# Supporting information: Size-dependent lattice symmetry breaking determines
the exciton fine structure of perovskite nanocrystals

Daniel Weinberg, Yoonjae Park, David Limmer, Eran Rabani

##  1 Nanocrystal Structures

###  1.1 Unrelaxed Nanocrystal Structures

The cubic and orthorhombic nanocrystal (NC) structures were generated to
correspond with the literature bulk crystal structures[1]. A large slab of the
bulk material was generated and the NC was and cut from the bulk. The cut was
made along the 100 facets of the cubic structures and the 110 and 001 facets
in the orthorhombic structures. The cuts were made to ensure that all lead-
iodide octahedra were complete and the surface termination was only cesium and
iodine. The Cs atoms at the corners of the NCs were found to be particularly
unstable when relaxed, so these were removed for all structures. The cubic NCs
considered in Figure 3 are listed in Table S1 and follow the general formula
that for a cubic NC with $N$ Pb atoms along the each axis, there will be
$(N+1)^{3}-8$ Cs atoms, $N^{3}$ Pb atoms, and $3(N^{3}+N^{2})$ I atoms. The
additional cuboidal NCs considered in Figure 4 have $N$ Pb atoms along the
$x$\- and $y$-axes and $Z$ Pb atoms along the $z$-axis. The configurations are
listed in Table S2 and follow the general formula that there will be
$Z(N+1)^{2}-8$ Cs atoms, $ZN^{2}$ Pb atoms, and $3ZN^{2}+N^{2}+2ZN$ I atoms.

###  1.2 Relaxed Nanocrystal Structures

For each size of NC, we take the unrelaxed orthorhombic structure as an
initial configuration. However, since the corresponding structure doesn’t
satisfy the condition of charge neutrality, different partial charge is
assigned for surface Cs atoms to stabilize the NC structure and reduce the
effect from surface boundaries of NC. Using the force field parameters adopted
from Ref.[2] which are parameterized to reproduce the energy difference
between different crystal structures, the partial charge of surface Cs atoms,
$q_{\textrm{surf}}$, is defined as

$q_{\textrm{surf}}=-\frac{N_{\textrm{Pb}}\,q_{\textrm{Pb}}\,+\,N_{\textrm{I}}\,q_{\textrm{I}}\,+\,N_{\textrm{Cs}}\,q_{\textrm{Cs}}}{N_{\textrm{surf}}}$
(1)

where $q_{\alpha}$ is a partial charge of atom $\alpha$ listed in Table S1 and
$N_{\alpha}$ is the number of $\alpha$ atoms in each NC with
$\alpha\in\\{\textrm{Pb},\,\textrm{I},\,\textrm{Cs}\\}$. The subscript surf is
used to refer to surface Cs atoms whereas the subscript Cs indicates the core
Cs atoms. With the modified initial configuration, the structure of each NC
size is minimized using conjugate gradient algorithm based on the pairwise
interaction $u_{ij}$ between atom $i$ and $j$ described by Lennard-Jones
potential with Coulombic interaction

$u_{ij}(r)=\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r}+4\varepsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r}\right)^{12}-\left(\frac{\sigma_{ij}}{r}\right)^{6}\right]\,,\
\ r<r_{c}$ (2)

where $i,j\in\\{\textrm{Pb},\,\textrm{I},\,\textrm{Cs},\,\textrm{surf}\\}$,
$\varepsilon_{0}$ is the vacuum electric permittivity, and the cutoff distance
$r_{c}$ is set to be the maximum length of each NC among three different axis
times $\sqrt{3}$ to take care of the fact that periodic boundary condition
cannot be applied for NC. Lennard-Jones parameters $\varepsilon$ and $\sigma$
of each atom are listed in Table S1. Where parameters not listed can be
derived using following combining rules
$\varepsilon_{ij}=\sqrt{\varepsilon_{i}\varepsilon_{j}}$ and
$\sigma_{ij}=(\sigma_{i}+\sigma_{j})/2$. Minimizations are performed using the
LAMMPS package [3].

A DFT structural minimization was also performed on a 1.26 nm NC using the PBE
exchange correlation functional. The electronic calculation used a kinetic
energy cutoff of 65 Rydbergs and a charge density cutoff of 530 Rydbergs. The
structure was optimized to a force threshold of less than $10^{-}4$ atomic
units using Quantum Espresso[4, 5, 6]

##  2 Single Particle Electronic States

###  2.1 Pseudopotential Method

The ground state electronic calculations for a single phase were carried out
using the semi-empirical pseudopotential method [7, 8, 9, 10], which assigns
to each atom in the NC an effective potential derived from bulk band
structures. The electronic Hamiltonian is a single electron operator

$\hat{H}=\hat{T}+\sum_{\alpha}^{\text{atoms}}\left[\hat{V}_{loc}^{\alpha}+\hat{V}_{nonloc}^{\alpha}+\hat{V}_{SO}^{\alpha}\right]$
(3)

where $\hat{T}$ is the kinetic energy operator, $\hat{V}_{loc}^{\alpha}$ is
the local part of the pseudopotential around atom $\alpha$,
$\hat{V}_{nonloc}^{\alpha}$ describes angular momentum-dependent corrections
to the local pseudopotential around atom $\alpha$, and $\hat{V}_{SO}^{\alpha}$
describes the spin orbit coupling around atom $\alpha$. The local part of the
potential is defined in by a reciprocal space function

$\tilde{v}_{loc}^{\alpha}\left(q\right)=a_{0}^{\alpha}\frac{q-a_{1}^{\alpha}}{a_{2}^{\alpha}\exp\left(a_{3}^{\alpha}q^{2}\right)-1}$
(4)

where $q$ is the reciprocal coordinate, and the parameters
$a_{0}^{\alpha}\dots\,a_{3}^{\alpha}$ are fit based on the atom $\alpha$. The
potential is defined in terms of the position-space counterpart of
$\tilde{v}_{loc}^{\alpha}\left(q\right)$, which we call
$v_{loc}^{\alpha}\left(r\right)$. The local part of the potential is given by

$\hat{V}_{loc}^{\alpha}=v_{loc}^{\alpha}\left(\left|\hat{\bm{r}}-\bm{R}_{\alpha}\right|\right)$
(5)

where $\hat{\bm{r}}$ is the position operator, $\bm{R}_{\alpha}$ is the
position of atom $\alpha$. The angular momentum-dependent part of the
pseudopotential gives a correction to the local part of the pseudopotential
for the electrons in p-type orbitals.

$\hat{V}_{nonloc}^{\alpha}=\delta
v_{l=1}^{\alpha}\left(\left|\hat{\bm{r}}-\bm{R}_{\alpha}\right|\right)\hat{P}_{l=1}^{\alpha}=\left[a_{4}^{\alpha}\exp\left(-\left|\hat{\bm{r}}-\bm{R}_{\alpha}\right|^{2}\right)+a_{5}^{\alpha}\exp\left(-\left(\left|\hat{\bm{r}}-\bm{R}_{\alpha}\right|-\rho\right)^{2}\right)\right]\hat{P}_{l=1}^{\alpha}$
(6)

where $\hat{P}_{l=1}^{\alpha}$ is the projector onto the $l=1$ angular
momentum subspace around atom $\alpha$, $\rho$ is a shift of 1.5 Bohr, and the
$a_{4}^{\alpha}$ and $a_{5}^{\alpha}$ parameters are fit based on atom
$\alpha$. The spin-orbit coupling acts only on the p-type orbitals as well and
has the form

$\hat{V}_{SO}^{\alpha}=v_{SO}^{\alpha}\left(\left|\hat{\bm{r}}-\bm{R}_{\alpha}\right|\right)\hat{\bm{L}}^{\alpha}\cdot\hat{\bm{S}}\hat{P}_{l=1}^{\alpha}=a_{6}^{\alpha}\exp\left(\frac{-\left|\hat{\bm{r}}-\bm{R}_{\alpha}\right|^{2}}{w^{2}}\right)\hat{\bm{L}}^{\alpha}\cdot\hat{\bm{S}}\hat{P}_{l=1}^{\alpha}$
(7)

where $\hat{\bm{L}}^{\alpha}$ is the vector of electron orbital angular
momentum operators around atom $\alpha$, $\hat{\bm{S}}$ is the vector of
electron spin operators, $w$ is a width of 0.7 Bohr and $a_{6}^{\alpha}$ is
fit based on atom $\alpha$. The total potential can be rewritten as three
separate, spherically symmetric potentials felt by $s$-type (along with
$d$-type and higher angular momentum) orbitals, $p_{\frac{1}{2}}$-type
orbitals and $p_{\frac{3}{2}}$-type orbitals. The $s$-type orbitals feel the
local potential only:

$v_{s}\left(r\right)=v_{loc}^{\alpha}\left(r\right)$ (8)

The $p_{\frac{1}{2}}$-type orbitals feel a combination of the angular momentum
dependent potential and the spin-orbit potential

$v_{p_{\frac{1}{2}}}\left(r\right)=v_{loc}^{\alpha}\left(r\right)+\delta
v_{l=1}^{\alpha}\left(r\right)-v_{SO}^{\alpha}\left(r\right)$ (9)

while the $p_{\frac{3}{2}}$-type orbitals feel a different combination

$v_{p_{\frac{3}{2}}}\left(r\right)=v_{loc}^{\alpha}\left(r\right)+\delta
v_{l=1}^{\alpha}\left(r\right)+\frac{1}{2}v_{SO}^{\alpha}\left(r\right)$ (10)

The single particle Hamiltonian was solved via the filter diagonalization
method [11, 12] on a real-space grid with a grid spacing of 0.5 Bohr. This
finer grid spacing was used to ensure sufficient convergence of the non-local
parts of the Hamiltonian. The non-local operators were implemented via a
modified Kleinman-Bylander representation[13, 14, 15]. On the order of a few
hundred states in the energy range near the band gap were converged.

###  2.2 Parameterization of Pseudopotentials

The pseudopotential parameters were fit for each atom in a particular crystal
structure in order to reproduce bulk band structures. For the perovskite
system we are investigating here, we generated best fit parameter sets for
both the cubic and the orthorhombic crystal structures. The best fit was
determined by using the pseudopotential Hamiltonian within a converged plane-
wave basis to generate the bulk band structures of the respective phases, with
care taken to properly describe the non-local and spin-orbit interactions [16,
8, 17, 18]. These pseudopotential band structures were then compared to
literature GW band structures [1]. In order to better compare with
experimental results, the band gap of the GW calculations for the cubic
structure, which differed significantly from measurements of the bulk band
gap, were corrected by a static shift of the valence band. The GW calculations
of the orthorhombic band structure agreed with experiment and were used
without modification. The valence band offsets between the cubic and
orthorhombic phase were calculated from DFT using Quantum Espresso [4, 5].

The seven pseudopotential parameters per atom were fit for each phase using a
Monte-Carlo fitting procedure where the objective function emphasized the
closeness of fit around the band gap as well as the effective mass of the
bands at the gaps. The parameter space was extensively searched to find the
best parameters. Initial fitting showed that contribution from the Cs ion was
nearly zero, consistent with the understanding across lead halide perovskite
materials that the valence and conduction bands are composed mainly of lead
and halide orbitals [19]. Thus the potential around the Cs atoms was set to
zero and the fits were further refined considering only the lead and iodine
parameters.

The results of the fitting are shown in Figure S2. The orthorhombic band
structure generated from our pseudopotentials has a hole effective mass of
0.320 $m_{e}$ and an electron effective mass of 0.386 $m_{e}$. The cubic band
structure generated from our pseudopotentials has a hole effective mass of
0.289 $m_{e}$ and an electron effective mass of 0.309 $m_{e}$. These slightly
overestimate the masses of the literature structures, leading to some small
additional confinement effects in the NCs.

###  2.3 Interpolation of Pseudopotentials for Relaxed Structures

For relaxed nanocrystal structures, there is no bulk phase that perfectly
matches the nanocrystal structure. This requires that the pseudopotential
around each atom adapt to the local structure. We do this by linearly
interpolating between the pseudopotentials for the cubic and orthorhombic
structures. The relaxed nanocrystal Hamiltonian is then

$\hat{H}=\hat{T}+\sum_{\alpha}^{\text{atoms}}\left(x_{\alpha}\right)\left[\hat{V}_{loc}^{\alpha,\text{ortho}}+\hat{V}_{nonloc}^{\alpha,\text{ortho}}+\hat{V}_{SO}^{\alpha,\text{ortho}}\right]+\left(1-x_{\alpha}\right)\left[\hat{V}_{loc}^{\alpha,\text{cubic}}+\hat{V}_{nonloc}^{\alpha,\text{cubic}}+\hat{V}_{SO}^{\alpha,\text{cubic}}\right]$
(11)

where $x_{\alpha}\in[0,1]$ denotes the extent of orthorhombic distortion. For
iodine atoms bonded to two lead atoms this is determined by the Pb-I-Pb bond
angle, $\theta_{\alpha}$. For the cubic phase that bond is straight,
$\theta=180^{\circ}$ while for the orthorhombic phase there are two bond
angles of $\theta_{\alpha}=150.8^{\circ}$ and $\theta_{\alpha}=160.6^{\circ}$.
Thus the orthorhombic distortion was calculated as

$x_{\alpha}=\frac{180-\theta_{\alpha}}{180-106.6}$ (12)

where if the angle was less than 106.6 degrees the atom was assigned a fully
orthorhombic pseudopotential. Iodine atoms not bonded to two lead atoms
(dangling iodine at the surface) were assigned the cubic pseudopotential. Lead
atoms were assigned the average of the distortion parameters of bonded iodine
atoms. It is important to note that for the local part of the pseudopotential,
$\hat{V}_{loc}$, a linear interpolation between the orthorhombic and cubic
operators does not mean a linear interpolation of the parameters listed in
Table S4.

The linear interpolation was deemed reasonable based on the closeness of the
pseudopotentials for the cubic and orthorhombic phases, as shown in Figure S3.
To ensure that this interpolation did not cause significant issues with with
electronic structure, band structures were calculated using unit cells with
distortions between those of the cubic and orthorhombic structures. The
energies of the valence and conduction band edges from these calculations are
shown in Figure S4 as the structure is distorted from the cubic to the
orthorhombic form. Because the orthorhombic structure differs from the cubic
structure only by slight rotations of the lead-iodide octahedra it was simple
to construct such structures and calculate their band structures both using
the interpolated pseudopotentials and DFT, again within the Quantum Espresso
[4, 5] package. The DFT results are known to significantly underestimate the
band gap of the cubic structure, but our pseudopotential method, being trained
on the corrected band structures, is able to match much better to experimental
results at the end-points. The interpolated pseudopotentials generate non-
monotonic trend in the band gap with overall a slight decrease in the band gap
over the sequence of structures from cubic to orthorhombic, while the DFT
shows a monotonic increase. However, the DFT increase seems to stem only from
the previously mentioned underestimation of the band gap in the cubic phase.
Without resorting to extremely computationally costly techniques like GW, we
are satisfied that the pseudopotentials are able to smoothly interpolate
between the cubic and orthorhombic phases without issue.

##  3 Excitonic States

The excitonic states were calculated using the Bethe-Salpeter equation (BSE)
within the Tamm-Dancoff approximation, which writes the excitonic states as
linear combinations of non-interacting electron-hole pair states [20]. The
$n$th excitonic state, $\left|\psi_{n}\right\rangle$, is written as

$\left|\psi_{n}\right\rangle=\sum_{ai}c_{a,i}^{n}\left|a,i\right\rangle$ (13)

where the indices $a,b,c,\dots$ refer to electron (unoccupied) states, and the
indices $i,j,k,\dots$ refer to hole (occupied) states,
$\left|a,i\right\rangle$ refers to the non-interacting pair state, and the
expansion coefficients, $c_{a,i}^{n}$, are determined by the eigenvalue
equation

$\left(E_{n}-\Delta\varepsilon_{a,i}\right)c_{a,i}^{n}=\sum_{bj}\left(K_{ai;bj}^{d}+K_{ai;bj}^{x}\right)c_{b,j}^{n}$
(14)

which also determines the energy of the $n$th excitonic state $E_{n}$. The
electron hole interaction kernel, which describes the binding of the
independent electron-hole states into correlated excitonic states, has two
parts: the direct interaction $K^{d}$ describes the coloumb attraction between
the electron-hole pair, while the exchange interaction $K^{x}$ controls the
details of the excitonic spectrum, crucially including the singlet triplet
splitting under consideration here. The direct interaction is calculated using
a screened coulomb interaction
$W\left(\left|\bm{r}-\bm{r^{\prime}}\right|\right)$ within a static screening
approximation with

$K_{ai;bj}^{d}=-\int
d\bm{x}d\bm{x^{\prime}}\phi_{a}^{*}\left(\bm{x}\right)\phi_{j}^{*}\left(\bm{x^{\prime}}\right)W\left(\left|\bm{r}-\bm{r^{\prime}}\right|\right)\phi_{i}\left(\bm{x^{\prime}}\right)\phi_{b}\left(\bm{x}\right)$
(15)

The static dielectric constant used in the calculation is $\epsilon=6.1$[21].
The exchange interaction is calculated with the bare coulomb interaction
$v\left(\left|\bm{r}-\bm{r^{\prime}}\right|\right)$ to be

$K_{ai;bj}^{x}=\int
d\bm{x}d\bm{x^{\prime}}\phi_{a}^{*}\left(\bm{x}\right)\phi_{j}^{*}\left(\bm{x^{\prime}}\right)v\left(\left|\bm{r}-\bm{r^{\prime}}\right|\right)\phi_{b}\left(\bm{x^{\prime}}\right)\phi_{i}\left(\bm{x}\right)$
(16)

The interaction kernel matrices are solved in the basis of band-edge states.
The binding energy of the $n$th excitonic state, $E^{n}_{\text{B}}$, is
calculated as

$E^{n}_{\text{B}}=\left\langle\hat{K}^{d}+\hat{K}^{x}\right\rangle_{n}=\sum_{abij}\left(c^{n}_{a,i}\right)^{*}\left(K^{d}_{ai;bj}+K^{x}_{ai;bj}\right)c^{n}_{b,j}.$
(17)

Generally 60-80 electron states and a similar number of hole states were
selected to form the basis for solving the BSE, which were sufficient to
converge the excitonic fine structure for the low energy states under
consideration here.

The effect of the exciton spin on the exchange interaction is the key factor
in energetically separating spin singlet and triplet states. The exciton spin
operator is the sum of the electron and hole spin operators,

$\hat{\bm{S}}_{\textrm{tot}}=\hat{\bm{S}}_{\textrm{e}}+\hat{\bm{S}}_{\textrm{h}}$
(18)

The total spin of an excitonic state can be calculated as,

$\displaystyle\left\langle\psi_{n}\right|\hat{S}_{\textrm{tot}}^{2}\left|\psi_{n}\right\rangle$
$\displaystyle=\left\langle\psi_{n}\right|\hat{S}_{\textrm{e}}^{2}+\hat{S}_{h}^{2}+2\hat{\bm{S}}_{\textrm{e}}\cdot\hat{\bm{S}}_{\textrm{h}}\left|\psi_{n}\right\rangle$
(19)
$\displaystyle=\sum_{ai;bj}c_{bj}^{n}\left(c_{ai}^{n}\right)^{*}\left\langle
a,i\right|\left(\hat{S}_{\textrm{e}+}^{2}+\hat{S}_{\textrm{h}}^{2}+2\hat{\bm{S}}_{\textrm{e}}\cdot\hat{\bm{S}}_{\textrm{h}}\right)\left|b,j\right\rangle$
(20)
$\displaystyle=\sum_{ai;bj}c_{bj}^{n}\left(c_{ai}^{n}\right)^{*}\left(\left\langle
a\right|\hat{S}_{\textrm{e}}^{2}\left|b\right\rangle\delta_{ij}+\left\langle
i\right|\hat{S}_{\textrm{h}}^{2}\left|j\right\rangle\delta_{ab}+2\left\langle
a\right|\hat{\bm{S}}_{\textrm{e}}\left|b\right\rangle\cdot\left\langle
i\right|\hat{\bm{S}}_{\textrm{h}}\left|j\right\rangle\right)$ (21)
$\displaystyle=\sum_{ai;bj}c_{bj}^{n}\left(c_{ai}^{n}\right)^{*}\left(3/4\delta_{ab}\delta_{ij}+3/4\delta_{ij}\delta_{ab}+2\left\langle
a\right|\hat{\bm{S}}_{\textrm{e}}\left|b\right\rangle\cdot\left\langle
i\right|\hat{\bm{S}}_{\textrm{h}}\left|j\right\rangle\right)$ (22)
$\displaystyle=3/2+2\sum_{ai;bj}c_{bj}^{n}\left(c_{ai}^{n}\right)^{*}\left\langle
a\right|\hat{\bm{S}}_{\textrm{e}}\left|b\right\rangle\cdot\left\langle
i\right|\hat{\bm{S}}_{\textrm{h}}\left|j\right\rangle$ (23)

Importantly, the spin of the hole states must be treated as that of time-
reversed electronic states, meaning that while the electron spin matrix
elements are simply,

$\left\langle
a\right|\hat{\bm{S}}_{\textrm{e}}\left|b\right\rangle=\frac{1}{2}\left\langle
a\right|\hat{\bm{\sigma}}\left|b\right\rangle$ (24)

the hole spin matrix elements are given by

$\displaystyle\left\langle
i\right|\hat{\bm{S}}_{\textrm{h}}\left|j\right\rangle$
$\displaystyle=\frac{1}{2}\left[\left\langle
i\right|\hat{\Theta}^{\dagger}\right]\hat{\bm{\sigma}}\left[\hat{\Theta}\left|j\right\rangle\right]$
(25) $\displaystyle=-\frac{1}{2}\left\langle
j\right|\hat{\bm{\sigma}}\left|i\right\rangle$ (26)

where $\hat{\bm{\sigma}}$ is the vector of Pauli spin matrices.

Simulated spectra were calculated from the oscillator strengths of the
transition from each excitonic state to the ground state.

$f_{(x,y,z)}\left(E\right)=\sum_{n}\delta\left(E-E_{n}\right)\sum_{ai}c_{a,i}^{n}\left|\left\langle
i\right|\hat{\mu}_{(x,y,z)}\left|a\right\rangle\right|^{2}$ (27)

where $c_{a,i}^{n}$ are the BSE expansion coefficients for electron state
$\left|a\right\rangle$ and hole state $\left|i\right\rangle$ into excitonic
state $n$ with energy $E_{n}$. The dipole operator $\hat{\mu}_{x,y,z}$ is
broken down along the three principal axes as shown in color in Figure S7, and
the average of the three polarizations is used for the total spectrum shown as
the black line in Figure S7.

Configuration | $N$ |  $L$ (nm)
---|---|---
Cs_19Pb_8I_36 | 2 | 1.26
Cs_56Pb_27I_108 | 3 | 1.89
Cs_117Pb_64I_240 | 4 | 2.52
Cs_208Pb_125I_450 | 5 | 3.14
Cs_335Pb_216I_756 | 6 | 3.77
Cs_504Pb_343I_1176 | 7 | 4.40
Cs_721Pb_512I_1728 | 8 | 5.03
Cs_992Pb_729I_2430 | 9 | 5.66
Table S1: Details of the cubic NC configurations Configuration | $N$ | $Z$ |  $L$ (nm) |  $L_{z}$ (nm) | aspect ratio
---|---|---|---|---|---
Cs_92Pb_48I_184 | 4 | 3 | 2.52 | 1.89 | 0.75
Cs_142Pb_80I_296 | 4 | 5 | 2.52 | 3.14 | 1.25
Cs_167Pb_96I_352 | 4 | 6 | 2.52 | 3.77 | 1.5
Cs_192Pb_112I_408 | 4 | 7 | 2.52 | 4.40 | 1.5
Cs_217Pb_128I_464 | 4 | 8 | 2.52 | 5.03 | 2.0
Cs_136Pb_75I_280 | 5 | 3 | 3.14 | 1.89 | 0.6
Cs_172Pb_100I_365 | 5 | 4 | 3.14 | 2.52 | 0.8
Cs_244Pb_150I_535 | 5 | 6 | 3.14 | 3.77 | 1.2
Cs_280Pb_175I_620 | 5 | 7 | 3.14 | 4.40 | 1.4
Cs_316Pb_200I_705 | 5 | 8 | 3.14 | 5.03 | 1.6
Cs_188Pb_108I_396 | 6 | 3 | 3.77 | 1.89 | 0.5
Cs_237Pb_144I_516 | 6 | 4 | 3.77 | 2.52 | 0.67
Cs_286Pb_180I_636 | 6 | 5 | 3.77 | 3.14 | 0.83
Cs_284Pb_252I_876 | 6 | 7 | 3.77 | 4.40 | 1.17
Cs_433Pb_288I_996 | 6 | 8 | 3.77 | 5.03 | 1.33
Table S2: Details of the cubiodal NC configuration Figure S1: Several sizes of cubic (top) and orthorhombic (bottom) NCs | Pb | I | Cs | Cs (Surface)
---|---|---|---|---
$q$ (e) | 0.85 | -0.57 | 0.86 | $q_{\text{surf}}$
$\sigma$ (Angstrom) | 3.210 | 4.014 | 3.584 | 3.584
$\varepsilon$ (eV) | 0.001086 | 0.06389 | 0.07728 | 0.07728
Table S3: Force field parameters from Ref. [2]

Figure S2: The band structures generated by the pseudopotential method (black
lines) and the literature band structures (red points).

Iodine | Orthorhombic | Cubic
---|---|---
$a_{0}$ | 117.89162750 | 113.75644925
$a_{1}$ | 2.12591587 | 2.14135468
$a_{2}$ | 2.91148249 | 2.84227140
$a_{3}$ | 0.58243028 | 0.56706414
$a_{4}$ | 0.16631532 | 0.04515637
$a_{5}$ | -0.02486412 | 0.00220169
$a_{6}$ | 2.23037552 | 3.03441874

Lead | Orthorhombic | Cubic
---|---|---
$a_{0}$ | 97.86083166 | 88.83030750
$a_{1}$ | 2.25710305 | 2.60529157
$a_{2}$ | 3.71951773 | 4.28692327
$a_{3}$ | 0.55872538 | 0.51950186
$a_{4}$ | 1.40412238 | 1.06405744
$a_{5}$ | -0.01566674 | -0.00808029
$a_{6}$ | 7.94721141 | 7.78990812

Table S4: Best pseudopotential parameters, in Hartree atomic units.

Figure S3: Pseudopotentials for iodine and lead atoms in the cubic (dotted
lines) and orthorhombic (dashed lines).  Figure S4: Comparison of DFT (blue)
and pseudopotential (red) band edges in bulk structures interpolated between
the cubic and orthorhombic phases. Figure S5: The total spin expectation value
(left) and he exchange energy as a fraction of the total exciton binding
energy (right) for the lowest dark (black) and bright (red) excitonic states
as a function of size.

Figure S6: The lowest bright (red) and dark (black) excitonic states for NCs
with orthorhombic and cubic structures as a function of size along with
experimental data [22, 23, 24, 25, 26].
Figure S7: Simulated absorption spectrum for a 3.8 nm NC with various crystal structures with the lowest excitonic state set to the energy zero. The polarized spectra along the z (green), y (blue) and x (red) are also shown with and offset for clarity.  NC Size (nm) | Excitation Energy (eV) | Ref.
---|---|---
$6.5$ | $1.849$ | [22]
$6.5\pm 1.1$ | $1.907$ | [23]
$10.8\pm 1.5$ | $1.864$ | [23]
$4.70$ | $1.83$ | [24]
$5.38$ | $1.80$ | [24]
$7.33$ | $1.79$ | [24]
$3.4$ | $2.07$ | [25]
$4.5$ | $2.01$ | [25]
$5.0$ | $1.95$ | [25]
$6.8$ | $1.91$ | [25]
$8.0$ | $1.88$ | [25]
$9.0$ | $1.86$ | [25]
$9.0\pm 1$ | $1.83$ | [26]
Table S5: Experimentally measured excitation energies NC Size (nm) | Splitting (meV) | Ref.
---|---|---
$4.9\pm 0.2$ | 1.61 | [27]
$5.4\pm 0.3$ | 1.4 | [27]
$6.2\pm 0.3$ | 1.25 | [27]
$6.7\pm 0.45$ | 1.20 | [27]
$7.4\pm 0.6$ | 0.85 | [27]
$7.9\pm 0.7$ | 0.70 | [27]
$9.3$ | 0.40 | [28]
$4.$ | 7.5 | [29]
$6.5$ | 2. | [29]
$10.$ | 0.40 | [30]
Table S6: Experimentally measured bright-bright splittings

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