Eﬀective Land e factors of electrons and holes in lead chalcogenide nanocrystals I.D. Avdeev1S.V. Goupalov1 2and M.O. Nestoklon1 1Ioﬀe Institute 194021 St. Petersburg Russia

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Eﬀective Land´e factors of electrons and holes in lead chalcogenide nanocrystals

I.D. Avdeev,1S.V. Goupalov,1, 2, ∗and M.O. Nestoklon1

1Ioﬀe Institute, 194021 St. Petersburg, Russia

2Department of Physics, Jackson State University, Jackson MS 39217, USA

The Land´e or g-factors of charge carriers in solid state systems provide invaluable information

about response of quantum states to external magnetic ﬁelds and are key ingredients in descrip-

tion of spin-dependent phenomena in nanostructures. We report on the comprehensive theoretical

analysis of electron and hole g-factors in lead chalcogenide nanocrystals. By combining symmetry

analysis, atomistic calculations, and extended k·ptheory, we relate calculated linear-in-magnetic

ﬁeld energy splittings of conﬁned electron states in nanocrystals to the intravalley g-factors of the

multi-valley bulk materials, renormalized due to the quantum conﬁnement. We demonstrate that

this renormalization is correctly reproduced by analytical expressions derived in the framework of

the extended k·pmodel.

Introduction. Lead salts nanocrystals (NCs) are en-

joying many practical applications in optoelectronics and

photovoltaics [1–5]. New devices built on NCs are pre-

dicted to enter the market in the nearest future [5–8]. All

these devices are based on the emission or absorption of

light by spatially conﬁned electron-hole pairs.

Applications of NCs in rapidly developing ﬁelds of

spintronics and quantum computing [9–12] would be im-

possible without control over the spin state of local-

ized carriers. Therefore, knowledge about carrier spin

relaxation and dynamics as well as their Land´e gfac-

tors becomes critically important. These properties

have been widely studied for CdSe NCs. The exciton

ﬁne structure relaxation dynamics was investigated in

Refs. 13–15, electron and exciton g-factors were mea-

sured, respectively, by the time-resolved Faraday rota-

tion [16–18] and single-dot magneto-photoluminescence

spectroscopy [19,20], and carrier g-factors were calcu-

lated within tight-binding [21–23] and eﬀective mass [24]

methods.

In the mean time, analogous studies for lead salts NCs

remain very scarce. Ultrafast exciton ﬁne structure relax-

ation dynamics was studied by Johnson et al. [25] Schaller

et al. measured averaged exciton g-factor in an ensem-

ble of PbSe NCs in magnetic-circular dichroism experi-

ments [26]. Turyanska et al. deduced exciton g-factors

of PbS NCs from magnetic ﬁeld dependences of photolu-

minescence circular polarization degree [27]. Single-NC

spectroscopy in external magnetic ﬁelds was performed

by Kim et al [28].

Yet, interpretation of these results is complicated by

the multi-valley band structure of lead salts compounds.

Bulk lead salts have extrema of the conduction and va-

lence bands at the four inequivalent L-points of the Bril-

louin zone. The widely used k·ptheory [31] treats these

L-valleys independently. An external magnetic ﬁeld leads

to the Zeeman splittings of the electron and hole states

characterized by certain magnetic quantum numbers in-

timately related to the spin degrees of freedom. Then the

main eﬀects of the quantum conﬁnement are renormaliza-

tion of the Zeeman splittings and their sensitivity to ori-

entation of the magnetic ﬁeld, which result in the renor-

malization and anisotropy of the carriers’ g-factors [32].

This kind of narrative is typical for nanostructures of II -

VI and III - V compound semiconductors with band ex-

trema at the Γ point of the Brillouin zone, and is adopted

by the conventional, or independent-valley, k·ptheory

developed for lead salt nanostructures [31]. This theory

is formulated in terms of the longitudinal and transverse

single-valley g-factors (Figure 1, (a)).

However, in lead salts nanostructures, due to the inter-

valley scattering on the surface, the zero-ﬁeld electron or

hole states represent combinations of the states originat-

ing from diﬀerent L-valleys. Thus, all atomistic band

structure calculations, basing on the symmetries of the

underlying crystal lattice and overall structure, automat-

ically take into account this alignment of the valley de-

grees of freedom [33,34]. The resulting zero-ﬁeld states

are classiﬁed with respect to irreducible representations

of the symmetry group. Application of the external mag-

netic ﬁeld further aﬀects the spin degrees of freedom, but

this narrative implies completely diﬀerent meaning and

deﬁnition of the magnetic quantum numbers as compared

to the single-valley case. Since, at weak magnetic ﬁelds,

states characterized by diﬀerent irreducible representa-

tions do not mix, the atomistic theories operate with the

g-factors associated with the corresponding irreducible

representations (Figure 1, (c)).

In this work we show that a solution to this ambiguity

comes from a symmetry-based construction of a transfor-

mation relating the basis of independent valley states and

the basis of valley combinations associated with certain

irreducible representations of the point group, as illus-

trated in Figure 1. This allows one to relate both kinds

of the g-factors and use a fusion of the two approaches to

get insight about conﬁnement eﬀect on carriers’ g-factors

in lead salts NCs.

Results and discussion. In PbX (X=S, Se) NCs with

cubic symmetry (point group Tdor Oh) the ground state

of conﬁned electron or hole splits into two doublets,

transforming under irreducible representations Γ6,Γ7

(Γ±

6,7) of group Td(Oh), and a quadruplet Γ8(Γ±

8) sepa-

arXiv:2210.13340v1 [cond-mat.mes-hall] 24 Oct 2022

FIG. 1. (a) Conduction and valence band extrema in bulk lead chalcogenide compounds occur at four ineqivalent L-points of the

Brillouin zone. Energy isosurfaces near these points form anisotropic valleys (red “cigars”). For valley states of charge carriers,

Zeeman splittings depend on the orientation of the magnetic ﬁeld with respect to the valley main axes. One can distinguish

longitudinal (Bl) and transverse (Bt) components of the magnetic ﬁeld in a given valley which deﬁne the longitudinal (gl)

and transverse (gt)g-factors. (b) Within the k·ptheory, the “spin” (total angular momentum) projections Fz=±1/2 (blue

arrows) are deﬁned in the valley coordinate frames adjacent to the valley wave vectors kν(red arrows). These vectors form an

irreducible star {k0}of some representation of the crystal space group [29]. One can analyze their transformation properties

[30] and form combinations of valley states transforming under irreducible representations of the NC point group. This allows

one to establish a relationship between the single-valley states described by the k·ptheory and multivalley combinations of

states, enforced by the symmetry and following from atomistic calculations of the NC band structure. (c) Zero-ﬁeld states of

charge carriers in NCs represent combinations of valley states transforming with respect to certain irreducible representations

of the symmetry group. The ground electron (or hole) level in a NC with no inversion center splits into two doublets and a

quadruplet transforming with respect to the Γ6, Γ7, and Γ8representations of the group Td. At low magnetic ﬁelds, the Zeeman

splittings are isotropic and determined by the eﬀective g-factors g6,g7,g8, and g0

rated by several meV as a result of valley mixing [33–35].

In the subspace of these states, interaction with a weak

magnetic ﬁeld Bis described by the following eﬀective

Hamiltonian, written as a block-diagonal matrix:

Hη

1(B) = µBB



2gη

6σ0 0

2gη

7σ0

0 0 gη

8J+g0η

8J0

,(1)

where η=c(v) for the conduction (valence) band states;

gη

6and gη

7are, respectively, the eﬀective g-factors of the

Γ6and Γ7doublets; gη

8and g0η

8are the two constants

describing Zeeman splitting of the quadruplet Γ8;σ=

(σx, σy, σz) are the Pauli matrices; J= (Jx, Jy, Jz) are

the matrices of the angular momentum j= 3/2,[30]µB

is the Bohr magneton, and the matrices J0are deﬁned

as[36]

γ=5

3J3

γ−41

12Jγ(2)

(γ=x, y, z).

In a strong magnetic ﬁeld, when µBBis compatible

with the valley splittings |EΓ7−EΓ8|and |EΓ8−EΓ6|,

two additional non-diagonal linear-in-Bterms should be

taken into account. They describe interaction of the

quadruplet Γ8with the doublets Γ6and Γ7and are dis-

cussed in Supplemental Material [30].

Tight-binding calculations. The eﬀective g-factors en-

tering equation (1) can be extracted from the tight-

binding calculations [30]. They are shown in Figure 2

for quasi-spherical PbS and PbSe NCs (see Supplemen-

tal Material [30] for deﬁnition of quai-spherical NCs).

The actual shapes of colloidal NCs can vary from cube

to truncated cube to cuboctahedron to truncated octa-

hedron to octahedron, depending on the synthesis condi-

tions [37,38]. Tight-binding calculations performed for

NCs of cubic, cuboctahedral, and octahedral shapes show

that the g-factors are almost shape-independent, in con-

trast to the zero-ﬁeld splittings of electron and hole levels

exhibiting strong dependencies on the NC shape [35] (see

Supplemetal Material [30] for energies of the Γ6,Γ7, and

Γ8levels in PbSe NCs. These for PbS NCs are given in

Ref. 35.) By dashed (solid) lines in Figure 2we show

the results of the isotropic (anisotropic) k·pmodel to be

discussed later.

Land´e factors in valleys. Before proceeding to the

results of the k·pmodel, we ﬁrst discuss the origin of

the parameters entering Eq. (1) in terms of anisotropic

g-factors describing linear magnetic ﬁeld dependence of

the phenomenological single-valley eﬀective Hamiltoni-

ans. We represent the Hamiltonian of the conﬁned

conduction- or valence-band ground state as a summa-

tion of the single-valley Hamiltonians over the valley in-

dex ν:

Hη=µB

[gη

t(σxBx,ν +σyBy,ν ) + gη

lσzBz,ν ],(3)

where the Pauli matrices σγ(γ=x, y, z) are deﬁned

in the coordinate frames of the corresponding valleys

FIG. 2. Calculated g-factors g6, g7, g8and g0

8(1) in conduc-

tion (top panel) and valence (second panel) bands of quasi-

spherical PbS NCs (ﬁlled red, blue, cyan, and empty purple

circles, respectively). The results are stable with respect to

NC shape variation, see Supplemental Material [30]. Solid

(dashed) lines show outcomes of the anisotropic (isotropic)

k·pmodel. Two lower panels show results for PbSe NCs.

(with the zaxis aligned along the valley C3axis) and

B= (Bx,ν , By,ν , Bz,ν ) is the magnetic ﬁeld written in

the same “local” basis. In particular, for the L0valley,

we choose the local basis as follows:

nx,0k[1¯

10] ,ny,0k[11¯

2] ,nz,0k[111] .(4)

The bases of the other valleys (ν= 1,2,3) are related via

C2rotations around the crystallographic axes of the “lab-

oratory” frame (xk[100], yk[010], zk[001]): 0 →1 via

C2z, 0 →2 via C2x, and 0 →3 via C2y(cf. Figure 1, (b)).

The Hamiltonian (3) can be transformed into the basis

of irreducible representations using an appropriate trans-

formation matrix [30], From a comparison of the trans-

formed Hamiltonian with Eq. (1) we obtain the follow-

ing set of the g-factors for the conﬁned conduction- and

valence-band states:

6=gc

l−2gc

3, gc

7=gc

l+ 2gc

3,(5a)

8=gc

l+ 4gc

15 , g0c

8= 2gc

l−gc

15 ,(5b)

6=gv

l+ 2gv

3, gv

7=gv

l−2gv

3,(5c)

8=gv

l−4gv

15 , g0v

8= 2gv

l+gv

15 .(5d)

Equations (5) can be inverted to extract the values of

gη

l(t)from the g-factors of the quantum conﬁned states in

a NC. We will use this procedure to obtain the eﬀective

g-factors gc(v)

l(t)from the tight-binding results presented in

Figure 2. One may notice that there are four independent

constants for each band in the eﬀective Hamiltonian (1)

but only two independent constants entering (3). There-

fore, we have some freedom in the choice of the extraction

procedure. In the present study, we will determine the

longitudinal and transverse g-factors as

gη

l= 3gη

8+ 6g0η

8,(6a)

t= 3gc

8−3

2g0c

8, gv

t=−3gv

8+3

2g0v

8.(6b)

The longitudinal and transverse g-factors obtained in this

manner from the tight-binding calculations are presented

in Figure 3. Also shown in Figure 3are results of the

single-valley k·pmodel.

As the numbers of independent constants in equa-

tions (1) and (3) are diﬀerent, the above procedure in-

troduces some error. In order to estimate it, in Supple-

mental Material [30] we compare the diﬀerence between

gη

6and gη

7calculated directly in the tight-binding with

the results of Eqs. (5a,5c). The diﬀerences are correlated

with the valley splittings and do not exceed 10% of the

values of the g-factors in the bulk.

k·pmodel. It is well known [32] that quantum con-

ﬁnement renormalizes electron g-factor in semiconduc-

tor nanostructures with respect to its bulk counterpart.

We will demonstrate the results of such renormalization

using the single-valley k·pmodel of Ref. 31 (see also

Refs. 39,40). In the isotropic approximation, the ef-

fective Hamiltonian of this model can be written as

Hiso =

Eg

2−αc∆−i¯hP

m0(σ∇)

−i¯hP

m0(σ∇)−Eg

2−αv∆

,(7)

where Egis the band gap, Pis the inter-band momen-

tum matrix element, m0is the free electron mass, the

coeﬃcients αc(v)stem from the contributions of the re-

mote bands to the conduction and valence bands energy

dispersion, and ∆ is the three-dimensional Laplace oper-

ator.

Within the isotropic model, the electron states can be

characterized by the value of the total angular momen-

tum Fand an additional quantum number prelated to

FIG. 3. The values of transverse (red circles) and longitu-

dinal (blue circles) Lvalley g-factors in conduction (positive

for large D) and valence (negative for large D) bands in PbS

(upper panel) and PbSe (lower panel) NCs extracted from the

tight-binding calculations using Eq. (6). Red and blue lines

show, respectively, longitudinal and transverse g-factors com-

puted within anisotropic (solid lines) and isotropic (dashed

lines) k·ptheory with parameters extracted from the tight-

binding model, Table I.

parity of the states. The dispersion equation for these

states is given in Supplemental Material [30]. The ground

electron states are characterized by the angular momen-

tum F= 1/2 and p= +1 (odd parity), the ground hole

states have F= 1/2 and p=−1 (even parity). All

the conﬁned states are (2F+ 1)-fold degenerate with re-

spect to the projection Fzof the total angular momen-

tum. Their wave functions written in the bispinor form

are

|F, p, n, Fzi=





fF−p

2,p r

Rˆ

ΩF−p

F,Fz

ip gF+p

2,p r

Rˆ

ΩF+p

F,Fz





,(8)

where ˆ

Ω`

F,Fzare the spherical spinors [41] and f`p,g`p

are the normalized radial functions [30].

To compute the g-factors we follow Ref. 32 and add to

the Hamiltonian (7) the following term

δH =e

2c(Av +vA)

=eB

2c



−2iαcm0

¯h2r×∇P

¯h(r×σ)

¯h(r×σ)2iαvm0

¯h2r×∇

,(9)

where vis the velocity operator [40]. We used the sym-

metric gauge A=B×r

2for the vector potential Aalong

with the additional electron and hole eﬀective g-factor

tensors with the non-zero components gη

0xx =gη

0yy =gη

0t,

gη

0zz =gη

0l, responsible for the contributions of remote

bands, to yield

Hiso(B)

=µBB



2ˆgc

0σ+2m0

¯h2αcLP

¯h(r×σ)

¯h(r×σ)1

2ˆgv

0σ−2m0

¯h2αvL

,(10)

where L=−ir×∇and Bˆgη

0σ=P

α,β

Bαgη

0αβσβ=

Bαgη

0αασα.

We use the variables gη

0t,gη

0las adjustable parameters

to reproduce the g-factors of the bulk PbS and PbSe, cal-

culated in the tight-binding model (see e.g. Supplemen-

tary information of Ref. 42), from the following relations

gη

l(t),bulk =gη

0l(t)±4P2

Egm0

,(11)

where the sign of the second term is positive for η=c

and negative for η=v. The values gη

l(t),bulk and Pare

given in Table I. Note that in Ref. 43 contributions of

remote bands to the g-factors were omitted, as the second

term in Eqs. (11) (13.1 for PbS and 33.4 for PbSe, by the

absolute value) prevails in determining the bulk g-factors.

However, in NCs, the g-factors are renormalized (in the

ﬁrst order, as a result of the quantum conﬁnement energy

being added to the band gap) and the contributions of

remote bands become important.

The matrix elements of the Hamiltonian (10) between

conduction (valence) band states |c, Fzi ≡ 1

2,+1,0; Fz

(|v, Fzi ≡ 1

2,−1,0; Fz), Eq. (8), can be calculated ex-

plicitly. They are reduced to one-dimensional integrals

containing the radial functions [30].Linear-in-Bterms

give the values of the renormalized g-factor in the con-

duction band

t,l =

drr2gc

0t,lf2

0+(r)−8P

3¯hrf0+(r)g1+(r)

−gv

0t,l

3+8αvm0

3¯h2g2

0+(r)(12)

and in the valence band

t,l =

drr2gv

0t,lg2

0−(r)−8P

3¯hrf1−(r)g0−(r)

−gc

0t,l

3−8αcm0

3¯h2f2

1−(r).(13)

In Figure 3the results of the calculations within the

k·pmodel according to equations (12,13) are shown in

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EectiveLandefactorsofelectronsandholesinleadchalcogenidenanocrystalsI.D.Avdeev,1S.V.Goupalov,1,2,andM.O.Nestoklon11IoeInstitute,194021St.Petersburg,Russia2DepartmentofPhysics,JacksonStateUniversity,JacksonMS39217,USATheLandeorg-factorsofchargecarriersinsolidstatesystemsprovideinvaluableinformat...

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Eﬀective Land e factors of electrons and holes in lead chalcogenide nanocrystals I.D. Avdeev1S.V. Goupalov1 2and M.O. Nestoklon1 1Ioﬀe Institute 194021 St. Petersburg Russia.pdf

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Eﬀective Land e factors of electrons and holes in lead chalcogenide nanocrystals I.D. Avdeev1S.V. Goupalov1 2and M.O. Nestoklon1 1Ioﬀe Institute 194021 St. Petersburg Russia

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