Density of one-particle states for 2-D electron gas in magnetic field
The density of states of a particle in a 2-D area is independent both of the energy and form of the area only at the region of large values of energy. If energy is small, the density of states in the rectangular potential well essentially depends on the form of the area. If the bottom of the potenti...
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Dubrovskyi, I.M. 2017-06-13T14:18:29Z 2017-06-13T14:18:29Z 2013 Density of one-particle states for 2-D electron gas in magnetic field / I.M. Dubrovskyi // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13001:1–10. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 05.30.Ch, 75.20.-g DOI:10.5488/CMP.16.13001 arXiv:1303.5206 https://nasplib.isofts.kiev.ua/handle/123456789/121068 The density of states of a particle in a 2-D area is independent both of the energy and form of the area only at the region of large values of energy. If energy is small, the density of states in the rectangular potential well essentially depends on the form of the area. If the bottom of the potential well has a potential relief, it can define the small eigenvalues as the discrete levels. In this case, dimensions and form of the area would not have any importance. If the conservation of zero value of the angular momentum is taken into account, the effective one-particle Hamiltonian for the 2-D electron gas in the magnetic field in the circle is the Hamiltonian with the parabolic potential and the reflecting bounds. It is supposed that in the square, the Hamiltonian has the same view. The 2-D density of states in the square can be computed as the convolution of 1-D densities. The density of one-particle states for 2-D electron gas in the magnetic field is obtained. It consists of three regions. There is a discrete spectrum at the smallest energy. In the intervening region the density of states is the sum of the piecewise continuous function and the density of the discrete spectrum. At great energies, the density of states is a continuous function. The Fermi energy dependence on the magnetic field is obtained when the field is small and the Fermi energy is located in the region of continuous spectrum. The Fermi energy oscillates and in the average it increases proportionally to the square of the magnetic induction. Total energy of electron gas in magnetic field also oscillates and increases when the magnetic field increases monotonously. Густина станiв частинки у 2D областi не залежить вiд енергiї i форми областi тiльки при великих значеннях енергiї. При малiй енергiї густина станiв у прямокутнiй потенцiальнiй ямi суттєво залежить вiд форми областi. Якщо дно потенцiальної ями має потенцiальний рельєф, то вiн може визначати малi власнi значення енергiї як дискретнi рiвнi. У цьому випадку розмiри i форма областi не мають значення. Якщо приймати до уваги збереження нульового значення кутового моменту, ефективний одночастинковий Гамiльтонiан для 2D електронного газу у магнiтному полi у колi є Гамiльтонiаном з параболiчним потенцiалом i вiдбиваючими границями. Припускається, що у квадратi Гамiльтонiан має такий самий вигляд. 2D густина станiв у квадратi може бути обчислена як згортка 1D густин. Обчислено густину станiв 2D еле-ктронного газу у магнiтному полi. Вона складається з трьох областей. Коли енергiї малi, спектр є дискретним. У промiжнiй областi густина станiв є сумою промiжково-неперервної функцiї i густини дискретного спектру. При великих значеннях енергiї густина станiв є неперервною функцiєю енергiї. Одержано залежнiсть енергiї Фермi вiд магнiтного поля, коли поле є слабким i енергiя Фермi знаходиться в областi неперервного спектру. Енергiя Фермi має доданок, який осцилює i, в середньому, зростає пропорцiйно квадрату магнiтної iндукцiї. Повна енергiя електронного газу у магнiтному полi також осцилює i зростає, коли магнiтне поле монотонно збiльшується. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Density of one-particle states for 2-D electron gas in magnetic field Густина одночастинкових станiв для 2D електронного газу у магнiтному полi Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Density of one-particle states for 2-D electron gas in magnetic field |
| spellingShingle |
Density of one-particle states for 2-D electron gas in magnetic field Dubrovskyi, I.M. |
| title_short |
Density of one-particle states for 2-D electron gas in magnetic field |
| title_full |
Density of one-particle states for 2-D electron gas in magnetic field |
| title_fullStr |
Density of one-particle states for 2-D electron gas in magnetic field |
| title_full_unstemmed |
Density of one-particle states for 2-D electron gas in magnetic field |
| title_sort |
density of one-particle states for 2-d electron gas in magnetic field |
| author |
Dubrovskyi, I.M. |
| author_facet |
Dubrovskyi, I.M. |
| publishDate |
2013 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Густина одночастинкових станiв для 2D електронного газу у магнiтному полi |
| description |
The density of states of a particle in a 2-D area is independent both of the energy and form of the area only at the region of large values of energy. If energy is small, the density of states in the rectangular potential well essentially depends on the form of the area. If the bottom of the potential well has a potential relief, it can define the small eigenvalues as the discrete levels. In this case, dimensions and form of the area would not have any importance. If the conservation of zero value of the angular momentum is taken into account, the effective one-particle Hamiltonian for the 2-D electron gas in the magnetic field in the circle is the Hamiltonian with the parabolic potential and the reflecting bounds. It is supposed that in the square, the Hamiltonian has the same view. The 2-D density of states in the square can be computed as the convolution of 1-D densities. The density of one-particle states for 2-D electron gas in the magnetic field is obtained. It consists of three regions. There is a discrete spectrum at the smallest energy. In the intervening region the density of states is the sum of the piecewise continuous function and the density of the discrete spectrum. At great energies, the density of states is a continuous function. The Fermi energy dependence on the magnetic field is obtained when the field is small and the Fermi energy is located in the region of continuous spectrum. The Fermi energy oscillates and in the average it increases proportionally to the square of the magnetic induction. Total energy of electron gas in magnetic field also oscillates and increases when the magnetic field increases monotonously.
Густина станiв частинки у 2D областi не залежить вiд енергiї i форми областi тiльки при великих значеннях енергiї. При малiй енергiї густина станiв у прямокутнiй потенцiальнiй ямi суттєво залежить вiд форми областi. Якщо дно потенцiальної ями має потенцiальний рельєф, то вiн може визначати малi власнi значення енергiї як дискретнi рiвнi. У цьому випадку розмiри i форма областi не мають значення. Якщо приймати до уваги збереження нульового значення кутового моменту, ефективний одночастинковий Гамiльтонiан для 2D електронного газу у магнiтному полi у колi є Гамiльтонiаном з параболiчним потенцiалом i вiдбиваючими границями. Припускається, що у квадратi Гамiльтонiан має такий самий вигляд. 2D густина станiв у квадратi може бути обчислена як згортка 1D густин. Обчислено густину станiв 2D еле-ктронного газу у магнiтному полi. Вона складається з трьох областей. Коли енергiї малi, спектр є дискретним. У промiжнiй областi густина станiв є сумою промiжково-неперервної функцiї i густини дискретного спектру. При великих значеннях енергiї густина станiв є неперервною функцiєю енергiї. Одержано залежнiсть енергiї Фермi вiд магнiтного поля, коли поле є слабким i енергiя Фермi знаходиться в областi неперервного спектру. Енергiя Фермi має доданок, який осцилює i, в середньому, зростає пропорцiйно квадрату магнiтної iндукцiї. Повна енергiя електронного газу у магнiтному полi також осцилює i зростає, коли магнiтне поле монотонно збiльшується.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121068 |
| citation_txt |
Density of one-particle states for 2-D electron gas in magnetic field / I.M. Dubrovskyi // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13001:1–10. — Бібліогр.: 7 назв. — англ. |
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AT dubrovskyiim densityofoneparticlestatesfor2delectrongasinmagneticfield AT dubrovskyiim gustinaodnočastinkovihstanivdlâ2delektronnogogazuumagnitnomupoli |
| first_indexed |
2025-11-25T22:33:24Z |
| last_indexed |
2025-11-25T22:33:24Z |
| _version_ |
1850570037484584960 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 1, 13001: 1–10
DOI: 10.5488/CMP.16.13001
http://www.icmp.lviv.ua/journal
Density of one-particle states for 2D electron gas in
magnetic field
I.M. Dubrovskyi∗
Institute for Metal Physics, 36 Vernadsky St., Kyiv 03680, Ukraine
Received July 3, 2012, in final form September 25, 2012
The density of states of a particle in a 2D area is independent both of the energy and form of the area only at
the region of large values of energy. If energy is small, the density of states in the rectangular potential well
essentially depends on the form of the area. If the bottom of the potential well has a potential relief, it can
define the small eigenvalues as the discrete levels. In this case, dimensions and form of the area would not
have any importance. If the conservation of zero value of the angular momentum is taken into account, the
effective one-particle Hamiltonian for the 2D electron gas in the magnetic field in the circle is the Hamiltonian
with the parabolic potential and the reflecting bounds. It is supposed that in the square, the Hamiltonian has
the same view. The 2D density of states in the square can be computed as the convolution of 1D densities. The
density of one-particle states for 2D electron gas in the magnetic field is obtained. It consists of three regions.
There is a discrete spectrum at the smallest energy. In the intervening region the density of states is the sum
of the piecewise continuous function and the density of the discrete spectrum. At great energies, the density
of states is a continuous function. The Fermi energy dependence on the magnetic field is obtained when the
field is small and the Fermi energy is located in the region of continuous spectrum. The Fermi energy has the
oscillating correction and in the average it increases proportionally to the square of the magnetic induction.
Total energy of electron gas in magnetic field also oscillates and increases when the magnetic field increases
monotonously.
Key words: density of states, electron gas, magnetic field, energy spectrum, Fermi energy, total energy
PACS: 05.30.Ch, 75.20.-g
1. Introduction
The spectrum of one-particle Hamiltonian in a bounded area is discrete. When values of energy are
more than maximum of potential energy (in the fourth section this criterion will be improved), the dis-
tances between levels are of the order ħ2/2mW 2/D . Here, m is themass of the particle,W is the volume of
the area, and D is the dimensionality of space. When the volume of the area is macroscopic, the spectrum
may be considered as quasicontinuous.Then, the density of states ND (E ) may be introduced:
ND (E )= ND (E +δE )−ND (E )
δE
. (1)
Here, ND (E ) is the number of states, energy eigenvalues of which are less than E , δE is a small interval
of energy which is larger than the distances between discrete levels. Function ND (E ) was considered
mathematically rigorously in the monograph [1]. The Schrödinger equation was multiplied by 2m/ħ2,
and the eigenvalue ε= 2mE/ħ2 has the dimension of physical quantity [L−2] (for brevity let us also refer
to it as “energy”). It was shown that the asymptotical formulae at ε→∞ for functions ND (ε) are:
N2(ε) < S
4π
ε+θc
p
ε, N3(ε) < V
6π2
ε3/2 +θcε, (2)
∗E-mail: dubrovskii@voliacable.com
© I.M. Dubrovskyi, 2013 13001-1
http://dx.doi.org/10.5488/CMP.16.13001
http://www.icmp.lviv.ua/journal
I.M. Dubrovskyi
when the wave functions are equal to zero at the boundaries of the area. Here, |θ| < 1 and c is indepen-
dent of ε. It is important that the first terms of these formulae are independent of the form of the area.
The functions ND (ε) take only integer values. If this is neglected when energy is great, the formula (1)
can be considered as ND = ∂ND /∂ε. Then, we can obtain from the formulae (2):
N2 =
S
4π
, N3 =
V
4π2
p
ε, (3)
if only the first terms of the asymptotical expansions are taken into account. Evidently, extrapolation of
the formulae (3) to small values of energy is incorrect even when there is no potential energy.
In the second section of this work, the density of states at small energy is considered in the absence
of a potential energy.
In the third section of this work the 2D density of states is considered in the potential well that is
created by the harmonic potential and the reflecting boundaries. As is shown in the work [2], this po-
tential takes place in the effective one-particle Hamiltonian of the electron gas in the magnetic field. The
dependencies on the magnetic field are investigated for the Fermi energy and for the total energy of the
gas.
The fourth section is a mathematical supplement. The problem on the linear harmonic oscillator with
condition zeroes of wave function at the segment ends is considered.
2. The density of states in a square
The derivation of the formula (2) for N2(ε) starts from consideration of the square with S = L2
(see monograph [1]). The states are determined by two integer numbers. The energy of state |kl〉 is
εkl = π2(k2 + l 2)/L2. Then, N2(ε) will be equal to the number of junctions of the net of squares that are
parallel to the coordinate axes and have the sides equal to unit, which fit into the interior of the positive
quadrant of the circle with radius (L/π)
p
ε. This quantity differs from the area of the quadrant Sε/4π by
the sum of areas of partial squares that are crossed by the circle. The second term in the formula (2) is
the approximate estimate of this amendment. The relative magnitude of this amendment will be smaller,
when the quadrant radius is larger.
Let us consider the other method of calculating the state density in the quadratic area that can be
used for small energy values too. The 2D Schrödinger equation for a free particle, provided that the wave
function is equal zero at the boundaries of the square, can be changed by two identical 1D equations. An
eigenvalue of the 2D equation is the sum of eigenvalues of 1D equations. Therefore, let us consider the
state density for the 1D equation.
The number of states, whose eigenvalues are less than ε for 1D equation, is as follows: N1(ε) =
⌈L
p
ε/π⌉. Here, ⌈α⌉ denotes the integer part of the number α. The density of states that is determined by
formula (1) is an interval function rather than a point function. The magnitude of the interval cannot be
taken arbitrarily small. In 1D, this magnitude is limited by the demand that one eigenvalue should be in
the interval at the greatest energy εm. Then,
L
π
(√
εm +δ− p
εm
)
Ê 1, δÊ
(π
L
)2
+ 2π
L
p
εm ≈ 2π
L
p
εm . (4)
In most cases, the density of states is used in integral formulae. Then, an interval function can be changed
by a piecewise continuous stepped function or a continuous differentiable function that is determined by
any method of interpolation. It is apparent that the consideration of peculiarities of the state density is
meaningless.
Let us determine the state density for the square at the values of energy that are smaller than εm. The
interval δ is determined by the formula (4), and it is accepted as the unity of energy. Non-dimensional
(ε/δ) eigenvalues of energy for 1D equation are denoted as λ1 and λ2, and non-dimensional eigenvalue of
energy for 2D equation is denoted as µ, µ=λ1+λ2. Let us consider the intervals [λ1,λ1+1] and [λ2−1,λ2]
where λ2 = µ−λ1. The eigenfunctions of the 2D equation that are products of the eigenfunctions 1D
equations, which are related to these intervals, have the eigenvalues that are located at interval [µ−1,µ+
1]. The number of these states is denoted as M2(µ−1,µ+1). In a similar way, the eigenvalues of the 2D
13001-2
Density of one-particle states for 2D electron gas in magnetic field
equation that are located at the interval [µ,µ+2] are obtained when λ2 =µ−λ1 +1. Then, the number of
the eigenvalues of the 2D equation at the interval [µ,µ+1] (when µÊ 2) is:
M2(µ,µ+1) =
1
2
µ−1∑
λ=0
M1(λ,λ+1)M1(µ−λ−1,µ−λ)
+ 1
2
µ∑
λ=0
M1(λ,λ+1)M1(µ−λ,µ−λ+1). (5)
It follows from the formula (4) that:
M1(λ,λ+1) = L
p
δ
π
(p
λ+1−
p
λ
)
. (6)
Then, M2(µ,µ+1) = (L2δ/2π2)Σ(µ), where
Σ(µ) = −
µ−1∑
i=1
(p
i
)(√
µ− i +1−
√
µ− i −1
)
+
µ−1∑
i=0
(p
i +1
)(√
µ− i +1−
√
µ− i −1
)
+
√
µ+1− p
µ. (7)
This function of integer argument can be written as:
Σ(µ) = S(µ+1)−S(µ)−S(µ−1)+S(µ−2), (8)
where
S(x) =
x−1∑
n=0
√
(n+1)(x −n), x Ê 1; S(0) = 0. (9)
The function S(x) can be computed using Euler-Maclaurin method:
S(x)=
(
x +1
2
)2
arcsin
x −1
x +1
+ x
p
x − x −1
12
p
x
. (10)
Based on the approximate formula
arcsin
x −1
x +1
=
π
2
−
2
p
x +1
−
1
3(x +1)3/2
+
9
4(x +1)5/2
, (11)
the asymptotical formula
Σ(µ) ≈ π
2
+ 3
4
p
µ
(12)
can be obtained. The function Σ(µ) can be calculated numerically. The results are obtained from the
formula (7) and from the asymptotical formula (12) tabulated in table 1. The values of the function Σ(µ)
approach π/2 = 1.57096 from above.
Table 1. The function Σ(µ) that is numerically calculated [formula (7)], and calculated using an asymptot-
ical formula (12).
µ 1 10 100 1000 10,000 100,000 1,000,000
formula (7) 1.81 1.577 1.571 1.5708 1.570797 1.570796
formula (12) 1.65 1.59 1.578 1.573 1.5715
For sufficiently large value µ the density of states in a square is described by formula:
N2(ε=µδ) = M2(µ,µ+1)
δ
= L2
2π2
Σ(µ) ≈ L2
4π
+ 3L2
p
δ
8π2
p
ε
. (13)
13001-3
I.M. Dubrovskyi
The first term in this formula coincides with the common expression for 2D system. It can be obtained by
differentiation N2(ε) [formula (2)], where S = L2. This formula can be used for derivation of the density of
states for a flat geometrical figure of arbitrary shape (see monograph [1]). In this process, L2 is changed
by the figure area S and amendments are proportional to ε−1/2, i.e., they alter the second term in the
formula (13). Therefore, the second term depends on the figure shape and on the interval magnitude δ.
The sign of the second term also depends on the figure shape. In the square this term is positive,
i.e., when the energy increases, the density of states decreases tending to the constant value from above.
This is explained by the fact that in 1D, the state density increases when the energy decreases. The eigen-
functions in a circle are the Bessel functions of the first kind Jn(r
p
ε). The eigenvalues in this case are
εnk = j 2
nk
/R2 where R is the circle radius, and jnk is the null of the function Jn that has the number k in
the order of increasing. There is no formula that describes these nulls when their numbers k are small,
but it is known that the distances between nulls increase when their numbers decrease. Therefore, the
density of states should decrease when the energy decreases, and the amendment should be negative.
In fact the spectrum at small energy values is formed by the potential relief of the bottom of the poten-
tial well. The distances between energy levels are determined by parameters of this relief. Therefore, this
spectrum cannot be considered as quasicontinuous. The density of states in this case can be described by
the set of δ-functions. By virtue of the fact that the determining factor is the potential relief, it is believed
that the figure shape is of no significance. Then, it may be helpful to obtain the state density for the square
in the whole region of energy values.
If the 2D Schrödinger equation with the potential energy can be solved by separating the variables in
the Cartesian coordinates, then the 2D density of states can be obtained. Every interval [ε1,ε1+dε] on the
axis of energy of 1D states contains N1(ε1)dε states, whose wave functions are ψ(x1). Products of these
functions with the wave functions ψ(x2) that have the energy values in the interval [ε2,ε2 +dε], where
ε2 = ε− ε1, are the wave functions of the 2D states, the energies of which are in the interval [ε,ε+dε].
The number of these states connected with the energy value ε1 is:
dM2(ε,ε1) =N1(ε1)N1(ε−ε1)dεdε. (14)
Then, the density of states in the square is:
N2(ε) = dM2(ε)
dε
=
ε∫
0
N1(α)N1(ε−α)dα, (15)
i.e., it is a convolution of 1D densities of states.
The form of the spectrum in the region of small energy values can play a significant role depending
on the form of the potential relief of quantities that are determined by integral formulae.
3. The density of states and energy of 2D electron gas in the magnetic
field
It is shown in the work [2] that the statistical operator of the electron gas in the magnetic field is de-
fined by effective Hamiltonian that is the sum of the same one-particle Hamiltonians. Each one-particle
Hamiltonian describes the particle in the potential well with a harmonic potential and reflecting bound-
aries. Electrons interact with each other and with the neutralizing background. The electron density in
the magnetic field should be distributed in such a way as to shield the harmonic potential. It is shown
in the work [2] that this shielding in the circle with radius R leads to renormalization of the electron
charge er ∽ e
p
a0/R , where (−e) is the electron charge, a0 is the Bohr radius. The residual harmonic po-
tential is proportional to ω2 where ω = erH/m is the cyclotron frequency, H is the magnetic induction,
m is the electron mass.
Let us suppose that in the square with the side 2L and zero of coordinate system in the center, the
effective one-particle Hamiltonian with the symmetrical gauge also has the view:
ĥ =−
ħ2
2m
(
∂2
∂x2
+
∂2
∂y2
)
+
mω2
8
(
x2 + y2
)
. (16)
13001-4
Density of one-particle states for 2D electron gas in magnetic field
Here, er = e
p
a0/L. Separating the variables and multiplying by 2m/ħ2, we obtain two identical 1D equa-
tions:
ψ′′+ 2mEν
ħ2
ψ−
m2ω2x2
ν
4ħ2
ψ= 0, ν= 1,2. (17)
The boundary conditions are:
ψ(±L) = 0. (18)
The problem on a linear oscillator with the boundary condition (18) is considered in the fourth section.
In this case, the 1D density of states is as follows:
N1(ε) =
n0∑
n=0
δ(ε−εn )+Θ(ε−εb)
L
π
p
ε
,
εn = ∆
(
n+ 1
2
)
, εb =∆
(
n0 +
1
2
)
, n0 ≈
⌈
4mωL2
π2ħ
⌉
, ∆= mω
ħ
. (19)
Here, the first term is the spectrum of the linear oscillator without reflecting boundaries. (We change n+
γ(n) to n because γ(n) are practically everywhere small). For ε> εb, the density of states is the density of
quasicontinuous spectrum. Boundarymagnitude n = n0 is determined approximately, but it is significant
that n0 is proportional to the magnetic induction and it is an integer.
The density of states of Hamiltonian (16) can be obtained by the formula (15). It consists of three
regions. At the smallest energy, the spectrum is discrete:
N2d(ε) =Θ
(
εb +
∆
2
−ε
)n0+1∑
n=1
nδ(ε−∆n). (20a)
In the intervening region, the density of states is the sum of a piecewise continuous function and the
density of a discrete spectrum:
N2pc(ε) = Θ
(
ε−
∆
2
−εb
)
Θ(2εb −ε)
×
{
2n0+1∑
n=n0+2
(2n0 +2−n)δ(ε−∆n)+ 2L
π
nε∑
i=0
[
ε−∆
(
i + 1
2
)]−1/2
}
,
nε =
⌈ ε
∆
⌉
− (n0 +1). (20b)
In the region of great energy values, the density of states is a continuous function:
N2c (ε) =Θ(ε−2εb)
{
2L
π
n0∑
i=0
[
ε−∆
(
i + 1
2
)]−1/2
+ 2L2
π2
arcsin
(
1− 2εb
ε
)}
. (20c)
These formulae are illustrated in figure 1. Let us represent the states by the points at the first quadrant.
The Cartesian coordinates of the point are the eigenvalues of 1D Hamiltonians that are the components of
the 2D Hamiltonian. The density of states in the point is the product of 1D densities in the projections of
the point multiplied by dε [see the derivation of formula (15)]. All the states with the same energy ε =α
are located at the line segment that cuts off the intercepts equal to α on the axes. The 2D density of states
that have the energy α is the sum of the densities over all points of this line segment. The coordinates
of the states that are located in the square Oε1bDε2b or on its sides are as follows: εi = ∆(ni + 1/2).
Therefore, the states that have energy eigenvalue εÉ εb+∆/2 (for example the point A) form the discrete
spectrum [formula (20a)]. The degeneracy multiplicity of the discrete state with energy εn = n∆, (n =
1, 2, . . . n0 + 1) is equal to n. These degenerate levels are transformed in zonule, if the quantities γ(n)
are taken into account, but this broadening is small everywhere except the immediate neighborhood
of εb. Among the states with energy εb +∆/2 < ε < 2εb [the section b1b2, formula (20b)] there should
be such ones that are located outside the square Oε1bDε2b . One of the projections of such a state is
located in the region of the continuous 1D spectrum (the point B). Therefore, the formula (20b) consists
of a discrete part and a piecewise continuous part. The degeneracy multiplicity of the discrete state with
13001-5
I.M. Dubrovskyi
Figure 1. The illustration of calculation of the density of states for 2D electron in the magnetic field in a
square.
energy εn = n∆, (n = n0+2, . . . 2n0+1) is equal to 2n0+2−n. When the state energy ε> 2εb [the section
c1c2, formula (20c)] the point C ′ is similar to the point B , and for the point C , the both projections are
located in the regions of a continuous 1D spectrum.
Let us consider the weak field case, when the Fermi energy εF > 2εb. If the density of states is deter-
mined by averaging over the interval ∆, and the amendments that take the form of the area into account
are neglected, the density of states �N2(ε) should be like the one obtained in the work [2]:
�N2(ε) =Θ
(
m2ω2S
4πħ2
−ε
)[ ε
∆2
]
+Θ
(
ε− m2ω2S
4πħ2
)[
S
4π
]
. (21)
The integral of the density of states that determines the number of states whose energies are smaller than
ε, can be represented as follows:
N (ε> 2εb) =
ε∫
0
�N2(α)dα+
ε∫
0
[
N2(α)− �N2(α)
]
dα, (22)
where N2(α) is taken from the formulae (20a, 20b, 20c). The equation for calculation of the Fermi energy
εF is obtained by equating N (εF) to the total number of electrons N0. (For the sake of simplicity, the
spin and the Pauli paramagnetism is not considered). This equation can be considered as the implicit
definition of the function εF(H). If the second term in the formula (22) is neglected, then:
εF(H) = εF0 +εF1(H) = εF0 +
1
2π
(
mωL
ħ
)2
= εF0 +
1
2π
(L∆)2 , (23)
13001-6
Density of one-particle states for 2D electron gas in magnetic field
where εF0 = 4πN0/S is the Fermi energy in the commonly used theory when there is no magnetic field.
The Fermi energy depends on the magnetic field due to the dependence on the magnetic induction of
two parameters: εF(H) = εF [∆(H),n0(H)]. These dependencies are qualitatively different: ∆(ω) = mω/ħ
is the linear function of H that can take on any values, and n0(ω) =
⌈
4mωL2/π2ħ
⌉
can be only integer.
Therefore, when the magnetic induction varies in an interval, in which the parameter n0 does not vary,
the singular points of the function N2(ε) [formulae (20a)–(20c)]move continuously and εF(H) = εF [∆(H)]
varies continuously. When the variation of themagnetic induction changes the parameter n0, the number
of the singular points changes, the spectrum reconstructs, and εF varies non-continuously. Formula (23)
that describes the function εF(H) when the density of states is smoothed, should be supplemented by the
oscillatory term εF2(∆,n0). By integration N2(α) between the limits 0 and εF > 2εb we obtain:
N (εF > 2εb) = n2
0 −
4L
p
∆
π
[
n3/2
0 −
n0∑
i=0
(εF
∆
− i
)1/2
]
+ 2L2
π2
[
εF arcsin
(
1− 2∆n0
εF
)
−2
√
∆n0(εF −∆n0)+2∆n0
]
. (24)
If n0(H)−n0(H ±h) = ±1 then ∆(H)−∆(H ±h) =±(π2/4L2). Then, the continuous change of εF on this
interval is:
δcεF = ∂εF
∂∆
(
± π2
4L2
)
=−∂N
∂∆
(
∂N
∂ε
)−1 (
± π2
4L2
)
=− 1
N2(εF0)
∂N
∂∆
(
± π2
4L2
)
≈−
π
L2
(
−
n2
0
∆
√
εF0
∆n0
)(
±
π2
4L2
)
≈±
2
L
p
εF0 =±
2
L2
√
πN0 . (25)
This change is much larger than the change that is described by the formula (23). It describes the oscilla-
tions of εF2(∆,n0). At the end of the interval of the continuous change of εF2(∆,n0), when n0 changes by
±1, the jump of the function εF is:
δ j εF =±
∂εF
∂n0
=∓
∂N
∂n0
(
∂N
∂εF
)−1
=∓
π
L2
∂N
∂n0
≈∓
π
L2
(n0εF0
∆
)
=∓
2
L
p
εF0 . (26)
When the magnetic field varies, the Fermi energy oscillates with the amplitude (ħ/L)
p
2EF0/m and the
period δH = π2ħ/4L2er. If L2
∽ 10−4 m2 and er ∽ 10−4e , this period is of the order of 10−6 T. The
monotonous change of the Fermi energy with the magnetic field is described by the formula (23).
It can be similarly proved that the energy of the electron gas in the magnetic field is described by
formula:
E = E0 +
S2ρ0e2
r H 2
16πm
−
S3e4
r H 4
1536π3ħ2m
+Eosc . (27)
Here, E0 is the energy of the electron gas in the absence of the magnetic field, ρ0 = N0/S is the density of
gas. This formula differs from the energy of gas that was calculated in the work [2] by an ultimate term
Eosc that describes the oscillation of the energy with the amplitude (4/3π)
√
(2mL2/ħ2)E 3
F0
.
We take into account that the number of discrete levels n0 is an integer and obtain the characteristics
of the gas that have the oscillating dependence on the magnetic field. In the commonly used theory, the
degenerate multiplicity of equidistant levels d(H) is an integer and is the same for all levels. This quantity
differs from n0 only by the numerical coefficient that is of the order of unity. When the magnetic field,
for example, increases, the gas energy should increase linearly until d is constant. When d increases by
1, the ⌈N0/d⌉ electrons should drop from the top level and the energy decreases by jumps. If the number
of electrons on the top level is less than ⌈N0/d⌉, the jump amplitude should be smaller. These decreased
jumps should recur, and only this oscillation is considered in the common theory. The 2D electron gas
in the magnetic field has been considered in the monographs [3, 4]. The fact that d(H) is an integer
is not taken into account in these works. In the work [2] it was shown that the system of equidistant
degenerated levels cannot be a correct description of the one-particle spectrum of the electron gas in the
magnetic field because in this theory, the angular momentum conservation and the Coulomb interaction
are not taken into account.
13001-7
I.M. Dubrovskyi
4. The linear oscillator with zero boundary condition
Let us change the variable in the equation (17) z = x
p
∆ and designate u =−2E/ħω. Then, the equa-
tion obtains the form of a standard equation for the function of parabolic cylinder (see handbook [5]):
d2ψ
dz2
−
(
z2
4
+u
)
ψ= 0. (28)
The even and odd solutions of this equation are:
ψe = Ae exp
(
− z2
4
)
Φ
(
u
2
+ 1
4
,
1
2
;
z2
2
)
;
ψo = Aoz exp
(
− z2
4
)
Φ
(
u
2
+ 3
4
,
3
2
;
z2
2
)
. (29)
Here, Φ(a,c; t) is the degenerate hypergeometric function (DHF), Ae(o) are the normalization constants.
The middle of the line segment is zero of the coordinate. The length of the segment is 2L. To satisfy the
boundary condition (18), the eigenvalue u should be such that
Φ
(
u
2
+
1
4
,
1
2
; z2
L
)
= 0, or Φ
(
u
2
+
3
4
,
3
2
; z2
L
)
= 0; z2
L =
mωL2
2ħ
. (30)
This problem could not be consequently considered because in the description of nulls of DHF in all
mathematical handbooks (see for example [5],[6]) an inaccuracy takes place. It is proved that, if a < 0 and
c > 0, the number of nulls of the function Φ(a,c; t) is equal to (−a), if it is integer, and is equal to ⌈−a⌉+1,
if (−a) is non-integer. It is also proved that nulls are described approximately by the formula:
ξi (a,c) = 1
2c −4a
j 2
c−1,i , (31)
if |a| ≫ 1. Here, ξi (a,c) is the null of DHF that has the number i É ⌈−a⌉+1 in the order of increasing,
j 2
c−1,i
is the square of the respective null of the Bessel function of the first kind Jc−1(x). If this were so, the
greatest nulls of the functions that are the solutions of the considered problem should be linear functions
of the eigenvalue u. Then, the smallest eigenvalues should be∼ mωL2/2ħ, i.e., the spectrum should begin
from very large values of energy. In the work [7] (see also [2]) it was shown that the formula (31) describes
each null of the DHF only when (−a) is integer. Then, the Kummer power series that describes DHF
terminates at the term with number 1− a, and the DHF in the formulae (30) are proportional to the
Laguerre polynomials L±1/2
(−a)
(z2
L). The number of nulls of these polynomials is k =−a. All these nulls are
real, positive, simple and are described by the formula (31). Let us consider the DHF Φ(a,1/2; t) when
a = −k −γ, where k is an integer and 0 < γ < 1. Then, this DHF has k +1 nulls of which k nulls come
out of the nulls of Laguerre polynomial that are diminished by quantities which are proportional to γ.
These nulls are described by the formula (31). The null that has the number k +1 is the largest, and its
dependence on γ should possess the following properties:
lim
γ→0
[
ξk+1(−k −γ,1/2)
]
→∞, lim
γ→1
[
ξk+1(−k −γ,1/2)
]
= ξk+1(−k −1,1/2). (32)
To calculate ξk+1(−k−γ,1/2) when γ is small, the DHFΦ(−k−γ,1/2; t) should be changed by the asymp-
totic expression. The Kummer power series would be represented as follows:
Φ(−k −γ,1/2; t) = P 1/2
k (γ, t)−γQ1/2
k (γ, t)−γ(1−γ)T 1/2
k (γ, t). (33)
Here, P 1/2
k
(γ, t) is a polynomial that can be obtained from the Laguerre polynomial L−1/2
k
(t) by changing
k → k +γ in its coefficients, γQ1/2
k
(γ, t) is the next term of the Kummer series that is proportional to γ,
γ(1−γ)T 1/2
k
(γ, t) is the remaining infinite series that also is proportional to γ. It can be shown that when
t is large and γ is small
γ(1−γ)T 1/2
k (γ, t) ≈ γ(−1)k (1+γ)k
p
π(t)−n−1/2 exp(t). (34)
13001-8
Density of one-particle states for 2D electron gas in magnetic field
Here, (b)i = b(b +1) · · · (b + i −1) is the Pochhammer symbol. The second term in the formula (33) can be
neglected. The polynomial P 1/2
k
(γ, t) can be changed by the last term, when t is large:
P 1/2
k (γ, t)≈
(−k −γ)k t k
(1/2)k k!
= (−1)k (1+γ)k t k
(1/2)k k!
. (35)
We set the obtained approximate expression for DHF equal to zero. This equation defines the quantity γ
as the function of ξk+1(−k −γ,1/2) = t0:
γ1/2(k, t0) =
[
k!Γ
(
k + 1
2
)]−1
t 2k+1/2
0 exp(−t0). (36)
For the boundary condition to be satisfied by the largest null of DHF Φ(−k −γ,1/2; t), this null should be
larger than ξk+1(−k −1,1/2), i.e.,
t0 =
mωL2
2ħ
> 1
1+4(k +1)
j 2
−1/2,k+1 , (37)
j−1/2,l = π(l −1/2), where l is integer. Therefore, the boundary condition can be satisfied by the largest
null of DHF, if the value of k is not greater than k0:
k0 ≈
⌈
2mωL2
π2ħ
⌉
. (38)
This is the approximate formula, but the operation of taking an integer part emphasizes that k0 is an
integer, and when the frequency changes, this quantity does not change continuously and takes only
integer values. The eigenvalues for the even solutions:
Ee =ħω
(
ke +
1
4
+γ1/2(ke)
)
. (39)
If k > k0, the boundary condition can be satisfied by one of the nulls of DHF that is described by the
formula (31) when (−a) > k0. To calculate these values a, i.e., the eigenvalues of energy, we use the
first term of the expansion DHF over the Bessel functions (see monograph [6]). This expansion is rapidly
convergent, when |a| is large. We obtain the following even solution of the equation (28):
ψe ≈ AeΓ
(
1
2
)(
−uz2
4
)1/4
J−1/2
(
2
√
−uz2/4
)
= Ae cos
(
z
√
2Ee/ħω
)
= Ae cos
( x
ħ
√
2mEe
)
. (40)
The eigenvalues of the energy for which these wave functions are equal to zero at the ends of a segment
are obtained:
Ee =
π2ħ2
2mL2
(
ke +
1
2
)2
= π2ħ2
8mL2
(2ke +1)2, ke > k0 . (41)
A similar computation for the odd wave function leads to the results:
γ3/2(k, t0) =
[
k!Γ
(
k + 3
2
)]−1
t 2k+3/2
0 exp(−t0), Eo =ħω
(
ko +
3
4
+γ3/2(ko)
)
; (42)
ψo ≈ Ao sin
( x
ħ
√
2mEo
)
, Eo =
π2ħ2
2mL2
k2
o = π2ħ2
8mL2
(2ko)2, ko > k0 .
The formulae for the spectrum can be unified, if it is taken into account that the amendments γ can be
neglected everywhere except the immediate neighborhood of k0:
En =
ħω
2
(
n+
1
2
)
, n < n0 = 2k0; En =
π2ħ2
8mL2
n2, n > n0 . (43)
If in the equation (17) we change ω = 2ω0, this equation will gain the form of a standard equation for
the linear oscillator with frequency ω0. If L → ∞, the boundary condition (18) should be changed by
the requirement of normability of the wave functions. Then, the obtained solution of the problem turns
into the common solution for the linear oscillator. When n > n0, the approximate solutions (40) and (42)
are the common solutions for the particle in the rectangular potential well. However, the energy Eb =
ħωn0/2 = 2mω2L2/π2 that is the boundary between the two kinds of solutions does not coincide with the
value of the potential energy at the boundary of the area as might be expected from the quasiclassical
consideration.
13001-9
I.M. Dubrovskyi
References
1. Courant R., Hilbert D., Methods of Mathematical Physics, Vol. 1, Interscience, New York, 1953.
2. Dubrovskyi I., In: Thermodynamics. Interaction Studies — Solids, Liquids and Gases, Moreno-Pirajan J.C. (Ed.),
InTech, 2011, Chap. 17; doi:10.5772/823.
3. Abrikosov A.A., Introduction to the Theory of Metals, Amsterdam, North-Holland, 1986.
4. Shoenberg D., Magnetic Oscillations in Metals, Cambridge University Press, Cambridge, 1984.
5. Handbook ofMathematical Functions with Formulas, Graphs, andMathematical Tables. National Bureau of Stan-
dards Applied Mathematics Series Vol. 55, Abramovitz M., Stegun I.A. (Eds.), U.S. Government Printing Office,
Washington, D.C., 1964.
6. Bateman H., Erdèlyi A., Higher Transcendental Functions, Vol. 1, Robert E. Krieger, 1981.
7. Dubrovskii I.M., Condens. Matter Phys., 2006, 9, 645.
Густина одночастинкових станiв для 2D електронного газу у
магнiтному полi
I.М. Дубровський
Iнститут металофiзики, бульв. Вернадського 36, Київ 03680, Україна
Густина станiв частинки у 2D областi не залежить вiд енергiї i форми областi тiльки при великих значен-
нях енергiї. При малiй енергiї густина станiв у прямокутнiй потенцiальнiй ямi суттєво залежить вiд форми
областi. Якщо дно потенцiальної ями має потенцiальний рельєф, то вiн може визначати малi власнi зна-
чення енергiї як дискретнi рiвнi. У цьому випадку розмiри i форма областi не мають значення. Якщо
приймати до уваги збереження нульового значення кутового моменту, ефективний одночастинковий
Гамiльтонiан для 2D електронного газу у магнiтному полi у колi є Гамiльтонiаном з параболiчним потен-
цiалом i вiдбиваючими границями. Припускається, що у квадратi Гамiльтонiан має такий самий вигляд.
2D густина станiв у квадратi може бути обчислена як згортка 1D густин. Обчислено густину станiв 2D еле-
ктронного газу у магнiтному полi. Вона складається з трьох областей. Коли енергiї малi, спектр є дискре-
тним. У промiжнiй областi густина станiв є сумою промiжково-неперервної функцiї i густини дискретного
спектру. При великих значеннях енергiї густина станiв є неперервною функцiєю енергiї. Одержано за-
лежнiсть енергiї Фермi вiд магнiтного поля, коли поле є слабким i енергiя Фермi знаходиться в областi
неперервного спектру. Енергiя Фермi має доданок, який осцилює i, в серед+ньому, зростає пропорцiйно
квадрату магнiтної iндукцiї. Повна енергiя електронного газу у магнiтному полi також осцилює i зростає,
коли магнiтне поле монотонно збiльшується.
Ключовi слова: густина станiв, електронний газ, магнiтне поле, енергетичний спектр, енергiя Фермi,
повна енергiя
13001-10
http://dx.doi.org/10.5772/823
Introduction
The density of states in a square
The density of states and energy of 2D electron gas in the magnetic field
The linear oscillator with zero boundary condition
|