus magnetic field. Dashed lines are drawn for clarity.

The s-shell band in Fig. 12b (QD8) shows different magnetic field behavior. First, unlike QD11, it has strong circular polarization and Zeeman splitting at zero magnetic field. Both circular polarization and Zeeman splitting disappear at 2T. Second, the band has a strong *low energy shift* (slope~0.8 meV/T) for magnetic fields 0-2T. For higher fields it has a high energy shift with a slope of 0.25 meV/T. Our observations of the circular polarization at zero magnetic field and negative magnetic field shift at B=0-2 T for QD8 reflect a strong

Molecular States of Electrons: Emission of

contrast to QD11).

**7.4 High-spatial resolution NSOM imaging**

Single Molecules in Self-Organized InP/GaInP Quantum Dots 143

For QD8 (Fig.13a and c) the observed magnetic-field-induced shell shifts follow the FD energy levels remarkably well. Here two magnetic-field induced transitions are observed. The first transition occurs at B=2T arising from an internal magnetic field (Bint) of 0T which was discussed above (see Fig12.b). In the unpolarized spectra presented in Fig.13a, Bint is responsible for the shift to lower energy of the s- and d-shell peaks at B<2. For B>2 (Bint>0) the peak positions shift to increasing energy (see Fig.13c). In the range B=2-7 T (Bint=0-5 T) the p-shell peak appears. At B>7 T (second transition; Bint=5 T) the p-shell peak transforms into a d-shell, and d into g-shell. As can be seen from Fig.13c the second transition (at 7T) corresponds to transition of electrons from the second (*υ*=1, p/d-shells) to the first Landau level (*υ*=0, d/g-shells). Such a transition was previously observed in the magnetic field dependence of Coulomb blockade and can be described as a molecular-droplet transition (Oosterkamp et al., 1999). It arises from formation of a maximum-density-droplet at B=2, and its decomposition is accompanied by a WM formation at B>4 T (Maksym et al., 2000; Reimann& Manninen, 2002; Szafran et al., 2003 ). In the spectra the WM formation at a high magnetic field corresponds to a strong increase of the intensity of the g-shell peak (in

We used the uncapped structure to perform ultra-high-spatial resolution imaging and spectroscopy of InP/GaInP QDs using uncoated fiber probes working in an apertureless regime. In Fig.14a and b we present monochromatic 5K-NSOM images (detection energy 1.713 eV) taken for the same area (size 500x500 nm2) in two separate scans. Fig.14c and d shows an LSS plot for the dashed line along the dot center shown in Fig.14a, and seven selected spectra from this scan (denoted 1-7), respectively. Fig.14e shows classical positions of the electrons and the trion for nine (cartoons I and II) and ten (cartoon III) electron WMs. In the NSOM image in Fig.14a and b in addition to the bright resonant QD denoted Qa (emission energy 1.713 eV), two weaker non-resonant QDs denoted Qb and Qc (emission energy 1.701 and 1.695 eV, respectively) appear in the images. Resonant Qa dot has ∆E=5.5 meV and three shells are filled, as can be seen from the spectra in Fig.14d. The image of Qa has size ~120 nm and it reveals a strong (up to 50%) intensity fluctuation on a length scale ~30 nm, which is of the order of the single electron separation in a WM in the classical limit. We determined, using LSS in Fig.14c and d, that the photon counting detection noise only partly contributes to the spatial intensity fluctuation in Fig.14a and b (note that photons are counted only for 20 ms in monochromatic imaging). In the image in Fig.14c plotted from spectra measured by a CCD with accumulation time 1 s we observed similar spatial fluctuations of the emission intensity. From Fig.14c and d it is seen that intensity fluctuations are accompanied by changes of the SU-part of the emission spectra and by the spectral diffusion of a few meV. Spectral diffusion indicates the effects of rearrangement of the charge distribution inside and/or around the QDs under near- field excitation. It is seen from Fig.14c and d that for positions 1, 2, 4 and 7 the SU-emission consists of two peaks SU1 and SU2, having energy shift from the s-peak of ~3 and 5 meV, respectively. For positions 3 and 6 the SU1 peak reveals a splitting of ~2 meV, and for position 5 a SU0 peak having energy shift ~2 meV appears. For position 5 the intensity of the SU0 emission peak is very strong: it is nearly the same intensity as the intensity of the s-peak. Using the analysis of the "electron-phonon" interaction discussed in section 4.2 and WM excited states discussed in section 3 we suppose that the SU1 and SU2 peaks can

internal magnetic field. It arises from optical pumping of the nuclear spins (Brown et al., 1996; Maleinsky et al., 2007; Tratakovskii et al., 2007), inducing an internal magnetic field (Bint) of 2T which is anti-parallel to the external field. Such a field can induce the Wigner localization even at zero external magnetic field and thus using optical pumping gives additional control of WM formation.

### **7.3.2 Observation of the molecular-droplet transition**

The measurements of the behavior of the multi-shell peaks in a magnetic field allow observation of a magnetic-field-induced phase transition of a WM using magneto-NSOM spectroscopy which is similar to that observed in Coulomb blockade measurements using nano-lithographically-defined GaInAs/AlGaAs QDs in a SET (Ashoori, 1996; Kastner, 1993; Tarucha, 1996). These measurements also confirm a shell type nature of an emission manifold of InP/GaInP QDs and estimate "valence" shell filling. In Fig.13a and b we show the unpolarized emission spectra versus magnetic field of QD8 and QD11 having ∆E=4.5 and 3.5 meV and N~10 and 20, respectively (see Fig.11a and b). In Fig.13c and d we compare the magnetic-field-induced shifts of the shell peaks for these QDs with the energy levels of the Fock-Darwin (FD) Hamiltonian. From Fig.13b and d one can see that for QD11 having larger size (i.e. smaller ∆E) than QD8, the f-shell peak is observed. Here the FD levels follow the experimentally observed shifts only for field up to 3T indicating filling of all f shell states, which gives electron number 19-20. At higher fields peak positions shift slightly to a lower energy and the d peak transforms to an x peak. This can indicate a transition of the electrons from the third (*υ*=2) to the second (*υ*=1) Landau levels. However, FD levels follow experiment only approximately and no distinct assignments of the shell peaks can be made at B>3T. We should point out that at 10T only a s-shell peak dominates the spectra and all other shells strongly suppressed.

Fig. 13. Near-field spectra at 10 K representing shell structure (peaks s-g) of QD8 (a) and QD11 (b) at magnetic fields 0, 1,..,10T. Peaks \* are contributions of neighboring QDs. Energy shift of emission peaks of QD8 (a) and QD11 (b) versus magnetic field (solid circles) are shown in (c) and (d), together with calculated Fock-Darwin energy levels (solid curves) and Landau levels *υ*=0 and 1 (dashed lines). In (c) and (d) the abscissa is the net *internal* magnetic field.

internal magnetic field. It arises from optical pumping of the nuclear spins (Brown et al., 1996; Maleinsky et al., 2007; Tratakovskii et al., 2007), inducing an internal magnetic field (Bint) of 2T which is anti-parallel to the external field. Such a field can induce the Wigner localization even at zero external magnetic field and thus using optical pumping gives

The measurements of the behavior of the multi-shell peaks in a magnetic field allow observation of a magnetic-field-induced phase transition of a WM using magneto-NSOM spectroscopy which is similar to that observed in Coulomb blockade measurements using nano-lithographically-defined GaInAs/AlGaAs QDs in a SET (Ashoori, 1996; Kastner, 1993; Tarucha, 1996). These measurements also confirm a shell type nature of an emission manifold of InP/GaInP QDs and estimate "valence" shell filling. In Fig.13a and b we show the unpolarized emission spectra versus magnetic field of QD8 and QD11 having ∆E=4.5 and 3.5 meV and N~10 and 20, respectively (see Fig.11a and b). In Fig.13c and d we compare the magnetic-field-induced shifts of the shell peaks for these QDs with the energy levels of the Fock-Darwin (FD) Hamiltonian. From Fig.13b and d one can see that for QD11 having larger size (i.e. smaller ∆E) than QD8, the f-shell peak is observed. Here the FD levels follow the experimentally observed shifts only for field up to 3T indicating filling of all f shell states, which gives electron number 19-20. At higher fields peak positions shift slightly to a lower energy and the d peak transforms to an x peak. This can indicate a transition of the electrons from the third (*υ*=2) to the second (*υ*=1) Landau levels. However, FD levels follow experiment only approximately and no distinct assignments of the shell peaks can be made at B>3T. We should point out that at 10T only a s-shell peak dominates the spectra and all

Fig. 13. Near-field spectra at 10 K representing shell structure (peaks s-g) of QD8 (a) and QD11 (b) at magnetic fields 0, 1,..,10T. Peaks \* are contributions of neighboring QDs. Energy shift of emission peaks of QD8 (a) and QD11 (b) versus magnetic field (solid circles) are shown in (c) and (d), together with calculated Fock-Darwin energy levels (solid curves) and Landau levels *υ*=0 and 1 (dashed lines). In (c) and (d) the abscissa is the net *internal* magnetic

additional control of WM formation.

other shells strongly suppressed.

field.

**7.3.2 Observation of the molecular-droplet transition**

For QD8 (Fig.13a and c) the observed magnetic-field-induced shell shifts follow the FD energy levels remarkably well. Here two magnetic-field induced transitions are observed. The first transition occurs at B=2T arising from an internal magnetic field (Bint) of 0T which was discussed above (see Fig12.b). In the unpolarized spectra presented in Fig.13a, Bint is responsible for the shift to lower energy of the s- and d-shell peaks at B<2. For B>2 (Bint>0) the peak positions shift to increasing energy (see Fig.13c). In the range B=2-7 T (Bint=0-5 T) the p-shell peak appears. At B>7 T (second transition; Bint=5 T) the p-shell peak transforms into a d-shell, and d into g-shell. As can be seen from Fig.13c the second transition (at 7T) corresponds to transition of electrons from the second (*υ*=1, p/d-shells) to the first Landau level (*υ*=0, d/g-shells). Such a transition was previously observed in the magnetic field dependence of Coulomb blockade and can be described as a molecular-droplet transition (Oosterkamp et al., 1999). It arises from formation of a maximum-density-droplet at B=2, and its decomposition is accompanied by a WM formation at B>4 T (Maksym et al., 2000; Reimann& Manninen, 2002; Szafran et al., 2003 ). In the spectra the WM formation at a high magnetic field corresponds to a strong increase of the intensity of the g-shell peak (in contrast to QD11).

### **7.4 High-spatial resolution NSOM imaging**

We used the uncapped structure to perform ultra-high-spatial resolution imaging and spectroscopy of InP/GaInP QDs using uncoated fiber probes working in an apertureless regime. In Fig.14a and b we present monochromatic 5K-NSOM images (detection energy 1.713 eV) taken for the same area (size 500x500 nm2) in two separate scans. Fig.14c and d shows an LSS plot for the dashed line along the dot center shown in Fig.14a, and seven selected spectra from this scan (denoted 1-7), respectively. Fig.14e shows classical positions of the electrons and the trion for nine (cartoons I and II) and ten (cartoon III) electron WMs. In the NSOM image in Fig.14a and b in addition to the bright resonant QD denoted Qa (emission energy 1.713 eV), two weaker non-resonant QDs denoted Qb and Qc (emission energy 1.701 and 1.695 eV, respectively) appear in the images. Resonant Qa dot has ∆E=5.5 meV and three shells are filled, as can be seen from the spectra in Fig.14d. The image of Qa has size ~120 nm and it reveals a strong (up to 50%) intensity fluctuation on a length scale ~30 nm, which is of the order of the single electron separation in a WM in the classical limit. We determined, using LSS in Fig.14c and d, that the photon counting detection noise only partly contributes to the spatial intensity fluctuation in Fig.14a and b (note that photons are counted only for 20 ms in monochromatic imaging). In the image in Fig.14c plotted from spectra measured by a CCD with accumulation time 1 s we observed similar spatial fluctuations of the emission intensity. From Fig.14c and d it is seen that intensity fluctuations are accompanied by changes of the SU-part of the emission spectra and by the spectral diffusion of a few meV. Spectral diffusion indicates the effects of rearrangement of the charge distribution inside and/or around the QDs under near- field excitation. It is seen from Fig.14c and d that for positions 1, 2, 4 and 7 the SU-emission consists of two peaks SU1 and SU2, having energy shift from the s-peak of ~3 and 5 meV, respectively. For positions 3 and 6 the SU1 peak reveals a splitting of ~2 meV, and for position 5 a SU0 peak having energy shift ~2 meV appears. For position 5 the intensity of the SU0 emission peak is very strong: it is nearly the same intensity as the intensity of the s-peak. Using the analysis of the "electron-phonon" interaction discussed in section 4.2 and WM excited states discussed in section 3 we suppose that the SU1 and SU2 peaks can

Molecular States of Electrons: Emission of

differences in the spectra.

in Fig.14 (see also Fig16b below).

the SU peak structure presented in Fig.16a.

Single Molecules in Self-Organized InP/GaInP Quantum Dots 145

in units of ∆E is equal to 0.2, 0.4 and 1.5 for QD2e, QD3e and QD5e. The right inserts in Fig. 15 shows our suggestions for the WM structure, which could explain the observed

We suppose that QD2e, QD3e, QD4e, and QD5e have ten, ten, nine, and two electrons, respectively. Three of the electrons in QD2e are centrally located, whereas only two are centrally located in QD3e. For QD2e the optical transition interacts equally with all three electrons, giving the high intensity of the p- and d-shells. The SU structure arises from the interaction with the other WM arrangement (QD3e ) having two central electrons, and the energy 0.2∆E is the splitting between these arrangements, which is the energy difference of the isomeric excitations. The intensity distribution for QD3e is similar to the metallic dot Qa

The QD4e dot must have electron number between 7 and 9 since the d shell is populated and the existence of the central electron suppresses the SU emission. The SU structure of QD5e allows us to assume that this is a two electron QD. This is supported from our analysis of

Fig. 15. Emission spectra of excitonic QDs molecules at 10 K. Right inserts show possible classical positions of electrons and trion of corresponding WMs in the excited state. Es energy (in eV) is 1.7073 for QD2e, 1.766 for QD3e , 1.687 for QD4e and 1.7438 for QD5e

We use six Gaussian peaks to model the SU structure of this QD5e (see Fig.16a). Our analysis has shown that the energy of these peaks can be grouped in two sequences – one is n\*∆E and the other is n\*1.5∆E, or nω0 and n\*1.5ω0=n\*ωS, where n is an integer. These energies are

**7.5.1 Vibronic structure of a two-electron Wigner molecule**

be related to the translational and breathing vibrational modes and that the splitting of the SU1 peak is related to a other normal mode of a 9e-WM. We assign the SU0 peak to isomeric excitation of a 10e-WM. The appearance of the different (from translational and breathing) modes for positions 3 and 6 can be explained by photoexcitation at the edge of the dot, in which recombination of an "edge" trion generates the WM rotational motion (cartoon II in Fig.14e). For photoexcitation at the center of the dot (position 4), no rotations are generated by symmetry (see cartoon I in Fig.14e). Positions 1 and 7 are non-contact positions for which the photexcitation can be considered uniform. We can expect that for such excitation the emission is generated by the trion bound to the central electron. We also suppose that the enhanced coupling to the vibrational mode arises in a 10e-WM due to the formation of a two-electron isomeric arrangement at the center see Fig.14e, cartoon III). We obtained further evidence of vibronic structure of the SU-emission from analysis of spectra of the excitonic dots.

Fig. 14. High-spatial resolution NSOM measurements of InP QDs at 5 K: (a and b) monochromatic NSOM image, (c) false-color LSS-plot for a scan along the QD (see dashed horizontal line in a); (d) seven selected spectra from LSS (see vertical arrows at the bottom of c) from the linear scan; (e) classical arrangements of electrons and trion in photoexcited state of nine (upper and central) and ten electron WMs. In (a) and (b) Qa, Qb and Qc dashed large circles outline different QDs, while the small circles in dot Qa represent possible WM electron positions.

### **7.5 Emission spectra of Wigner molecules**

Fig. 15a-d show spectra of four excitonic dots (QD2e, QD3e QD4e and QD5e) having different intensity distributions of the shell and SU peaks. The ∆E for these dots have values between 1 and 1.9 meV. Three shell peaks are observed for QD2e, QD3e, and QD4e and two for QD5e. The p-shell peaks for QD2e and QD4e reveal two components having splitting 0.4-0.5 meV whereas no components are observed for QD3e and QD5e. The p- and d-shell peaks are strong for QD2e, QD4e and QD5e and weak for QD3e. The intensity of the SU emission is strong for QD2e, QD3e and QD5e and very weak for QD4e. The energy shift of the SU peaks

be related to the translational and breathing vibrational modes and that the splitting of the SU1 peak is related to a other normal mode of a 9e-WM. We assign the SU0 peak to isomeric excitation of a 10e-WM. The appearance of the different (from translational and breathing) modes for positions 3 and 6 can be explained by photoexcitation at the edge of the dot, in which recombination of an "edge" trion generates the WM rotational motion (cartoon II in Fig.14e). For photoexcitation at the center of the dot (position 4), no rotations are generated by symmetry (see cartoon I in Fig.14e). Positions 1 and 7 are non-contact positions for which the photexcitation can be considered uniform. We can expect that for such excitation the emission is generated by the trion bound to the central electron. We also suppose that the enhanced coupling to the vibrational mode arises in a 10e-WM due to the formation of a two-electron isomeric arrangement at the center see Fig.14e, cartoon III). We obtained further evidence of vibronic structure of the SU-emission from analysis

Fig. 14. High-spatial resolution NSOM measurements of InP QDs at 5 K: (a and b)

monochromatic NSOM image, (c) false-color LSS-plot for a scan along the QD (see dashed horizontal line in a); (d) seven selected spectra from LSS (see vertical arrows at the bottom of c) from the linear scan; (e) classical arrangements of electrons and trion in photoexcited state of nine (upper and central) and ten electron WMs. In (a) and (b) Qa, Qb and Qc dashed large circles outline different QDs, while the small circles in dot Qa represent possible WM

Fig. 15a-d show spectra of four excitonic dots (QD2e, QD3e QD4e and QD5e) having different intensity distributions of the shell and SU peaks. The ∆E for these dots have values between 1 and 1.9 meV. Three shell peaks are observed for QD2e, QD3e, and QD4e and two for QD5e. The p-shell peaks for QD2e and QD4e reveal two components having splitting 0.4-0.5 meV whereas no components are observed for QD3e and QD5e. The p- and d-shell peaks are strong for QD2e, QD4e and QD5e and weak for QD3e. The intensity of the SU emission is strong for QD2e, QD3e and QD5e and very weak for QD4e. The energy shift of the SU peaks

of spectra of the excitonic dots.

electron positions.

**7.5 Emission spectra of Wigner molecules**

in units of ∆E is equal to 0.2, 0.4 and 1.5 for QD2e, QD3e and QD5e. The right inserts in Fig. 15 shows our suggestions for the WM structure, which could explain the observed differences in the spectra.

We suppose that QD2e, QD3e, QD4e, and QD5e have ten, ten, nine, and two electrons, respectively. Three of the electrons in QD2e are centrally located, whereas only two are centrally located in QD3e. For QD2e the optical transition interacts equally with all three electrons, giving the high intensity of the p- and d-shells. The SU structure arises from the interaction with the other WM arrangement (QD3e ) having two central electrons, and the energy 0.2∆E is the splitting between these arrangements, which is the energy difference of the isomeric excitations. The intensity distribution for QD3e is similar to the metallic dot Qa in Fig.14 (see also Fig16b below).

The QD4e dot must have electron number between 7 and 9 since the d shell is populated and the existence of the central electron suppresses the SU emission. The SU structure of QD5e allows us to assume that this is a two electron QD. This is supported from our analysis of the SU peak structure presented in Fig.16a.

Fig. 15. Emission spectra of excitonic QDs molecules at 10 K. Right inserts show possible classical positions of electrons and trion of corresponding WMs in the excited state. Es energy (in eV) is 1.7073 for QD2e, 1.766 for QD3e , 1.687 for QD4e and 1.7438 for QD5e

#### **7.5.1 Vibronic structure of a two-electron Wigner molecule**

We use six Gaussian peaks to model the SU structure of this QD5e (see Fig.16a). Our analysis has shown that the energy of these peaks can be grouped in two sequences – one is n\*∆E and the other is n\*1.5∆E, or nω0 and n\*1.5ω0=n\*ωS, where n is an integer. These energies are

Molecular States of Electrons: Emission of

having frequency ω0.

**8. Conclusions**

experiments are in progress.

**9. Acknowledgment**

measurements.

**10. References**

June), pp.2765-2768

Single Molecules in Self-Organized InP/GaInP Quantum Dots 147

energy of the weak structure is ~2*ħω*0, which fits well to the high frequency vibrational modes. These modes include the breathing mode, having frequency 1.7ω0 (see Fig.3). Reduction of this energy down to *ħω*0 for a metallic dot reflects the loosing of the molecular character of the electron distribution for *rs*~ 1.2 and failure of the classical approximation. In this Fermi liquid regime the electron motion is described only by translation vibration

We should also point out the stronger contribution of p- and d-shell peaks in the smaller Qa

We presented the results of an experimental (photoluminescence) study of correlated states of electrons in a WM in self-organized InP/GaInP quantum dots. The unique properties of these QDs are their relatively large lateral size (~80-200 nm) and their ability to accommodate up to 20 electrons, providing electron density up to 2x1011 cm-2 and strong emission intensity. Using high-spatial-resolution low-temperature near-field scanning optical microscopy (NSOM) having spatial resolution up to 30 nm in combination with a high magnetic field, we were able to resolve emission spectra of single QDs, and to observe crossover from a Fermi liquid to WM behavior at a critical density of 5x1010 cm-2. A magnetic-field-induced molecular-droplet transition has been observed in the Fermi liquid regime. In the Wigner molecule regime we observed a rich vibrational structure, which opens the way to identify electron arrangement in WMs. Further

Development of the NSOM technique was supported by a National Science Foundation NIRT grant No. ECS-0609249. The authors wish to thank Nikolay A. Kalugnyy, Vladimir. M. Lantratov, Sergei A. Mintairov and Dmitrii A. Vinokurov from the Ioffe Physical-Technical Institute (St. Petersburg, Russia) for growth InP/GaInP structures. The authors wish to thank Alexei Vlasov and Tatiana Prutskih for making fiber tips and for help in NSOM

Andrei, E.Y.; Deville, G.; Williams, F.; Paris, I.B. & Etienne, B. (1988). Observation of

Ashoori, R.C. (1996). Electrons in artificial atoms, *Nature,* Vol.379, (1 February), pp.413-419 Betzig, E. & Trautman, J. K. (1992). Near-Field Optics: Microscopy, Spectroscopy, and Surface Modification Beyond Diffraction Limit, *Science* Vol.257, (10 July), pp. 189-195 Blome, P. G.; Wenderoth, M.; Hubaner, M. & Ulbrich, R. G. (2000). Temperature-dependent

Magnetically Induced Wigner Solids, *Physical Review Letters*,Vol.60, No.26 ,(27

linewidth of single InP/GaxIn1-xP quantum dots: Interaction with surrounding charge configuration, *Physical Review B,* Vol.61, No.12, (15 March) pp.8382-8387

dot, which reflect stronger overlap of hole and electrons wave functions.

naturally assigned to bending and stretching vibrations of a 2e-WM (see Eq. 1), which according to the theoretical calculations have values ω0 and 1.7ω0. The smaller value of the frequency of the stretching vibration observed experimentally (1.5ω0 instead of 1.7ω0) can result from deviations of the real confinement potential from the ideal 2D potential. The anti-Stokes peak ω0 arises from the thermal activation and its relative intensity is well described by the thermal factor exp(-∆E/kT), which for T=10K is equal to 0.3. From the intensities of the peaks we can estimate Huang-Rhys factors for bending and stretching vibrations to be ~0.2 and ~0.1.

Fig. 16. Gaussian contour decomposition of emission spectra of QD5e (a) and comparison of emission spectra of two QDs having shell splitting ∆E=1.7 and 5.5 meV (b). The horizontal axis in b has reduced units (Es-E)/∆E.

#### **7.5.2 Size-dependent structure**

Fig. 16b shows the comparison of the emission spectra of the metallic and the excitonic dots having similar intensity distributions of the SU peaks. These are the Qa dot (lower spectrum in Fig. 16b), the spectrum of which is also shown in Fig.14d for tip position 5, and the QD3e dot (upper spectrum), the spectrum of which is shown in Fig.15b. The shell splitting of these two dots is 1.7 and 5.5 meV, which corresponds to sizes 120 and ~70 nm, respectively. We suppose that these two dots have the same electron occupation equal to ten, which corresponds to *rs*~2 and ~1.2. The horizontal axis in Fig.16b is in reduced units n=(ES-E)/∆E or n=(Es-E)/*ħω0* and from Fig.16b a very good coincidence of the spectral shape is seen. This coincidence is remarkable, accounting for a nearly three times difference in the absolute spectral range, which is equal 15.3 and 45.5 meV for QD3e and Qa, respectively, for n values from -5 to 4 in Fig.16b. These dots have a strong SU peak having the same intensity as the intensity of the "zero-phonon" line, i.e. the s-peak. The weaker structure also observed near n=2 for QD3e and n=1 for Qa. The SU peak can be assigned to the isomeric excitation between the ground isomer having [2, 8] classical electron geometry and excited isomer having [3,7] geometry. This follows from the fact that this is the only excitation which linearly scales with *rs* (see Fig.2). The weaker structure does not scale with *rs* and it shows shrinkage of the energy. Such shrinkage is expected for a transition from partial Wigner localization to Fermi liquid discussed in section 3.1 for 2e-WM. For the excitonic dot the energy of the weak structure is ~2*ħω*0, which fits well to the high frequency vibrational modes. These modes include the breathing mode, having frequency 1.7ω0 (see Fig.3). Reduction of this energy down to *ħω*0 for a metallic dot reflects the loosing of the molecular character of the electron distribution for *rs*~ 1.2 and failure of the classical approximation. In this Fermi liquid regime the electron motion is described only by translation vibration having frequency ω0.

We should also point out the stronger contribution of p- and d-shell peaks in the smaller Qa dot, which reflect stronger overlap of hole and electrons wave functions.

### **8. Conclusions**

146 Fingerprints in the Optical and Transport Properties of Quantum Dots

naturally assigned to bending and stretching vibrations of a 2e-WM (see Eq. 1), which according to the theoretical calculations have values ω0 and 1.7ω0. The smaller value of the frequency of the stretching vibration observed experimentally (1.5ω0 instead of 1.7ω0) can result from deviations of the real confinement potential from the ideal 2D potential. The anti-Stokes peak ω0 arises from the thermal activation and its relative intensity is well described by the thermal factor exp(-∆E/kT), which for T=10K is equal to 0.3. From the intensities of the peaks we can estimate Huang-Rhys factors for bending and stretching

Fig. 16. Gaussian contour decomposition of emission spectra of QD5e (a) and comparison of emission spectra of two QDs having shell splitting ∆E=1.7 and 5.5 meV (b). The horizontal

Fig. 16b shows the comparison of the emission spectra of the metallic and the excitonic dots having similar intensity distributions of the SU peaks. These are the Qa dot (lower spectrum in Fig. 16b), the spectrum of which is also shown in Fig.14d for tip position 5, and the QD3e dot (upper spectrum), the spectrum of which is shown in Fig.15b. The shell splitting of these two dots is 1.7 and 5.5 meV, which corresponds to sizes 120 and ~70 nm, respectively. We suppose that these two dots have the same electron occupation equal to ten, which corresponds to *rs*~2 and ~1.2. The horizontal axis in Fig.16b is in reduced units n=(ES-E)/∆E or n=(Es-E)/*ħω0* and from Fig.16b a very good coincidence of the spectral shape is seen. This coincidence is remarkable, accounting for a nearly three times difference in the absolute spectral range, which is equal 15.3 and 45.5 meV for QD3e and Qa, respectively, for n values from -5 to 4 in Fig.16b. These dots have a strong SU peak having the same intensity as the intensity of the "zero-phonon" line, i.e. the s-peak. The weaker structure also observed near n=2 for QD3e and n=1 for Qa. The SU peak can be assigned to the isomeric excitation between the ground isomer having [2, 8] classical electron geometry and excited isomer having [3,7] geometry. This follows from the fact that this is the only excitation which linearly scales with *rs* (see Fig.2). The weaker structure does not scale with *rs* and it shows shrinkage of the energy. Such shrinkage is expected for a transition from partial Wigner localization to Fermi liquid discussed in section 3.1 for 2e-WM. For the excitonic dot the

vibrations to be ~0.2 and ~0.1.

axis in b has reduced units (Es-E)/∆E.

**7.5.2 Size-dependent structure**

We presented the results of an experimental (photoluminescence) study of correlated states of electrons in a WM in self-organized InP/GaInP quantum dots. The unique properties of these QDs are their relatively large lateral size (~80-200 nm) and their ability to accommodate up to 20 electrons, providing electron density up to 2x1011 cm-2 and strong emission intensity. Using high-spatial-resolution low-temperature near-field scanning optical microscopy (NSOM) having spatial resolution up to 30 nm in combination with a high magnetic field, we were able to resolve emission spectra of single QDs, and to observe crossover from a Fermi liquid to WM behavior at a critical density of 5x1010 cm-2. A magnetic-field-induced molecular-droplet transition has been observed in the Fermi liquid regime. In the Wigner molecule regime we observed a rich vibrational structure, which opens the way to identify electron arrangement in WMs. Further experiments are in progress.

### **9. Acknowledgment**

Development of the NSOM technique was supported by a National Science Foundation NIRT grant No. ECS-0609249. The authors wish to thank Nikolay A. Kalugnyy, Vladimir. M. Lantratov, Sergei A. Mintairov and Dmitrii A. Vinokurov from the Ioffe Physical-Technical Institute (St. Petersburg, Russia) for growth InP/GaInP structures. The authors wish to thank Alexei Vlasov and Tatiana Prutskih for making fiber tips and for help in NSOM measurements.

### **10. References**

Molecular States of Electrons: Emission of

(19 March) pp.795-798

(15 November) pp.12386-12384

*Review B* Vol.64, pp. 233308-1-11

(22 December) pp. 328-332

quantum dot *Nature Physics* 4 469

4237-42400

pp.362-364

pp.235325-8

7792

334

Single Molecules in Self-Organized InP/GaInP Quantum Dots 149

Grimes, C.C.& Adams, G. (1979) Evidence of Liquid-to-crystal Phase Transition in a

Hawrylak, P. *(*1992). Many electron effects in acceptor-related radiative recombination of

Hawrylak, P. & Potemski, M. (1997). Theory of photoluminescence from an interacting two-

Hirose, K. & Wingreen, N. S. (1999*).* Spin-density-functional theory of circular and elliptical quantum dots, *Physical Review B,* Vol.59, No. 7, (15 February) pp. 4604

Hessman, D.; Persson, J.; Pistol, M-E.; Pryor, C.& Samuelson, L. (2001). Electron

Hosaka, N. & Saiki, T. (2001) Near-field fluorescence imaging of single molecules with

Huang, K.& Rhys, A. (1950). Theory of light absorption and non-radiative transitions in F-

Ilani, S.; Martin, J.; Teitelbaum, E.; Smet, J.H.; Mahalu, D.; Umansky, V. & Yacoby, A. (2004).

Janssens, K. L.; Partoens, B. & Peeters F. M. (2003). Effect of strain on the magnetoexciton

Kalliakos, S. Rontani, M. Pellegrini, V. GARC´IA, C. P. Pinczuk, A. Goldoni, G. Molinari, E.

Kukushkin, I. V.; Klitzing, K. von; Ploog, K. & V. B. Timofeev (1989). Radiative

Laughlin, R.B. (1983). Quantized motion of three two-dimensional electrons in a strong magnetic field, *Physical Review,* Vol.27, No.6, (15 March) pp. 3383-3389. Lewis, A.; Isaacson, M.; Harootunian, A. & Muray, A. (1984). Development of a 500 Å spatial

Maksym, P.A.; Imamura, H.; Mallon, G.P. & Aoki, H. (2000). Molecular aspects of electron

Jacak, L.; Hawrylak, P. & Wojs, A. (1998)Quantum dots (Springer, Berlin, p. 176)

*Sciences,* Vol.204, No. 1078 (22 December), pp. 406-423

Kastner, M.A. (1993). Artificial atoms, *Physics Today*, (January) pp.24-31

apertures, *Ultramicroscopy,* Vol. 13, pp.227-232

Classical Two-Dimensional Sheet of Electrons *Physical Review Letters Vol.*42, No.12,

quasi-two-dimensional electrons, *Physical Review B,* Vol.45, No*.*8 (15 February), pp.

dimensional electron gas in strong magnetic fields, *Physical Review B* , Vol.65, No.19

accumulation in single InP quantum dots observed by photoluminescence *Physical*

spatial resolution in a range of 10 nm, *Journal of Microscopy,* Vol.202 No.2 (May),

centers, *Proceedings of the Royal Society of London. Series A, Mathematical and Physical*

The microscopic nature of localization in the quantum Hall effect, *Nature*, Vol.427,

ground state in InP/GaxIn1ÀxP quantum disks *Phisical Review B* Vol.67,

N. Pfeiffer, L. West K. W. (2008) A molecular state of correlated electrons in a

recombination of two-dimensional electrons in acceptor δ-doped GaAs-AlxGa1 xAs single heterojunctions, *Physical Review B,* Vol.40, No.11, (15 October) pp.7788-

resolution light microscope: I. light is efficiently transmitted through λ/16 diameter

correlation in quantum dots, *Journal of Physics: Condensed. Matter,* Vol.12, R299-

Blundell, S. A. & Chacko, S. (2011). Excited states of incipient Wigner molecules *Physical*

Blundell, S. A. & Joshi, K. (2010). Precise correlation energies in small parabolic quantum dots from configuration interaction, *Physical Review B,* Vol.81, No.11, pp.115232-1-11 Bolton, F. & Rössler, U. (1993). Classical model of a Wigner crystal in a quantum dot,

Brown, S. W.; Kennedy, T. A.; Gammon, D. & Snow, E. S. (1996). Spectrally resolved

Butov, L. V.; Grinev, V. I. ; Kulakovskii, V. D. & T. G. Anderson (1992) Direct observation of

Chu, Y.; Mintairov, A. M. ; He, Y.; Merz, J. L.; Kalyuzhnyy, N. A.; Lantratov, V. M.&

Coutjon, D.; Sarayeddine, K.& Spajer, M. (1989). Scanning tunneling optical microscopy,

Egger, R.; Häusler, W.; Mak, C. H. & Grabert, H. (1999). Cossover from Fermi Liquid to Wigner Molecule Behavior in Quantum Dots *Physical Review Letters* 82 3320 Ellenberger, C. Ihn, T. Yannouleas, C Landman, U. Ensslin, K. Driscoll, D. & Gossard A. C.

Fischer, U. C. & Pohl, D. W. (1989). Observation of Single-Particle Plasmons by Near-Field

Flack, F. ; Samarth, N.; Nikitin, V.; Crowell, P. A.; Shi, J.; Levy, J.& Awschalom, D. D. (1996).

Govorov, A. O.; Schulhauser, C.; Haft, D.; Kalameitsev, A. V.; Chaplik, A.; Warburton, R. J.;

Ghosal, A.;. G¨uc¸l¨u, A. D.; Umrigar, C. J.; Ullmo, D. & Baranger H. U. (2006). Correlation-

dots, *Physical Review B,*Vol.54, No.24 (15 december) pp.R17312-R17315 Georgsson, K.; Carlsson, N.; Samuelson, L.; Seifert, W.& Wallenberg L. R. (1995)

with Negligible Zeeman Splitting*, Physical Review Letters* 96, 126806 Finkelstein, G.; Shtrikman, H. & Bar-Joseph, I. (1995). Optical Spectroscopy of Two-

Overhauser shifts in single GaAs/AlxGa1-xAs quantum dots, *Physical Review B,*

magnetoplasmon-phonon coupled modes in the magnetophotoluminescence spectra of the two-dimensional electron gas in InxGa1-xAs/GaAs quantum wells,

Mintairov S. A. (2009). Lasing of whispering-gallery modes in asymmetric waveguide GaInP micro-disks with InP quantum dots, P*hysics Letters A*, Vol.373,

(2006) Excitation Spectrum of Two Correlated Electrons in a Lateral Quantum Dot

Dimensional Electron Gas near the Metal-Insulator Transition, *Physical Review*

Optical Microscopy, *Physical Review Letters,* Vol.62, No.4 (23 January), pp.458-461

Near-field optical spectroscopy of localized excitons in strained CdSe quantum

Transmission electron microscopy of the morphology of InP Stranski-Krastanov islands grown by metalorganic chemical deposition *Appl. Phys. Lett.* Vol.67, No.20

Karrai, K.; Schoenfeld, W.; Garcia, J. M.& Petroff P. M. (2002). Magneto-optical properties of charged excitons in quantum dots, (26 February, 2002) Available from

induced inhomogeneity in circular quantumdots, *Naure Physics.* Vol.2, No.5 (23

*Superlattices and Miucrostructures*, Vol.13, No.2 (March) pp.139-145

*Physical Review B,* Vol.46, No.20 (15 November) pp. 13627-13630

*Optics Communication,* Vol.71, No.1,2 (1 May) pp.23-28

*Letters* Vol.74, No.6 (6 February) pp.976-979

(13 November) pp. 2981-2982

arXiv:cond-mat/0202480 v1

April) pp. 336-340

*Review B,* Vol.83, No.19, (31 May) pp.195444-11

*Vol.*54, No.24 (15 December) pp.R17339-R17342

1185-1188

Molecular States of Electrons: Emission of

507-510

pp.235309-1-10

Single Molecules in Self-Organized InP/GaInP Quantum Dots 151

Reddick, R. C.; Warmack, R. J. & Ferrell, T. L. (1989). New form of scanning optical microscopy, *Physical Review B,* Vol.39, No.1, (1 January), pp. 767-770 Reimann, S. M.; Koskinen, M. &. Manninen, M. (2000). Formation of Wigner molecules in

Reimann, S.M. & Manninen, M. (2002). Electronic structure of quantum dots, *Review of*

Ren, H.-W.; Sugizaki, M.; Lee, J.-S.; Sugou, S.&Masumoto, Y.,Highly Uniform and Small

Saint Jean, M; Even, C. & Guthmann, C. (2001). Macroscopic 2D Wigner islands, *Europhysics*

Singha A, V. Pellegrini, A. Pinczuk, L. N. Pfeiffer, K. W. West, and M. Rontani,

Sugizaki, M.; Ren, H.-R.; Nair, S. V.; Nishi, K&Masumoto, Y. (2002) External-field effects on

Synge, E. H. (1928). A suggested method for extending microscopic resolution into the ultramicroscopic region, *Phyloofical Maazine Series 7,* Vol.6, No.35, pp.356-362

Szafran, B.; Bednarek, S. & Adamowski, J. (2003). Magnetic-field-induced phase transitions in Wigner molecules, *Journal of Physics: Condensed Matter*, Vol.15, pp.4189-4205. Tratakovskii, A. I.; Wright, T.; Russell, A.; Fal'ko, V. I.; Van'kov, A. B.; Skiba-Szymanska, J.;

Tarucha, S.; Austing, D.G.; Honda, T.; Van der Hage, R.J.; Kouwenhoven, L.P. (1996). Shell

Toda, Y.; Kourogi, M.; Otsu, M.; Nagamune, Y.& Arakawa Y. (1996). Spatially and spectrally

Toda, Y.; Sugimoto, T.; Nishioka, M. & Arakawa, Y. (2000). Near-field coherent excitation

Toda, Y.; Moriwaki, O.; Nishioka, M. & Arakawa, Y. (1999). Efficient Carrier Relaxation

technique, *Applied Physics Letters* Vol.69, No.6, (5 August) pp.827-829 Toda, Y.; Shinomori, S.; Suzuki, K. & Arakawa, Y. (1998). Polarized photoluminescence

*Modern Physics*, Vol.74, No.4, (October), pp.1283-1342

Electron Dimer Molecule *Phys. Rev. Lett.* 104, 246802

Szabo, A. & Ostlund, N. (1996) Modern Quantum Chemistry, *Dover, New York*

*Physical Review Letters,* Vol.98, No.2 (12 January) pp. 026806-4

*Letters,* Vol.55, No.1 (1 July), pp. 45–51

Vol.77, No.17 (21 October), pp.3613-3616

No.16 (15 October) pp.R10147-10150

Vol.76, No.26 (26 June) pp.3887-3889

4117

small quantum dots, *Physical Review B* Vol.62, No.12 (15 September) pp.8108-8113

InP/GaInP Self-Assembled Quantum Dots Grown by Metal-Organic Vapor Phase Epitaxy (1999) *Japanese Journal of Applied Physics* Vol.38, No.1B, Part 1 (January) pp.

(2010)Correlated Electrons in Optically Tunable Quantum Dots: Building an

the optical spectra of self-assembled InP quantum dots. *Physical Review B,*Vol.66,

Drouzas, I.; Kolodka, R. S.; Skolnik, M. S.; Fry, P. W.; Taharaoui, A.; Lui, H-Y. & Hopkinson, M. (2007). Nuclear Spin Switch in Semiconductor Quantum Dots,

Filling and Spin Effects in a Few Electron Quantum Dots, *Physical Review Letters*,

resolved imaging of GaAs quantum-dot structures using near-field optical

spectroscopy of single self-assembled InAs quantum dots, *Physical Review B*, Vol.58,

spectroscopy of InGaAs/GaAs self-assembled quantum dots, *Applied Physics Letters*

Mechanism in InGaAs/GaAs Self-Assembled Quantum Dots Based on the Existence of Continuum States, *Phys. Rev. Lett.* Vol.82, No.20 (17 May) pp.4114-

Maksym, P. A. (1996). Eckardt frame theory of interacting electrons in quantum dots,

Maleinsky P, Lai C W, Bodalato A, and Imamoglu, A. (2007). Nonlinear dynamics of

Matsuda, K.; Matsumoto, T.; Saito, H.; Nishi, K.& Saiki, T. (2000). Direct observation of

Mintairov, A. M.; Kosel, T. H.; Merz, J. L.; Blagnov, P. A.; Vlasov, A. S.; Ustinov, V. M.

Mintairov, A. M.; Sun, K.; Merz, J. L.; Li, C.; Vlasov, A. S.; Vinokurov, D. A.; Kovalenkov, O.

Mintairov, A. M.; Herzog, J.; Kuno, M. & Merz, J. L. (2010). Near-field scanning optical

Nash, K. J.; Scolnick, M. S.; Saker, M. K. & Bass, S. J. (1993). Many Body Shakeup in

Oosterkamp, T. H.; Janssen, J. W.; Kouwenhoven, L. P.; Austing, D. G.; Honda, T.& Tarucha,

Ortner, G.; Bayer, M.; Larionov, A.; Timofeev, V. B.; Forchel, A.; Lyanda-Geller, Y. B.;

Coupling, *Physical Review Letters,* Vol.90, No.8, (28 February) pp. 086404-4 Otsu, M. (1998) Near-Field Nano/Atom Optics and Technology, *Springer-Verlag, Tokyo.*  Paskov, P.P.; Holtz, P.O.; Wongmanerod, S.; Monemar, B.; Garcia, J.M.; Schoenfeld, W.V.&

Pohl, D. W.; Denk, W. & Lanz, M. (1984). Optical stethoscpy: Image recording with resolution *λ*/20, *Applied Physics Letters,* Vol.44, No.&, (1 April), pp.651-653 Pryor, C.; Pistol, M-E. & Samuelson, L. (1997) Elecronic structure of strained

Pudalov, V.M.; D'Iotio, M.; Kravchenko, S.V.& Campbell J.W. (1993) Zero-Magnetic-Field

variations of optical properties in single quantum dots by using time-resolved nearfield scanning optical microscope *Physica E,* Vol.7, No.3-4, (May), pp.377-382 Matsuda, K.; Saiki, T.; Nomura S., Mihara, M.; Aoyagi, Y.; Nair S. & Takagahara, T. (2003)

"Near-Field Optical Mapping of Exciton Wave Functions in a GaAs Quantum Dot"

&Cook, R. E. (2001). Near-Field Magnetophotoluminescence Spectroscopy of Composition Fluctuations in InGaAsN, *Physical Review Letters,* Vol.87, No.27 (31

V.;Tokranov, V. & Oktyabrsky, S. (2004). Nanoindentation and near-field spectroscopy of single semiconductor quantum dots, *Physical Review B,* Vol.69,

microscopy of colloidal CdSe nanowires, *Phys. Status Solidi B* Vol.247, No. 6,

Quantum Well Luminescence Spectra, *Phys. Rev. Lett.* Vol.70, No.20 (17 May),

S. *(*1999). Maximum-Density Droplet and Charge Redistributions in Quantum Dots at High Magnetic Fields, *Physical Review Letters,* Vol. 82, No.14 (5 April) pp.2931-

Reinecke, T. L.; Hawrylak, P.; Fafard, S. & Wasilewski, Z. (2003). Fine Structure of Excitons in InAs/GaAs Coupled Quantum Dots: A Sensitive Test of Electronic

Petroff, P.M. (2000). Auger processes in InAs self-assembled quantum dots, *Physica*

InP/Ga0.51In0.49P quantun dots*. Physical Review B,* Vol.56, No.16, (October 15) pp.

Collective Insulator Phase in a Dilute 2D Electron System, *Physical Review Letters*,

*Physical Review B,* Vol. 53, No. 16 (15 April) pp.10871-10886

*Phys. Rev. Lett.* Vol.91, No.17 (24 october), pp.177401-4

December), pp.277401-14

pp.1416–1419

pp.3115-3118

*E*, Vol.6, 440-443

10404-10411

Vol.70, No.12, (22 March) pp. 1866-1869

2934

No.15 (15 April), pp.155306-12

quantum dot nuclear spins, *Physical Review B,* Vol.75, 035409-7

**6**

*Mexico*

**InAs Quantum Dots in Symmetric**

The self-assembled InAs quantum dots (QDs) are the subject of substantial interest during last fifteen years due to both fundamental scientific and application reasons. In these systems, the strong localization of an electronic wave function leads to an atomic-like electronic density of states and permits to realize the novel and improved photonic and electronic devices. Microlectronic and optoelectronic devices based on quantum wells (QWs) with InAs QDs have been the subject of investigation for the applications in semiconductor lasers for the optical fiber communication [1-3], infrared photo-detectors [4- 6], electronic memory devices [7,8], as well as single electron devices and single photon light sources on the base of single-QD structures for the quantum information applications [9-12]. QDs are especially attractive for the applications in semiconductor lasers. For laser or photodiode applications the surface density of QDs has to be high, but for single-QD devices the QD density has to be low. As a result, there is an extensive effort to manipulate

Self-assembled InAs/GaAs QDs in lattice-mismatched systems can be achieved by using the Stranski–Krastanow (S–K) growth mode [20-22]. In the process of InAs/GaAs QDs growth using the S–K growth mode, the InAs mismatched layer growths two dimensionally on the GaAs substrate during the initial stage; then, above a critical thickness, the strain increased and the dislocation-free QDs with a three-dimensional shape are formed on a residual twodimensional wetting layer (WL) [23,24]. InAs/GaAs QD structures grown using the S–K mode have inherent several problems: the density random distribution, the large temperature-dependent variation of the photoluminescence (PL) intensity and line width resulting from the nonuniform size and density of InAs QDs [25] etc. Precise control of the QD size and the homogeneity of InAs QD distribution is necessary for achieving the high-

InAs/GaAs QDs are especially attractive for the applications in lasers because the QD based lasers have a higher differential gain, lower threshold current density, and improved temperature performance in comparison with QW lasers [1-3]. The band structures of InAs/GaAs QDs are well suited for covering the 1.30 and 1.55 µm spectral ranges, important

and control the position, size, shape and density of QDs [13-19].

**2. Advances of InAs QDs in InGaAs/GaAs quantum wells**

**1. Introduction**

performance devices.

**InGaAs/GaAs Quantum Wells**

*ESFM-National Polytechnic Institute, Mexico D.F.,*

Tetyana V. Torchynska

## **InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells**

Tetyana V. Torchynska *ESFM-National Polytechnic Institute, Mexico D.F., Mexico*

### **1. Introduction**

152 Fingerprints in the Optical and Transport Properties of Quantum Dots

Vinokurov, D. A.; Kapitonov, V. A.; Nikolaev, D. N.; Sokolova, Z. N. & Tarasov I. S. (1999).

Yannouleas, C. & Landman, U. (2000). Collective and Independent-Particle Motion in Two-

Wigner, E.P. (1934) On the Interaction of Electrons in Metals, *Physical Review*, Vol.46, (1

InGaAs/InP, *Semiconductors,* Vol.35, No.7, (July) pp.788-791

1729

December), pp.1002-1011.

Self-organized nanoscale InP islands in an InGaP/GaAs host and InAs islands in an

Electron Artificial Atoms *Physical Review Letters,* Vol.85, No.8, (21 August) pp.1726-

The self-assembled InAs quantum dots (QDs) are the subject of substantial interest during last fifteen years due to both fundamental scientific and application reasons. In these systems, the strong localization of an electronic wave function leads to an atomic-like electronic density of states and permits to realize the novel and improved photonic and electronic devices. Microlectronic and optoelectronic devices based on quantum wells (QWs) with InAs QDs have been the subject of investigation for the applications in semiconductor lasers for the optical fiber communication [1-3], infrared photo-detectors [4- 6], electronic memory devices [7,8], as well as single electron devices and single photon light sources on the base of single-QD structures for the quantum information applications [9-12]. QDs are especially attractive for the applications in semiconductor lasers. For laser or photodiode applications the surface density of QDs has to be high, but for single-QD devices the QD density has to be low. As a result, there is an extensive effort to manipulate and control the position, size, shape and density of QDs [13-19].

Self-assembled InAs/GaAs QDs in lattice-mismatched systems can be achieved by using the Stranski–Krastanow (S–K) growth mode [20-22]. In the process of InAs/GaAs QDs growth using the S–K growth mode, the InAs mismatched layer growths two dimensionally on the GaAs substrate during the initial stage; then, above a critical thickness, the strain increased and the dislocation-free QDs with a three-dimensional shape are formed on a residual twodimensional wetting layer (WL) [23,24]. InAs/GaAs QD structures grown using the S–K mode have inherent several problems: the density random distribution, the large temperature-dependent variation of the photoluminescence (PL) intensity and line width resulting from the nonuniform size and density of InAs QDs [25] etc. Precise control of the QD size and the homogeneity of InAs QD distribution is necessary for achieving the highperformance devices.

### **2. Advances of InAs QDs in InGaAs/GaAs quantum wells**

InAs/GaAs QDs are especially attractive for the applications in lasers because the QD based lasers have a higher differential gain, lower threshold current density, and improved temperature performance in comparison with QW lasers [1-3]. The band structures of InAs/GaAs QDs are well suited for covering the 1.30 and 1.55 µm spectral ranges, important

InAs Quantum Dots in Symmetric InGaAs/GaAs Quantum Wells 155

In DWELL structures the introduction of surrounding InxGa1-xAs/GaAs QWs changes the QD density, the elastic strain in structures, the height of potential barriers for exciton capture and thermal escape in/from QDs as well as can increase in some cases the density of nonradiative (NR) centres. In these structures the mechanism of PL intensity thermal decay could depend strongly on parameters of DWELL. Improved understanding of the operation and design of InAs/InGaAs QD-based devices could be emanated from studies involving

Note the highest emission intensity of QDs was achieved in the symmetric wells with the composition of quantum well (QW) capping/buffer layers of In0.15Ga0.85As/GaAs [3]. One of the best methods of manipulating the InAs QD density and sizes inside of QWs related to controlling the QD growth temperatures [3,12-15]. But even for the optimal QD growth parameters and capping/buffer layer compositions, the InAs QD structures are

The inhomogeneity of InAs QD parameters across the wafer due to the size, chemical composition and stress variations leads to a broadening of emission spectra. This type of problems in QD structures was investigated using scanning PL spectroscopy [23, 26-29], high resolution transmission electron microscopy [30], scanning tunneling microscopy [31], as well as spatially resolved scanning tunneling luminescence [32]. However, in dot-in-a well structures, where the InAs QDs coupled with InGaAs/GaAs QWs, the physical reasons of emission inhomogeneity still have to be discussed. The technology of growth of InAs QD structures has become more reliable recently enabling the systematic studies their physical

**3. PL spectra of symmetric GaAs/In0.15Ga0.85As/GaAs QWs with the different**

The solid-source molecular beam epitaxy (MBE) in V80H reactor was used to grow the waveguide structures consisting of the layer of InAs self-organized QDs inserted into 12 nm In0.15Ga0.85As/GaAs QWs. The thickness of the buffer In0.15Ga0.85As layer was 2nm, which was grown on the 200.nm GaAs buffer layer and the 2 inch (100) GaAs SI substrate (Fig.1). Then an equivalent coverage of 2.4 monolayers of InAs QDs were confined by the capping (10nm) In0.15Ga0.85As layer and by the 100nm GaAs barrier. Investigated structures are grown under As-stabilized conditions at five different temperatures: 470C (#1), 490C (#2), 510C (#3), 525C (#4) and 535C (#5), during the deposition of the InAs active region and InGaAs wells, and at 590-610C for the rest of layers. All layers were grown with the growth rate of 0.30 ML/s, but for the QD formation the process provides deposition of 2.4 ML with

characterized by photoluminescence (PL) inhomogeneity along the wafers [16-19].

the excitation and temperature dependences of PL in such structures.

properties and emission inhomogeneity of QD ensembles.

Fig. 1. The schematic design of studied QD structures

**density of InAs QDs**

the growth rate of 0.053 ML/s.

in optical fiber communications. These wavelengths are impossible to achieve in quantum well (QW) InAs/GaAs structures due to the strain-limited thicknesses. It is shown that the QD density in laser structures can be enlarged significantly by growing the dots within InxGa1-xAs/GaAs QWs, in so called dot–in-a-well (DWELL) structures [3, 4]. In these structures photoluminescence has been enhanced due to better crystal quality of the layer surrounding QDs and more effective exciton capture into the QW and into QDs.

A crucial aspect for the realization of efficient light-emitting devices operated at room temperature is the understanding of a temperature dependence of QD photoluminescence. PL intensity decay in InAs QDs as a rule attributed to thermal escape of carriers from QDs into the wetting layer or into the GaAs barrier [5-8], as well as to thermally activated capture of excitons by nonradiative defects in the GaAs barrier or at the GaAs/InAs interface [5, 9, 10]. The unusual variations of emission energy and the full width at half maximum (FWHM) of PL bands in the InAs/GaAs self-assembled QDs have been investigated earlier as well [11, 12]. A decrease in the FWHM, together with a red shift of the emission wavelength were explained by the re-localization of carriers between dots caused by their inhomogeneous sizes. A set of theoretical works has been devoted to these questions, which described the carrier dynamics in QD systems using a rate equation model [13-19].

However, it is not still clear the details of exciton capture and thermal escape in high quality QD structures. Two main ways are discussed: a) thermal escape (capture) of carriers (electron and hole) in/from QDs takes place as an exciton or correlated electron-hole pairs [17, 18]; or b) the excitons dissociate and single electrons (holes) thermally escape or capture independently [19-21].

The first mentioned mechanism is the subsequence of the fact that the activation energy of ground state PL thermal decay measured in the regime of strong quenching matches the difference between the GaAs band gap and QD ground state (GS) energy (the sum of barrier heights for electrons and holes) [17-19]. The same effect was earlier revealed in strained InGaAs/GaAs QWs, where the activation energy of PL thermal decay corresponded to the difference between the GS energies of QW and the GaAs barrier, defined as total confinement energy, i.e. the sum of electron and hole potential depths. The main conclusion follows from this fact: excitons or electron-hole pairs are emitted from QWs into the barrier [22].

The motivation for the second mechanism was presented in [19-21]. It was shown in [19] at the InAs/GaAs QD PL investigation under low excitation and high temperature the ground state PL intensity varies quadratically with excitation density. This fact was explained as an evidence of independent occupation of QDs by electrons and holes with the probability which is proportional to the multiplication of their concentrations (nenp).

Other evidences for supporting the second approach were shown in [20] at the investigation of cathodoluminescence thermal quenching in InAs/GaAs self-assembled QDs at highexcitation conditions. The significant reductions in the thermal activation energies in the 230–300o K temperature range for the ground (GS) and excited states (ES) are found. It was suggested that excitons dissociate and the thermal reemission of single holes from QD states into the GaAs is responsible for the observed temperature dependence. The same explanation was supposed in [21] at the joint investigation of InAs QD and InGaAs QW PL thermal decays.

in optical fiber communications. These wavelengths are impossible to achieve in quantum well (QW) InAs/GaAs structures due to the strain-limited thicknesses. It is shown that the QD density in laser structures can be enlarged significantly by growing the dots within InxGa1-xAs/GaAs QWs, in so called dot–in-a-well (DWELL) structures [3, 4]. In these structures photoluminescence has been enhanced due to better crystal quality of the layer

A crucial aspect for the realization of efficient light-emitting devices operated at room temperature is the understanding of a temperature dependence of QD photoluminescence. PL intensity decay in InAs QDs as a rule attributed to thermal escape of carriers from QDs into the wetting layer or into the GaAs barrier [5-8], as well as to thermally activated capture of excitons by nonradiative defects in the GaAs barrier or at the GaAs/InAs interface [5, 9, 10]. The unusual variations of emission energy and the full width at half maximum (FWHM) of PL bands in the InAs/GaAs self-assembled QDs have been investigated earlier as well [11, 12]. A decrease in the FWHM, together with a red shift of the emission wavelength were explained by the re-localization of carriers between dots caused by their inhomogeneous sizes. A set of theoretical works has been devoted to these questions, which

surrounding QDs and more effective exciton capture into the QW and into QDs.