Conductivity of layered structures with a strongly degenerate electron gas
The conductivity of periodical multilayer structures with a degenerate electron gas is calculated. It is shown that for smooth tunnelly transparent barriers the resistivity of structure coincides with the corresponding value of the bulk material. In the presence of a random potential at the interfa...
Збережено в:
| Дата: | 1999 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
1999
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/119919 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Conductivity of layered structures with a strongly degenerate electron gas / P.P. Petrov // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 143-148. — Бібліогр.: 3 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-119919 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1199192017-06-11T03:03:13Z Conductivity of layered structures with a strongly degenerate electron gas Petrov, P.P. The conductivity of periodical multilayer structures with a degenerate electron gas is calculated. It is shown that for smooth tunnelly transparent barriers the resistivity of structure coincides with the corresponding value of the bulk material. In the presence of a random potential at the interface between the layers, the conductivity of the system decreases with the increase of a ratio of the mean free path to the period of the structure. Обчислена електропровiднiсть перiодичних багатошарових структур з виродженим електронним газом. Показано, що для гладких тунельно прозорих бар’єрiв опiр структури спiвпадає з вiдповiдним значенням для об’ємного матерiалу. У випадку, коли на границях роздiлу iснує випадковий розсiюючий потенцiал, провiднiсть системи зменшується зi збiльшенням вiдношення довжини вiльного пробiгу електронiв до перiоду структури. 1999 Article Conductivity of layered structures with a strongly degenerate electron gas / P.P. Petrov // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 143-148. — Бібліогр.: 3 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.143 PACS: 73.20.H, 82.65.Y http://dspace.nbuv.gov.ua/handle/123456789/119919 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The conductivity of periodical multilayer structures with a degenerate electron gas is calculated. It is shown that for smooth tunnelly transparent barriers the resistivity of structure coincides with the corresponding value of
the bulk material. In the presence of a random potential at the interface
between the layers, the conductivity of the system decreases with the increase of a ratio of the mean free path to the period of the structure. |
| format |
Article |
| author |
Petrov, P.P. |
| spellingShingle |
Petrov, P.P. Conductivity of layered structures with a strongly degenerate electron gas Condensed Matter Physics |
| author_facet |
Petrov, P.P. |
| author_sort |
Petrov, P.P. |
| title |
Conductivity of layered structures with a strongly degenerate electron gas |
| title_short |
Conductivity of layered structures with a strongly degenerate electron gas |
| title_full |
Conductivity of layered structures with a strongly degenerate electron gas |
| title_fullStr |
Conductivity of layered structures with a strongly degenerate electron gas |
| title_full_unstemmed |
Conductivity of layered structures with a strongly degenerate electron gas |
| title_sort |
conductivity of layered structures with a strongly degenerate electron gas |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119919 |
| citation_txt |
Conductivity of layered structures with a strongly degenerate electron gas / P.P. Petrov // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 143-148. — Бібліогр.: 3 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT petrovpp conductivityoflayeredstructureswithastronglydegenerateelectrongas |
| first_indexed |
2025-07-08T16:54:37Z |
| last_indexed |
2025-07-08T16:54:37Z |
| _version_ |
1837098526078664704 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 143–148
Conductivity of layered structures with
a strongly degenerate electron gas
P.P.Petrov
State University “Lvivska Politechnika”,
12 S.Bandera Str., 290646 Lviv, Ukraine
Received August 31, 1998
The conductivity of periodical multilayer structures with a degenerate elec-
tron gas is calculated. It is shown that for smooth tunnelly transparent bar-
riers the resistivity of structure coincides with the corresponding value of
the bulk material. In the presence of a random potential at the interface
between the layers, the conductivity of the system decreases with the in-
crease of a ratio of the mean free path to the period of the structure.
Key words: multilayer structure, conductivity.
PACS: 73.20.H, 82.65.Y
The study of the effect of the boundaries on the electrical conductivity of thin
films is a well-known problem of solid state physics. An interest in it is aroused
due to the latest intense investigations of multilayer periodical structures with the
conductivity of a metallic type. When considering the conductivity of spatially
restricted systems on the basis of the Boltzman equation, the problem of correct
boundary conditions for distribution functions is of essential importance.
We consider a structure consisting of many layers of the same material with
a degenerate electron gas, divided either by thin tunnelly transparent barriers or
by planes with randomly distributed scattering centres on them. This role can be
played by any defects of the interface. A similar situation is observed, for example,
in metallic superlattices or semiconductor structures with delta-doped layers. The
width of the conducting layers z0 is assumed to be large enough comparing to the
electron wavelength λ, so that the effect of size quantization could be neglected. To
calculate the conductivity of the structure we use a method based on the solution
of a kinetic equation in the layer with periodical boundary conditions. At the
interface the boundary conditions for the distribution functions are determined by
joining the electron wave functions.
Let the electric field E = (E, 0, 0) be directed along the conducting layers
plane which is perpendicular to the z-axis of the structure. The kinetic equation
for distribution functions inside each layer under a weak electric field has the form
c© P.P.Petrov 143
P.P.Petrov
eEvx
∂f0
∂ǫ
+ vz
∂f
∂z
+
f − f0
τ
= 0, (1)
and takes into account the scattering from impurities in the bulk of the layer within
the framework of the relaxation time τ approximation. Here f0 is an equilibrium
distribution function.
Let us consider first an electric current transport in the layer 0 < z < z0. The
solution of equation (1) can be written separately for carriers moving along the
axis of the structure and in the opposite direction:
f+ = f0 + C1e
−z/l − eEvxτ
(
1− e−z/l
) ∂f0
∂ ε
, (2)
f− = f0 + C2e
z/l − eEvxτ
(
1− ez/l
) ∂f0
∂ ε
.
Here l = vzτ and C1, C2 are certain functions of velocity determined by joining
conditions of the electron wave functions at z = 0 and z = z0.
In the case of an electron transport along the interface, the electric field is a
constant in the region 0 < z < z0, and the longitudinal conductivity is determined
by the mean value of the current
j̄x =
1
z0
z0
∫
0
jx (z) dz, (3)
where
jx (z) = 2e
∫
d3k
(2π)3
vx
(
f+ + f−
)
. (4)
Using formulas (2), for the integrals in (3) and (4) one obtains:
z0
∫
0
(
f̃+ + f̃−
)
dz =
= −lC̃1
(
e−z0/l − 1
)
+ l C̃2
(
ez0/l − 1
)
+ 2Evzτ (z0 − l sinh (z0/l)) , (5)
where the new constants
C1 =
(
−e
∂ f0
∂ ε
)
C̃1, C2 =
(
−e
∂ f0
∂ ε
)
C̃2 (6)
are introduced. Their values should be determined for a specific structure of inter-
faces between the layers.
Apart from the conditions of periodicity for the functions f±
f± (±0) = f± (z0 ± 0) , (7)
the conditions of joining at any interface are also imposed.
144
Conductivity of layered structures . . .
When obtaining the equations of sewing, we follow the method proposed in [1]
for a problem of electron scattering at a rough metal-vacuum interface and worked
out in [2] for the problem of conductivity of a contact of two metals.
At the interface, the carriers are quantum reflected and scattered from defects.
Accounting for elasticity of these processes, the wave function in the region 0 <
z < z0 can be written as a superposition of states on the isoenergetic surface:
Ψ1 (ε, ρ ) =
∫
dkzd
2kρ
(2π)3
a1 (kz,kρ) e
i(kzz+kρ·ρ) δ
ε −
(
k2
z + k2
ρ
)
2m
, (8)
where ρ = (x, y) is a vector perpendicular to the axis of the system. After evalu-
ating the integral over kz, we obtain the expression:
Ψ1 (ε, ρ) =
∫ d2k
(2π)2
1
v′
[
a>1ke
i k′zz + a<1ke
−i k′zz
]
eik·ρ, (9)
where the notations k ≡ kρ, k′
z =
√
2mε− k2, v′ = k′
z/m are introduced (from
here on we put h̄ = 1). The upper indices (>,<) refer to two possible signs of
k′
z. Therefore, a
>
1k and a<1k are amplitudes of the waves moving in the opposite
directions.
For the wave function in the region z0 < z < 2z0, we obtain an analogous
expression:
Ψ2 (ε, ρ) =
∫
d2k
(2π)2
1
v′
[
a>2ke
i k′
z
z + a<2ke
−i k′
z
z
]
eik·ρ. (10)
The Lagrangian of the system of two layers divided by a barrier has the form:
L =
2z0
∫
0
dz d2ρ
{
|∇Ψ1|2
2m
[1− Θ(z − z0)] −
|∇Ψ2|2
2m
Θ(z − z0) (11)
− ε |Ψ1|2 (1−Θ(z − z0)) + ε |Ψ2|2 Θ(z − z0) + V (ρ) |Ψ(z0,ρ)|2 δ (z − z0)
}
,
where V (ρ) is a scattering potential at the interface, Θ (z) is a step function,
Ψ = Ψ1 at z < z0; Ψ = Ψ2 at z > z0; Ψ = Ψ1,Ψ2 at z = z0. After a variation over
the variable Ψ in (11) we obtain:
− 1
2m
∂i (1−Θ(z − z0)) ∂iΨ1 −
1
2m
∂iΘ(z − z0) ∂iΨ2
+ δ (z − z0) V (ρ) Ψ− εΨ = 0. (12)
The integration of (12) over an infinitely small region δ → +0 near the point
z = z0 gives an equation of joining for wave functions:
1
2m
dΨ1
dz
∣
∣
∣
∣
z=z0−δ
− 1
2m
dΨ2
dz
∣
∣
∣
∣
z=z0+δ
+ V (ρ)Ψ
∣
∣
∣
∣
z0
= 0, (13)
145
P.P.Petrov
which, after substitution of formulas (9) and (10), can be rewritten in the form:
a>1k − a<1k + a<2k − a>2k − 2i
∑
q
Vq
v′
(
a>1k−q + a<1k−q
)
= 0. (14)
The second equation for the amplitudes is a consequence of the continuity
condition for wave functions Ψ1 (z0, ρ) = Ψ2 (z0, ρ) at the interfaces
a>1k + a<1k = a>2k + a<2k . (15)
The equations for distribution functions follow from their relation to the diag-
onal components of the density matrix
| a<1 |2 = vf−, | a>1 |2 = v f+, (16)
and from analogous expressions for the second region z0 < z < 2z0 (factor v in
these formulas is related to the normalization conditions).
Further we consider two cases corresponding to different model descriptions of
the interface between the conducting layers.
For the tunnelly transparent barrier (no impurities or defects at the interface
when V (r) =const), from formulas (14)–(16) there follows the result:
{
f− (z0 − 0) = Rf+ (z0 − 0) + (1−R) f− (z0 + 0)
f+ (z0 + 0) = Rf− (z0 + 0) + (1−R) f+ (z0 − 0) ,
(17)
where R (v) is a reflection coefficient at a barrier. Substituting the values of func-
tions f± from (2) in (17) and taking into account their periodicity (7), we obtain
a set of equations for the determination of the coefficients C̃1, C̃2. Solving the ob-
tained set of equations we become assured that in the absence of scattering from
the interface C̃1 = C̃2 = Evxτ . Using this value in (3)–(5), after integration we
obtain an expression for the longitudinal conductivity:
σ
(1)
|| = σ0 =
e2nτ
m
, (18)
where σ0 is a bulk conductivity of the material of the conducting layers.
In the presence of impurities or defects at the interface we assume the corre-
sponding random potential to be Gaussian:
〈V (ρ)〉 = 0, 〈V (ρ) V (ρ′)〉 = γ δ (ρ− ρ′) , (19)
where the angled brackets refer to the averaging over realizations of the random
potential. In this case the equations of sewing at the interface have the form:
f−
k (z0 − 0) = f−
k (z0 + 0)
(
1− 4πγ
v
∑
q
1
v′
)
+
+
2πγ
v
∑
q
1
v′
f+
k′ (z0 − 0) +
2πγ
v
∑
q
1
v′
f−
k′ (z0 + 0) ,
f+
k (z0 + 0) = f+
k (z0 − 0)
(
1− 4πγ
v
∑
q
1
v′
)
+ (20)
+
2πγ
v
∑
q
1
v′
f+
k′ (z0 − 0) +
2πγ
v
∑
q
1
v′
f−
k′ (z0 + 0) ,
146
Conductivity of layered structures . . .
where index k corresponds to the z-component of the momentum, whereas the
sum over q means summation over the longitudinal components of the momentum
(i.e. along the interface).
Substituting in (20) the boundary values of the distribution functions from (2)
we get for the coefficients C̃1, C̃2:
C̃1 = E lx
(
1− 2γ̃ (vF/v)
1− e−z0/l
)
, C̃2 = E lx
(
1 +
2γ̃ (vF/v)
1− ez0/l
)
, (21)
where the dimensionless scattering parameter γ̃ = 2m2γ is introduced.
Using (21) in (3)–(5) for the electric current, we obtain an expression for con-
ductivity in the presence of electron scattering from the nonidealities of the inter-
face:
σ
(2)
|| = σ0
(
1− 3 γ̃ lF
2z0
Φ (ξ)
)
, (22)
where
Φ (ξ) =
1
∫
0
(
1− y2
) (
1− e−ξ/y
)
dy =
2
3
− ξ Γ (−1, ξ) + ξ3 Γ (−3, ξ) , (23)
ξ = z0/lF,
and Γ (a, ξ) is an incomplete gamma-function [3]. Note that the function Φ (ξ) has
the following asymptotic properties: Φ (ξ) ≈ ξ ln (1/ξ) at ξ << 1, and Φ (ξ) ≈ 2/3
at ξ >> 1.
Summarizing, we would like to make several remarks on the received results.
The first of them refers to the applicability of the expressions. It was assumed
everywhere above that the dimensionless scattering parameter γ̃ is small, so that
the condition γ̃ l/z0 << 1 holds. Then, as it is seen from (18), for smooth tunnel
barriers the conductivity of the structure coincides with σ0, which is a consequence
of specular scattering at the boundaries of the conducting layer. At the same time,
the presence of impurity scattering at the interface leads to a dependence of the
structure conductivity upon the parameter l/z0. An additional factor pointing to
the nonideality of the interfaces is a change in the character of the dependence
Φ (ξ) at l/z0 ∼ 1.
147
P.P.Petrov
References
1. Falkovsky L.A. Transport phenomena at metal surfaces. // Adv. Phys., 1983, vol. 32,
No. 5, p. 753–789.
2. Dugaev V.K., Litvinov V.I., Petrov P.P. Electric-current transmission through the
contact of two metals. // Phys. Rev. B., 1995, vol. 52, No. 7, p. 5306–5312.
3. Abramowitz M., Stegun I.A. Handbook of Mathematical Functions. Moscow, Nauka,
1979 (in Russian).
Провiднiсть у багатошарових структурах з сильно
виродженим електронним газом
П.П.Петров
Державний університет “Львівська політехніка”,
290646 Львів, вул. С.Бандери, 12
Отримано 31 серпня 1998 р.
Обчислена електропровiднiсть перiодичних багатошарових структур
з виродженим електронним газом. Показано, що для гладких тунель-
но прозорих бар’єрiв опiр структури спiвпадає з вiдповiдним значен-
ням для об’ємного матерiалу. У випадку, коли на границях роздiлу iс-
нує випадковий розсiюючий потенцiал, провiднiсть системи зменшу-
ється зi збiльшенням вiдношення довжини вiльного пробiгу електро-
нiв до перiоду структури.
Ключові слова: багатошарова структура, провiднiсть.
PACS: 73.20.H, 82.65.Y
148
|