Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі
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irk-123456789-1657422020-02-17T01:25:47Z Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі Алі, Г. Манзіні, Ч. Фросалі, Г. Статті 3a допомогою масштабного перетворення одержано гідродинамічні рівняння у квазікласичному наближенні з двокомпонентного рівняння Шредінгера. By using a scale transformation, we obtain hydrodynamic equations in the quasiclassical approximation from the two-band Schrodinger equation. 2005 Article Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі / Г. Алі, Ч. Манзіні, Г. Фросалі // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 723–730. — Бібліогр.: 15 назв. — укр. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165742 517.9 + 531.19 uk Український математичний журнал Інститут математики НАН України |
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Статті Статті |
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Статті Статті Алі, Г. Манзіні, Ч. Фросалі, Г. Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі Український математичний журнал |
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3a допомогою масштабного перетворення одержано гідродинамічні рівняння у квазікласичному наближенні з двокомпонентного рівняння Шредінгера. |
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| author |
Алі, Г. Манзіні, Ч. Фросалі, Г. |
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Алі, Г. Манзіні, Ч. Фросалі, Г. |
| author_sort |
Алі, Г. |
| title |
Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі |
| title_short |
Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі |
| title_full |
Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі |
| title_fullStr |
Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі |
| title_full_unstemmed |
Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі |
| title_sort |
про модель переносу дифузії для двозонної квантової рідини при нульовій температурі |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165742 |
| citation_txt |
Про модель переносу дифузії для двозонної квантової рідини при нульовій температурі / Г. Алі, Ч. Манзіні, Г. Фросалі // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 723–730. — Бібліогр.: 15 назв. — укр. |
| series |
Український математичний журнал |
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2025-07-14T19:46:45Z |
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2025-07-14T19:46:45Z |
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| fulltext |
UDC 517.9 + 531.19
G. Alı̀ (Ist. Appl. Calcolo “M. Picone”, Napoli, Italy),
G. Frosali (Univ. Firenze, Italy),
Ch. Manzini (Scuola Normale Superiore, Pisa, Italy)
ON THE DRIFT-DIFFUSION MODEL
FOR A TWO-BAND QUANTUM FLUID
AT ZERO-TEMPERATURE
PRO MODEL\ PERENOSU DYFUZI}
DLQ DVOZONNO} KVANTOVO} RIDYNY
PRY NUL\OVIJ TEMPERATURI
By using a scale transformation, we obtain hydrodynamic equations in the quasiclassical approximation from
the two-band Schrödinger equation.
Za dopomohog masßtabnoho peretvorennq oderΩano hidrodynamiçni rivnqnnq u kvaziklasyçnomu nably-
Ωenni z dvokomponentnoho rivnqnnq Íredinhera.
1. Introduction. In the recent literature there is a growing interest for diodes in which the
valence band electrons play a relevant role in the current flow, such as Interband Resonant
Tunneling Diodes [1 – 3]. Correspondingly, the effort in the theoretical study of multiband
models has increased [4 – 8]. The typical band diagram structure of a tunneling diode is
characterized by a band alignment such that the valence band at the positive side of the
semiconductor device lies above the conduction band at the negative one.
Correspondingly, one of the simplest multiband model, introduced by E. O. Kane in
the early 60’s [9], includes only two energy bands of the device material, separated by a
forbidden region. It consists of two coupled Schrödinger-like equations for the conduc-
tion and the valence band wave (envelope) functions. The coupling term is derived with
the k ·p perturbation method [10, 11], which relies on the assumption that, for a reli-
able description, it is sufficient solve of the single-electron Schrödinger equation in the
neighbourhood of the bottom and the top of the conduction and the valence bands, respec-
tively, since most of the electrons and holes are located there. This model is successfully
employed for simulations [2, 3]; in particular, it is suitable for investigations on the bulk
properties of semiconductors, such as band nonparabolicity and optical properties.
Nevertheless, the approximation of the original multiband problem by the two-band
Kane model has not a clear physical interpretation; indeed, the corresponding equations
result to be coupled even in absence of an external potential. Moreover the choice of the
envelope functions is subtle: in the literature are present various methods based on par-
tial diagonalization of the Kane Hamiltonian (such as Luttinger effective-mass models);
however, they don’t give satisfactory results when nonperiodic potentials are present [12].
The method proposed in [8], instead, is based on the use of the Wannier envelope func-
tions, and the “multiband envelope function” model obtained is reliable even when the
symmetry of the crystal is broken by an external potential (standing for heterostructures,
impurities, e.g.), since the elements of the basis are located at the crystal sites.
Already, in the (semi-)classical framework, a hydrodynamic formulation is recom-
mended, because of the lower computational cost of the implementation. In the quantum
framework, many works in the literature are devoted to quantum hydrodynamic formu-
lation, [13, 14] e.g. In a recent work [15], Alı̀ and Frosali have developed a method to
extend the derivation of quantum hydrodynamic models from a (single-band) Schrödinger
c© G. AlI, G. FROSALI, CH. MANZINI, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6 723
724 G. ALI, G. FROSALI, CH. MANZINI
equation [13] to the Kane model. There, they have obtained a closed system of hydrody-
namic equations for a two-band quantum fluid.
The method formulated in [15] is suitable to be applied to the multiband envelope
function model in [8] as well, and that is precisely the content of the present work. Af-
ter introducing (Section 2) the two-band envelope function model, in Section 3 we de-
rive the corresponding fluiddynamical system for particle and current densities, using
the Madelung transform. In Section 4, we perform a drift-diffusive scaling and we end up
with a closed set of equations which are the analog of the zero-temperature quantum drift-
diffusion model for a two-band envelope function system. In the last section, we compare
our model with the one obtained in [15] and we discuss briefly many open problems in the
two-band quantum hydrodynamical model, such as closure and numerical experiments.
2. A two-band envelope function system. Let ψc(x, t) be the conduction band
envelope function and ψv(x, t) be the valence band envelope function. The multiband
envelope function model in the two-band time-dependent case reads as follows:
i�
∂ψc
∂t
= − �
2
2m∗∆ψc + (Vc + V )ψc −
�
2
m
P · ∇V
Eg
ψv,
i�
∂ψv
∂t
=
�
2
2m∗∆ψv + (Vv + V )ψv − �
2
m
P · ∇V
Eg
ψc.
(2.1)
This model will be considered in R
3. Here, i is the imaginary unit, � is the reduced
Planck constant, m∗ is the isotropic effective mass of both the conduction and valence
band electrons, which we suppose to be equal, and m is the bare mass of the carriers.
Moreover, V is the electrostatic potential, Vc and Vv are the minimum and the max-
imum of the conduction and the valence band energy, respectively. The last two quan-
tities depend, through the x-coordinate, on the layer composition, while their difference
Eg = Vc − Vv, which is called gap energy, is supposed to be constant. The coupling
coefficient between the two bands P represents the momentum operator matrix element
between the corresponding Wannier functions.
For the derivation of model (2.1) in the framework of the Bloch theory, we refer the
reader to [8].
We recall that, for anisotropic materials, the inverse of the isotropic effective mass
should be replaced by an inverse mass tensor. We make use of the following scaling: after
choosing a (scalar) characteristic length scale xR and a characteristic time scale tR, we
introduce the rescaled Planck constant
ε =
�
α
,
with the dimensional parameter
α =
m∗x2
R
tR
,
and the rescaled time and space variables t′ =
t
tR
, x′ =
x
xR
componentwise.
In the adimensional version of (2.1), the masses m and m∗ are kept unchanged,
since they appear in a ratio, while the band energy can be rescaled by VR =
m∗x2
R
t2R
.
A dimensional argument shows that the original coupling coefficient is a reciprocal of a
characteristic lenght, thus P ′ = PxR, componentwise.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
ON THE DRIFT-DIFFUSION MODEL FOR A TWO-BAND QUANTUM FLUID . . . 725
Hence, dropping the prime, we get the following two-band envelope function model,
which will be the object of our study:
iε
∂ψc
∂t
= −ε
2
2
∆ψc + (Vc + V )ψc − ε2K ψv,
iε
∂ψv
∂t
=
ε2
2
∆ψv + (Vv + V )ψv − ε2K ψc,
(2.2)
where K =
m∗
m
P · ∇V
Eg
.
3. Derivation of the fluiddynamical model. The simplest way to derive a flu-
iddynamical formulation of the evolution equations for particle and current densities is
the (classically used) Madelung transform. Since our model consists of two coupled
Schrödinger equations, we decompose the wave (envelope) functions for conduction and
valence bands into their amplitudes
√
nc,
√
nv and phases Sc, Sv, defined by the
relations
ψc(x, t) =
√
nc(x, t) exp
(
iSc(x, t)
ε
)
,
ψv(x, t) =
√
nv(x, t) exp =
(
iSv(x, t)
ε
)
.
(3.1)
For more details on the notation see Section 2 of [15], where the same procedure is applied
to the two-band Kane model.
By using the first equation of the two-band envelope function system (2.2), we find
∂nc
∂t
= ψc
∂ψc
∂t
+ ψc
∂ψc
∂t
= −∇· Im
(
εψc∇ψc
)
− 2K Im
(
εψcψv
)
.
In a similar way, we get the equation for the evolution of nv. Then, the previous equations
become
∂nc
∂t
+ ∇· Im
(
εψc∇ψc
)
= −2K Im
(
εψcψv
)
,
∂nv
∂t
−∇· Im
(
εψv∇ψv
)
= 2K Im
(
εψcψv
)
,
(3.2)
and by using the definition of current density, we get
∂nc
∂t
+ ∇·Jc = −2K Im
(
εψcψv
)
,
∂nv
∂t
−∇·Jv = 2K Im
(
εψcψv
)
.
(3.3)
Summing the equations in (3.3), we obtain the balance law for the total density,
∂
∂t
(nc + nv) + ∇·(Jc − Jv) = 0. (3.4)
We remark that, in contrast with the Kane model, currents due to the interband terms do
not appear in the conservation of the total density.
Next, we derive equations for phases Sc, Sv. Using systems (2.2) and (3.2), we get
∂Sc
∂t
= −iε ∂
∂t
ln
(
ψc√
nc
)
= −iε
(
1
ψc
∂ψc
∂t
− 1
2nc
∂nc
∂t
)
=
=
ε2
2nc
(
∇· Re (ψc∇ψc) −∇ψc ·∇ψc
)
− (Vc + V ) +
ε2
nc
K Re (ψcψv).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
726 G. ALI, G. FROSALI, CH. MANZINI
It is possible to rewrite the previous equation as
∂Sc
∂t
= −1
2
|∇Sc|2 +
ε2∆
√
nc
2
√
nc
− (Vc + V ) +
ε2
nc
K Re (ψcψv).
A similar equation can be derived for Sv. The resulting system is
∂Sc
∂t
+
1
2
|∇Sc|2 −
ε2∆
√
nc
2
√
nc
+ (Vc + V ) =
ε2
nc
KRe(ψcψv),
∂Sv
∂t
− 1
2
|∇Sv|2 +
ε2∆
√
nv
2
√
nv
+ (Vv + V ) =
ε2
nv
KRe (ψcψv).
(3.5)
Equations (3.2) and (3.5) are equivalent to the coupled Schrödinger equations in (2.2).
We would like to replace system (3.5) with one of coupled equations for the currents.
We can evaluate
∂Jc
∂t
= ε Im
(
ψc∇
∂ψc
∂t
+ ∇ψc
∂ψc
∂t
)
=
=
∑
j
ε2
2
∂
∂xj
Re
(
ψc∇
∂ψc
∂xj
−∇ψc
∂ψc
∂xj
)
− ψcψc∇Vc+
+ ε2∇K Re (ψcψv) + ε2K Re
[
∇(ψcψv) − 2∇ψc ψv
]
. (3.6)
Using standard identities, eq. (3.6) can be rewritten in the more familiar form
∂Jc
∂t
+ div
(
Jc ⊗ Jc
nc
+ ε2∇√
nc ⊗∇√
nc −
ε2
4
∇⊗∇nc
)
+ nc(∇Vc + ∇V ) =
= ε2∇K Re (ψcψv) + ε2K Re
[
∇(ψcψv) − 2∇ψcψv
]
. (3.7)
Similarly, for Jv we find
∂Jv
∂t
− div
(
Jv ⊗ Jv
nv
+ ε2∇√
nv ⊗∇√
nv − ε2
4
∇⊗∇nv
)
+ nv(∇Vv + ∇V ) =
= ε2∇K Re (ψcψv) + ε2K Re
[
∇(ψvψc) − 2ψc∇ψv
]
. (3.8)
The left-hand sides of the equations for the currents can be reformulated by the following
identity:
div
(
∇√
ni ⊗∇√
ni −
1
4
∇⊗∇ni
)
= −ni
2
∇
[
∆
√
ni√
ni
]
, i = c, v.
The correction terms
ε2
2
∆
√
ni√
ni
i = c, v ,
can be interpreted as internal self-potentials for each band and are called quantum Bohm
potentials.
In addition, the right-hand sides of equations (3.7), (3.8) can be expressed in terms
of the hydrodynamic quantities, by using the relations and definitions we recall here
(cf. [15]):
εψi∇ψj =
√
ni
√
nj exp
(
i
Sj − Si
ε
) (
ε
∇√
nj√
nj
+ i∇Sj
)
, (3.9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
ON THE DRIFT-DIFFUSION MODEL FOR A TWO-BAND QUANTUM FLUID . . . 727
ncv := ψcψv =
√
nc
√
nv e
iσ, (3.10)
where σ is the phase difference defined by σ :=
Sv − Sc
ε
,
uc :=
ε∇ψc
ψc
=
ε∇√
nc√
nc︸ ︷︷ ︸
uos,c
+i∇Sc︸︷︷︸
uel,c
, uv :=
ε∇ψv
ψv
=
ε∇√
nv√
nv︸ ︷︷ ︸
uos,v
+i∇Sv︸︷︷︸
uel,v
, (3.11)
ε∇ncv = ncv(uv + uc). (3.12)
Thus,
∂Jc
∂t
+ div
(
Jc ⊗ Jc
nc
)
− nc∇
(
ε2∆
√
nc
2
√
nc
)
+ nc(∇Vc + ∇V ) =
= ε2∇K Re (ψcψv) + εK Re (ncv(uv − uc)) ,
∂Jv
∂t
− div
(
Jv ⊗ Jv
nv
)
+ nv∇
(
ε2∆
√
nv
2
√
nv
)
+ nv(∇Vv + ∇V ) =
= ε2∇K Re (ψcψv) − εK Re (ncv(uv − uc)) .
(3.13)
By exploiting, instead, the definition (3.11) of osmotic velocities (uos,c, uos,v) and cur-
rent velocities (uel,c, uel,v) , and the relation (3.9), we get
∂Jc
∂t
+ div
(
Jc ⊗ Jc
nc
)
− nc∇
(
ε2∆
√
nc
2
√
nc
)
+ nc(∇Vc + ∇V ) =
= ε2∇K Re ncv + εK
√
nc
√
nv(cosσ(uos,v − uos,c) − sinσ(uel,c + uel,v)),
∂Jv
∂t
− div
(
Jv ⊗ Jv
nv
)
+ nv∇
(
ε2∆
√
nv
2
√
nv
)
+ nv(∇Vv + ∇V ) =
= ε2∇K Re ncv − εK
√
nc
√
nv(cosσ(uos,v − uos,c) − sinσ(uel,c + uel,v)).
(3.14)
Analogously to Ref. [15] and at variance with the uncoupled model, systems (3.3) and
(3.14) are not equivalent to the original system (2.2), due to the presence of σ. The way
to close systems (3.3) and (3.14) is not unique; one possibility is to use system (3.5) to
derive an evolution equation for σ = (Sv − Sc)/ε , namely
ε
∂σ
∂t
− 1
2
(∣∣∣∣Jc
nc
∣∣∣∣2 +
∣∣∣∣Jv
nv
∣∣∣∣2
)
+
ε2
2
(
∆
√
nc√
nc
+
∆
√
nv√
nv
)
− Vc + Vv =
=
ε
2
K Re (εψcψv)
(
1
nv
− 1
nc
)
. (3.15)
Equation (3.15) must be supplemented with the constraint
ε∇σ =
Jv
nv
− Jc
nc
. (3.16)
It is possible to prove that equations (3.15) and (3.16) are equivalent. Indeed, if we con-
sider equation (3.16), then we can recover σ as a function of the other variables by
solving the elliptic equation
ε∆σ = ∇·
(
Jv
nv
− Jc
nc
)
, (3.17)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
728 G. ALI, G. FROSALI, CH. MANZINI
which can be obtained immediately by derivation of the constraint (3.16).
Another possibility is to regard ncv in system (3.13) as an independent variable,
rather than σ. From definition (3.10) and the two-band envelope function system (2.2),
we find
∂ncv
∂t
= ψv
∂ψc
∂t
+ ψc
∂ψv
∂t
=
= − iε
2
∇ ·
(
ψv∇ψcψc∇ψv − 2∇ψv∇ψc
)
+
+
i
ε
(Vc − Vv)ψcψv − iεK
(
ψvψv − ψcψc
)
,
which, using (3.9) and the definitions of osmotic and current velocities, leads to
ε
∂ncv
∂t
= − i
2
∇·∇ncv − i
ε
ncv(ucuv) +
i
ε
ncv(Vc − Vv) + iεK (nc − nv) . (3.18)
In addition to (3.18), the complex function ncv must satisfy the constraints
ncvncv = ncnv, (3.19)
ε∇ncv = (uv + uc)ncv. (3.20)
Alternatively, we can use the identity (3.20) to derive a nonlinear elliptic equation for
ncv,
div
(
ε∇ncv
ncv
)
= div(uv + uc), (3.21)
which must be solved together with the constraint (3.19).
Now we are in position to rewrite the hydrodynamic system as follows:
∂nc
∂t
+ divJc = −2εK Imncv,
∂nv
∂t
− divJv = 2εK Imncv,
∂Jc
∂t
+ div
(
Jc ⊗ Jc
nc
)
− nc∇
(
ε2∆
√
nc
2
√
nc
)
+ nc(∇Vc + ∇V ) =
= ε2∇K Re ncv + εK Re (ncv(uv − uc)) , (3.22)
∂Jv
∂t
− div
(
Jv ⊗ Jv
nv
)
+ nv∇
(
ε2∆
√
nv
2
√
nv
)
+ nv(∇Vv + ∇V ) =
= ε2∇K Re ncv − εK Re (ncv(uv − uc)) ,
ε∇σ =
Jv
nv
− Jc
nc
,
where ncv, uv, uv are expressed in the terms of the hydrodynamic quantities nc, nv,
Jc, Jv, σ by (3.10) and (3.11). System (3.22) is the extension of the classical Madelung
fluid equations to a two-band quantum fluid.
4. The drift-diffusive scaling. In the following we will consider a modified version
of the system (3.3), (3.13) and (3.17), with additional relaxation terms for the currents. It
is convenient to rewrite this system as
∂nc
∂t
+ divJc = −2εK Imncv,
∂nv
∂t
− divJv = 2εK Imncv,
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ON THE DRIFT-DIFFUSION MODEL FOR A TWO-BAND QUANTUM FLUID . . . 729
∂Jc
∂t
+ div
(
Jc ⊗ Jc
nc
)
− nc∇
(
ε2∆
√
nc
2
√
nc
)
+ nc(∇Vc + ∇V ) =
= ε2∇K Re ncv + εK Re (ncv(uv − uc)) −
Jc
τ
, (4.1)
∂Jv
∂t
− div
(
Jv ⊗ Jv
nv
)
+ nv∇
(
ε2∆
√
nv
2
√
nv
)
+ nv(∇Vv + ∇V ) =
= ε2∇K Re ncv − εK Re (ncv(uv − uc)) −
Jv
τ
,
ε∇σ =
Jv
nv
− Jc
nc
,
where τ is a relaxation time, which we assume the same for the two bands. As customary
in semiconductor theory, we perform the diffusive limit by introducing the scaling
t→ t
τ
, Jc → τJc, Jv → τJv . (4.2)
Consequently, from definition (3.17), the phase difference σ has to be rescaled as
ε∇σ → τε∇σ ,
and hence
σ → τσ + constant.
Then, by choosing the constant equal to zero, we have
ncv → √
nc
√
nv +O(τ),
uc → ε∇√
nc√
nc
+ i
Jc
nc
τ,
uv → ε∇√
nv√
nv
+ i
Jv
nv
τ.
The coupling term has to be tackled with much care, by writing
ncvuv → √
nc
√
nvuos,v + i
√
nc
√
nv (εσuos,v + uel,v) τ +O(τ2).
Formally, as τ tends to zero, after expressing the osmotic and current velocities in terms
of the other hydrodynamic quantities, (4.1) reduces to
∂nc
∂t
+ divJc = −2εσK
√
nc
√
nv,
∂nv
∂t
+ divJv = 2εσK
√
nc
√
nv,
Jc = nc∇
(
ε2∆
√
nc
2
√
nc
)
− nc(∇Vc + ∇V ) + ε2∇K√
nc
√
nv +
+ε2K(
√
nc∇
√
nv −√
nv∇
√
nc) , (4.3)
Jv = −nv∇
(
ε2∆
√
nv
2
√
nv
)
− nv(∇Vv + ∇V ) + ε2∇K√
nc
√
nv +
+ε2K(
√
nv∇
√
nc −
√
nc∇
√
nv ),
ε∇σ =
Jv
nv
− Jc
nc
.
This system represents the analog of the zero-temperature, quantum drift-diffusion
model for a two-band envelope function system.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
730 G. ALI, G. FROSALI, CH. MANZINI
5. Conclusions. We can summarize the considerations done during the derivation of
our fluiddynamical model (3.22), by saying that the two-band envelope function model
[8] seems to be more suitable for the formulation of a hydrodynamic system and, con-
sequently, of a drift-diffusion model for a two-band quantum system. As a further con-
firmation, we remark that in the scaled equations (4.3) the interband current terms have
disappeared, making more evident the physical meaning of the model presented. Both
the system (3.22) and the its scaled version (4.3) refer to quantum systems described by
pure states; however, the procedure can be easily repeated for an appropriate combination
of pure states in order to get the corresponding systems for mixed states (cf. [15]). The
closure relation chosen is the one proposed for one-band system by Gasser – Markowich
[13]; however, a deeper discussion about this problem would be needed and has to be
postponed for further investigations. Thus, this contribution is meant to be a preliminary
step of a bigger project in which thermal effects will be taken into account as well, and
numerical validations will be included.
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Received 08.11.2004
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