Different Approaches for Multiband Transport in Semiconductors
We compare the well-known Kane model with a new multiband envelope function model, which presents many advantages with respect to the first one.
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509762908585984 |
|---|---|
| author | Borgioli, G. Frosali, G. Modugno, M. Morandi, O. Борґіолі, Г. Фросалі, Г. Модуньо, М. Моранді, О. |
| author_facet | Borgioli, G. Frosali, G. Modugno, M. Morandi, O. Борґіолі, Г. Фросалі, Г. Модуньо, М. Моранді, О. |
| author_sort | Borgioli, G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:55Z |
| description | We compare the well-known Kane model with a new multiband envelope function model, which presents many advantages with respect to the first one. |
| first_indexed | 2026-03-24T02:46:16Z |
| format | Article |
| fulltext |
UDC 517.9 + 531.19
G. Borgioli, O. Morandi, G. Frosali (Univ. Firenze, Italy),
M. Modugno (Univ. Firenze and INFM, Italy)
DIFFERENT APPROACHES FOR MULTIBAND TRANSPORT
IN SEMICONDUCTORS
RIZNI PIDXODY DO BAHATOZONNOHO PERENOSU
V NAPIVPROVIDNYKAX
We compare the well-known Kane model with a new multiband envelope function model, which presents many
advantages with respect to the first one.
Dobre vidomu model\ Kane porivnqno z novog modellg bahatokomponentno] obvidno] funkci] i prode-
monstrovano bahato perevah ostann\o].
1. Introduction. Multiband [1, 2] models are of paramount importance in nanomet-
ric semiconductor devices. In particular, we refer here to modeling interband tunneling,
which is the main mechanism of working for some heterostructure devices (superlattices)
like RITDs (Resonant Interband Tunneling Devices). In these superlattices the contact of
heterogeneous semiconductor materials enables to obtain potential barriers that interface
the conduction and the valence bands [3 – 6].
A quantum mechanics based model, able to treat a multiband dynamics [7 – 10] is the
Kane model. Here we deal with the two-band case, to which some literature has been
devoted (Wigner formulation, hydrodynamic formulation, . . . [11 – 13]. The physical
assumptions foresee a simplified environment, where magnetic and spin effects are disre-
garded. Dissipative phenomena like electron-phonon collisions are not taken into account.
The dynamics of charge carriers is considered as confined in the two highest energy bands
of the semiconductor, i.e., the conduction and the (nondegenerate) valence bands, around
the point k = 0, where k is the “crystal” wave vector. The point k = 0 is assumed
to be a minimum for the conduction band and a maximum for the valence band, in the
parabolic approximation.
The Hamiltonian introduced in the Schrödinger equation is
H = H0 + V, H0 = − �
2
2m0
∆ + Vper , (1)
where Vper is the periodic potential of the crystal and V an “external” potential, which
takes in account different effects, like the device energy-band offset for the heterojunc-
tions, the bias voltage applied across the device, the contribution from the doping impuri-
ties and from the self-consistent field produced by the mobile electronic charge.
In this paper we compare the well-known Kane model to a new multiband envelope
function model, which presents many advantages with respect to the first one. In Section
2 we recall the procedure of derivation of the Kane model from the Schrödinger equation
and in Section 3 we describe the k · P method, which is the classical way of studying
multiband systems governed by an Hamiltonian perturbed by an external potential [2,
14]. In Section 4 we propose a different procedure of approximation for the Schrödinger
equation. Using again an expansion founded on the classical Bloch basis, the new strategy
is separating the intraband dynamics terms from the interband coupling ones. The model
obtained in such a way eventually contains interband terms and uses envelope functions
which have, beyond other advantages, a direct physical interpretation. We observe that
c© G. BORGIOLI, O. MORANDI, G. FROSALI, M. MODUGNO, 2005
742 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
DIFFERENT APPROACHES FOR MULTIBAND TRANSPORT IN SEMICONDUCTORS 743
the present paper is only a preliminary study for modeling interband tunneling effects.
The reader interested in more details is referred to the forthcoming paper [15].
Finally, we claim that this paper, devoted to the modeling topics, is the first part of
a wider project which foresees both analytical and numerical study of Schrödinger-like
models, kinetic-like models (Wigner transform) and quantum hydrodynamic models for
multiband systems.
2. Kane model. The equations system proposed (in a 3D infinite spatial domain) is
the following:
i�
∂ψc
∂t
= − �
2
2m0
∆ψc + (Ec + V )ψc −
�
m0
P · ∇ψv,
i�
∂ψv
∂t
= − �
2
2m0
∆ψv + (Ev + V )ψv +
�
m0
P · ∇ψc,
(2)
where ψc (ψv) is a conduction (valence) envelope function, m0 is the bare mass of the
carriers, Ec(Ev) is the minimum (maximum) of the conduction (valence) band energy
and P is the coupling term between the two bands, which represents the momentum
operator matrix element between the corresponding (conduction and valence) Wannier
functions. In general P depends on the effective mass tensor and on the energy gap
between the bands. Introducing
Ψ ≡
(
ψc
ψv
)
, HK ≡
− �
2
2m0
∆ + Ec + V − �
m0
P · ∇
�
m0
P · ∇ − �
2
2m0
∆ + Ev + V
, (3)
we can rewrite (2) in a vectorial notation
i�
∂Ψ
∂t
= HKΨ . (4)
HK is known as the Kane – Hamiltonian and the extension of (4) to the n-band case is
straightforward.
We recall here briefly the procedure of derivation of (2) from the Schrödinger equation
for an electron subjected to a periodic potential plus an external potential V, which writes
as
i�
∂Ψ
∂t
= (H0 + V )Ψ , (5)
where Ψ represents the wave function of the electron.
We perform the expansion of a generic solution Ψ on a Bloch basis, considering all
the bands
Ψ(x) =
∑
n
∫
B
dkϕn(k)bn(k, x), n = 1, 2, . . . , (6)
where B is the first Brillouin zone and the eigenfunctions of H0, i.e., the Bloch eigen-
functions, can be written as
bn(k, x) = ei k·xun(k, x) ≡ 〈x|n, k〉 . (7)
We consider as a new basis the periodic part of Bloch functions, i.e., un(k, x), at k = 0
and perform the expansion
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
744 G. BORGIOLI, O. MORANDI, G. FROSALI, M. MODUGNO
un(k, x) =
∑
n′
Cn,n′(k)u0
n′(x), (8)
where u0
n′(x) = un′(0, x) .
Using as a basis u0
n′(x) in (6), Ψ(x) reads as
Ψ(x) =
∑
n
ψn(x)u0
n(x) , (9)
where the functions
ψn(x) =
∑
n′
∫
B
dkϕn′(k)Cn,n′(k)eik·x, n, n′ = 1, 2, . . . , (10)
are “Kane” envelope functions, which do not depend on k anymore. In a two-band
dynamics ( n,m = c, v, where c stands for “conduction band” and v stands for “valence
band”) expansion (6) takes the form
Ψ(x) = ψc(x)u0
c(x) + ψv(x)u0
v(x) , (11)
which is the envelope function expansion of the wave function Ψ that leads to Kane
model (2).
Summarizing the preceding considerations, we remark that the Kane model is derived
introducing a new basis (constructed around k = 0). The “Kane” basis functions are
not eigenfunctions of the unperturbed Hamiltonian. The “Kane” basis corresponds to a
rotation of the Bloch basis
u0
n(x) =
∑
n′
C−1
n,n′(k)un′(k, x) . (12)
We conclude the section highlighting some defect and shortcoming produced by the ap-
proximations performed on the way of Kane model’s derivation. First, the potential V
affects the band energy terms, but it does not appear in the coupling term P; second,
there is an interband coupling even in absence of an external potential; third, the inter-
band term P increases when the energy gap between the two bands Eg increases (the
opposite of physical evidence); fourth there is no direct physical interpretation of “con-
duction” and “valence” electron envelope functions. These considerations convince us,
beyond the sure merits of Kane model, to investigate a different approximation approach
to the Schrödinger equation, in order to attain a more physically consistent model.
3. Luttinger – Kohn model. We now briefly describe the k ·P technique [2], which
is successfully used to analyze the electronic properties of a wide type of semiconductor.
The spirit of a k · P model is to use momentum k as a perturbation parameter of the
Hamiltonian.
If we write the eigenvalue equation for the semiconductor without external field, we
have
H0bn(k, x) = En(k)bn(k, x),
where bn(k, x) are the Bloch functions in (7). It easy to show that the eigenvalue equation
solved by the periodic part of Bloch function un(k, x) is now:
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
DIFFERENT APPROACHES FOR MULTIBAND TRANSPORT IN SEMICONDUCTORS 745
(H0 +H ′)un(k, x) =
=
[
− �
2
2m0
∆ − i
�
2
m0
(k · ∇) +
�
2k2
2m0
+ Vper(x)
]
un(k, x) =
= En(k)un(k, x) ,
where k = |k|. Here H0 + H ′ is the so-called k · P Hamiltonian, where H0 is the
single electron Hamiltonian (1) and
H ′ = − i
�
2
m0
(k · ∇ ) +
�
2k2
2m0
is an additional term which is treated as a perturbation, since it vanishes when k → 0.
Thus, for states localized near the center of the (first) Brillouin zone, H ′ is small and the
original problem is faced first solving the unperturbed problem
H0un(0, x) = En(0)un(0, x)
and then using the perturbation H ′ to get some correction on En(0) and un(0, x) when
k
= 0. The non degenerate perturbation theory provides that, at the first order of the
perturbation H ′, the eigenfunctions for the perturbed problem are
u1
n(0, x) = u0
n(x) + k · ∇kun(k, x)
k=0
=
= u0
n(x) − �
m0
∑
n′ �=n
k · Pn,n′
�En,n′
un′(0, x),
with
�En,n′ = En(0) − En′(0)
and
Pn,n′ ≡ (2π)3
Ω
∫
u−cell
dxu∗n(0, x) ∇ un′(0, x), (13)
where Ω is the volume of the unitary cell (u− cell). In this way, using perturbation the-
ory, we are driven to work with a (non orthonormal) basis, that arises from the following
quasiunitary rotation operator Θ, applied to the unperturbed (u0
n(x)) basis
u1
n(k, x) =
∑
n′
Θn,n′(k)u0
n′(x),
with
Θn,n′(k) =
(
δn,n′ − �
m0
k · Pn,n′
�En,n′
)
.
Luttinger and Kohn [14] proposed to apply the previous procedure to the Kane Hamilto-
nian HK (3) in the n-band case (where the basis elements are the unperturbed elements
u0
n(x) ), diagonalizing it to the first order in k. Using Fourier transform, we can recast
the system (4) in the following way:(
En +
�
2k2
2m0
)
ψ̃n(k) +
�
m0
∑
n′
k · Pn,n′ ψ̃n′(k) +
∫
B
dk′ Ṽ (k − k′)ψ̃(k′) = 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
746 G. BORGIOLI, O. MORANDI, G. FROSALI, M. MODUGNO
where ψ̃n(k), Ṽ (k) are the Fourier transform of ψn(x), V (x) respectively. If we
operate the following change of variable
χ̃n =
∑
n′
Θ−1
n,n′ ψ̃n′
and we go back to the coordinate space, we recover the Luttinger – Kohn system. The
authors proposed to neglect all off-diagonal term, and thus they achieved the following
uncoupled equations set, for a 3D spatial domain (conduction-valence, i.e., the two-band
case):
i�
∂χc
∂t
= Ecχc −
�
2
2m∗
c
∆χc + V χc,
i�
∂χv
∂t
= Evχv +
�
2
2|m∗
v|
∆χv + V χv ,
(14)
where m∗
c and m∗
v are, respectively, the isotropic effective mass in the conduction and
valence bands, given by the following expression:
m0
m∗
n
= 1 − 2�
2
3m0
∑
n′ �=n
Pn,n′ · Pn′,n
�En,n′
,
where m0 is the bare mass of the electron, n denotes the band index (in the two-band
case n = c, v ) and n′ runs on all the other bands. As it is manifest, disregarding the
off-diagonal terms implies the achievement of two uncoupled equations for the envelope
functions in the two bands. This means that the model, at this stage of approximation, is
not able to describe an interband tunneling dynamics.
4. A different approach. In this section we propose a different procedure of ap-
proximation for the Schrödinger equation (5), under the same physical assumptions used
in the Kane model [15].
If we expand equation (5) using the Bloch basis bn(k, x) like in (6), we obtain n-
equations for the expansion coefficients
i�
∂ϕn
∂t
(k) = En(k)ϕn(k) +
+
∑
n′
∫
B
dk′ 〈n,k|V |n′, k′〉ϕn(k′), n = 1, 2, . . . . (15)
Equations (15) are exact but of no practical utility. In order to provide suitable approxi-
mations and attain a model which still maintain interband dynamics terms, but which is
less tough to handle, we separate the intraband dynamics from the interband coupling.
After some algebra we get
i�
∂ϕn
∂t
(k) = En(k)ϕn(k) +
∫
B
dk′ Ṽ (k − k′)ϕn(k′) −
−i �
2
m0
∑
n′ �=n
∫
B
dk′ Ṽ (k − k′)ϕn′(k′)
(2π)3
Ω
×
×
∫
u−cell
dxu∗n(k, x)
k − k′
�En,n′
· ∇un′(k′, x), (16)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
DIFFERENT APPROACHES FOR MULTIBAND TRANSPORT IN SEMICONDUCTORS 747
where Ṽ denotes the Fourier transform of the potential V and
�En,n′(k,k′) ≡ En′(k′) − En(k) +
�
2
2m0
(
k′
2 − k2
)
. (17)
This new set of n-equations is so far very general and only relies on the assumption that
the potential V has no appreciable variation on the scale of a single lattice cell. The
approximation procedure that we have chosen is based on three steps.
First, simplify the interband term to the lowest order in k :
−i
∑
n′ �=n
�
2Pn,n′
m0�En,n′
∫
B
dk′ (k − k′) Ṽ (k − k′)ϕn′(k′). (18)
Second, introduce the effective mass approximation
En(k) = En +
�
2k2
2m∗
n
+ · · · , (19)
where m∗
n is the isotropic effective mass in the n-band.
Third and final, obtain the equations for the envelope functions in the coordinates
space by inverse Fourier transform.
This result can be attained by projection of the wave function on the Wannier basis
φW
n which depends on (x − Ri), where Ri are the atomic sites positions, i.e.,
Ψ(x) =
∑
n
∑
Ri
χn(Ri)φW
n (x − Ri) , (20)
where the Wannier basis functions can be expressed in terms of Bloch functions as
φW
n (x − Ri) =
√
Ω
(2π)3
∫
B
bn(k, x − Ri)dk . (21)
The use of the Wannier basis has some advantages. As a matter of fact the amplitudes
χn(Ri), that play the role of envelope functions on the new basis, can be obtained from
the Bloch coefficients (see (15)) by a simple Fourier transform
χn(Ri) =
√
Ω
(2π)3
∫
B
ϕn(k)eik·Ridk. (22)
Moreover they can be interpreted as the actual wave function of an electron in the n-band.
In fact, “macroscopic” properties of the system, like charge density and current, can be
expressed in term of χn(Ri), averaging on a scale of the order of the lattice cell.
Performing the limit to the continuum by extending the dependence of the χn(Ri) to
the whole space (Ri −→ x) and by using standard properties of the Fourier transform,
equations for the coefficients χn(x) are achieved.
In case of only two bands (“conduction” and “valence”) the equations for the envelope
functions take the form
i�
∂χc
∂t
= − �
2
2m∗
c
∆χc + V χc + Ecχc −
�
2P · ∇V
m0Eg
χv,
i�
∂χv
∂t
=
�
2
2|m∗
v|
∆χv + V χv + Evχv − �
2P · ∇V
m0Eg
χc.
(23)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
748 G. BORGIOLI, O. MORANDI, G. FROSALI, M. MODUGNO
These equations describe the intraband dynamics in the effective mass approxima-
tion in the same fashion as the Luttinger – Kohn model, but also contain an interband
coupling, proportional to the momentum matrix element P, that is responsible for tun-
neling between different bands induced by the applied electric field proportional to the
x-derivative of V. As discussed above, the envelope functions χc and χv are the pro-
jections of the full wavefunction ψ on the Wannier basis, and therefore represent the
(cell-averaged) probability amplitude for finding an electron on the conduction or valence
bands (of the unperturbed problem) respectively. This “natural” choice of the basis allows
in principle for a clear and systematic expansion at higher orders in k, and has impor-
tant advantages with respect to the Kane approach. Indeed, as one would naively expect,
in this case the interband coupling term reduces as the energy gap Eg increases, and
vanishes in the absence of the external field V.
This opens the interesting perspective of comparing the predictions of the Kane model
and the model in equation (23) for semiconductor devices where interband tunneling ef-
fects plays a major role, like in RITDs.
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Received 08.11.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
|
| id | umjimathkievua-article-3640 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:16Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c4/3bfe849592ebafee9b64a3ade2a63ac4.pdf |
| spelling | umjimathkievua-article-36402020-03-18T20:00:55Z Different Approaches for Multiband Transport in Semiconductors Різні підходи до багатозонного переносу в напівпровідниках Borgioli, G. Frosali, G. Modugno, M. Morandi, O. Борґіолі, Г. Фросалі, Г. Модуньо, М. Моранді, О. We compare the well-known Kane model with a new multiband envelope function model, which presents many advantages with respect to the first one. Добре відому модель Кане порівняно з новою моделлю багатокомпонентної обвідної функції i продемонстровано багато переваг останньої. Institute of Mathematics, NAS of Ukraine 2005-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3640 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 6 (2005); 742–748 Український математичний журнал; Том 57 № 6 (2005); 742–748 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3640/4010 https://umj.imath.kiev.ua/index.php/umj/article/view/3640/4011 Copyright (c) 2005 Borgioli G.; Frosali G.; Modugno M.; Morandi O. |
| spellingShingle | Borgioli, G. Frosali, G. Modugno, M. Morandi, O. Борґіолі, Г. Фросалі, Г. Модуньо, М. Моранді, О. Different Approaches for Multiband Transport in Semiconductors |
| title | Different Approaches for Multiband Transport in Semiconductors |
| title_alt | Різні підходи до багатозонного переносу в напівпровідниках |
| title_full | Different Approaches for Multiband Transport in Semiconductors |
| title_fullStr | Different Approaches for Multiband Transport in Semiconductors |
| title_full_unstemmed | Different Approaches for Multiband Transport in Semiconductors |
| title_short | Different Approaches for Multiband Transport in Semiconductors |
| title_sort | different approaches for multiband transport in semiconductors |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3640 |
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