Electron structure of topologically disordered metals
Here two methods for calculating the density of states of electrons in conduction band of disordered metals are investigated. The first one is based on the usage of one-parameter trial electron wave function. The equation for density of states gotten within this method is more general as compared...
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irk-123456789-1197492017-06-09T03:05:05Z Electron structure of topologically disordered metals Yakibchuk, P. Here two methods for calculating the density of states of electrons in conduction band of disordered metals are investigated. The first one is based on the usage of one-parameter trial electron wave function. The equation for density of states gotten within this method is more general as compared to the results of perturbation theory. Electron-ion interaction is applied in the form of electron-ion structure factor, which makes it possible to use this method for a series of systems where potential form factor is not a small value and the perturbation theory fails. It also gives us well-known results of Relel-Schrodinger and Brilliuen-Vigner perturbation theory in case of small potential. Basically, the second approach is a common perturbation theory for pseudo-potential and Green’s function method. It considers the contributions up to the third order. The results of computation for density of states in some non-transition metals are presented. The deviation of density of states causing the appearance of pseudo-gap is clearly recognized. В даній роботі розглядаються два різних підходи до визначення густини станів та енергетичного спектру електронів провідності у невпорядкованих металах. Перший з них грунтується на варіаційному принципі з використанням однопараметричної пробної хвильової функції електронів провідності. Для енергетичного спектру отримано рівняння, яке має більш загальний вигляд у порівнянні з результатами теорії збурень. Електрон-іонна взаємодія входить в теорію через електрон-іонний структурний фактор, що дає змогу застосувати теорію і в тих випадках, коли формфактор потенціалу не є малою величиною і теорія збурень не може бути застосована. Якщо формфактори екранованого потенціалу є малими, то із виведеного варіаційного виразу в часткових випадках отримуються відомі результати теорії збурень Релея-Шредінгера та Бріллюена-Вігнера. Другий підхід пов’язаний з використанням методу функцій Гріна та стандартної теорії збурень за псевдопотенціалом з урахуванням членів до третього порядку включно. Для ряду металів виконані чисельні розрахунки густини електронних станів. Виявлено помітне відхилення відносно вільноелектронного наближення на залежностях густини станів від енергії, що обумовлює появу псевдощілини. 2005 Article Electron structure of topologically disordered metals / P. Yakibchuk // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 537–546. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 43.38.Kw DOI:10.5488/CMP.8.3.537 http://dspace.nbuv.gov.ua/handle/123456789/119749 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
Here two methods for calculating the density of states of electrons in conduction
band of disordered metals are investigated. The first one is based
on the usage of one-parameter trial electron wave function. The equation
for density of states gotten within this method is more general as compared
to the results of perturbation theory. Electron-ion interaction is applied in
the form of electron-ion structure factor, which makes it possible to use this
method for a series of systems where potential form factor is not a small
value and the perturbation theory fails. It also gives us well-known results
of Relel-Schrodinger and Brilliuen-Vigner perturbation theory in case of
small potential. Basically, the second approach is a common perturbation
theory for pseudo-potential and Green’s function method. It considers the
contributions up to the third order. The results of computation for density of
states in some non-transition metals are presented. The deviation of density
of states causing the appearance of pseudo-gap is clearly recognized. |
| format |
Article |
| author |
Yakibchuk, P. |
| spellingShingle |
Yakibchuk, P. Electron structure of topologically disordered metals Condensed Matter Physics |
| author_facet |
Yakibchuk, P. |
| author_sort |
Yakibchuk, P. |
| title |
Electron structure of topologically disordered metals |
| title_short |
Electron structure of topologically disordered metals |
| title_full |
Electron structure of topologically disordered metals |
| title_fullStr |
Electron structure of topologically disordered metals |
| title_full_unstemmed |
Electron structure of topologically disordered metals |
| title_sort |
electron structure of topologically disordered metals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119749 |
| citation_txt |
Electron structure of topologically disordered metals / P. Yakibchuk // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 537–546. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT yakibchukp electronstructureoftopologicallydisorderedmetals |
| first_indexed |
2025-07-08T16:31:55Z |
| last_indexed |
2025-07-08T16:31:55Z |
| _version_ |
1837097096270839808 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 3(43), pp. 537–546
Electron structure of topologically
disordered metals
P.Yakibchuk
Ivan Franko Lviv National University
Received March 29, 2005, in final form June 16, 2005
Here two methods for calculating the density of states of electrons in con-
duction band of disordered metals are investigated. The first one is based
on the usage of one-parameter trial electron wave function. The equation
for density of states gotten within this method is more general as compared
to the results of perturbation theory. Electron-ion interaction is applied in
the form of electron-ion structure factor, which makes it possible to use this
method for a series of systems where potential form factor is not a small
value and the perturbation theory fails. It also gives us well-known results
of Relel-Schrodinger and Brilliuen-Vigner perturbation theory in case of
small potential. Basically, the second approach is a common perturbation
theory for pseudo-potential and Green’s function method. It considers the
contributions up to the third order. The results of computation for density of
states in some non-transition metals are presented. The deviation of den-
sity of states causing the appearance of pseudo-gap is clearly recognized.
Key words: density of states, mass operator, form factor, structure factor,
pseudo-gap
PACS: 43.38.Kw
Introduction
Disorder metallic systems, including liquid, amorphous metals and metallic glass-
es are still investigated intensively due to the prospects of their industrial use. Some
unique properties of such systems cannot be explained within crystalline physics
methods. The problem of disorder remains to be one of the most urgent and has not
been ultimately solved in condensed matter physics. Disordered state differs from
crystalline one by the absence of long range ordering and makes it impossible to
use the translation symmetry methods [1]. Nonequilibrium of the structure makes
us apply the configuration averaging over all accidentally occupied ion locations [2].
Accented features of disordered metals bring about a number of anomalous char-
acteristics for many thermodynamic properties. One of the most interesting problems
of disordered condensed matter physics for quantum systems is the description of the
c© P.Yakibchuk 537
P.Yakibchuk
formation of pseudo-gap in the density of electron state. Two methods of calculating
these characteristics are presented below.
1. Variation approach
Another method for calculation of density of states is the variation one. The
calculation scheme for this method was proposed in [3]. Setting the wave function
and energy of the ground state of an electron in disordered metal media, where Ri
is the ith ion coordinate, as the known values φ0 and E0 respectively, we assume the
wave function of the excited state
ψk = χkψ0 , (1.1)
where
χk =
1√
Ck
eikr
{
1 +
1√
N
∑
q 6=0
uk(q)ρqe
iqr
}
(1.2)
and
ρq =
1√
N
∑
j
eiqRi .
It corresponds to an eigenstate of both full impulse operator for eigenvalue ~k and
Hamiltonian – Ek. Obviously, we have the one parameter trial function for Ritz’s
condition of the variation problem for energy spectrum of an electron in the con-
duction band. We use the following condition to define Ck
Ck = |χk|2 . (1.3)
Now, the variation problem transforms into the next functional equation
Ek = E0 +
~
2〈|∇χk|2〉
2m
. (1.4)
Calculating the energy spectrum of the electrons in the conduction band, we consider
the one-sum contributions and neglect many-particle correlations. In this case, we
get minimum magnitude of Ritz’s functional in case
uk(q) = −Sei(q){Ek − ~
2
2m
k(k + q)}
Sq{Ek − ~2
2m
(k + q)2}
. (1.5)
Here Sq = 〈ρqρ−q〉 – is a structure factor of disordered metal and Sei(k, q) = 〈ρqe
−ikr〉
– electron-ion structure factor of the system. The latter can be presented in Born
approximation in case of small potential [4] as
Sei(k, q) = −2Sq〈k + q|w|q〉
~2q2/2m
. (1.6)
538
Electron structure of disordered metals
Applying this result to equations (1.1)–(1.4) and substituting Ek = ~
2/2m(k2 +
∆k) we define
∆k =
Ω0
8π2Ik
∫ ∞
0
dqq2Sei(k, q)2
Sq
{
q2 − q4 − ∆2
k
4kq
ln
|q2 − ∆k + 2kq|
|q2 − ∆k − 2kq|
+
∆k(q
2 + ∆k)
2
(q2 − ∆k)2 − 4k2q2
}
. (1.7)
Here
Ik = 1 +
Ω0
8π2
∫ ∞
0
dqq2Sei(k, q)2
Sq
{
−3 +
q2 + ∆k
2kq
ln
|q2 − ∆k + 2kq|
|q2 − ∆k − 2kq|
+
(q2 + ∆k)
2
(q2 − ∆k)2 − 4k2q2
}
. (1.8)
This result can be easily transformed into Relel-Schrodinger theory expression [6]
for energy spectrum by assuming ∆k = 0 and Ik = 1. Moreover, it can also be
transformed by the same assumption, but saving ∆k 6= 0 in logarithm, to Brilliuen-
Vigner theory result [6]. Thus, for density of states reduced by its free-electron
approximation N0(E) = Ω0k/π2 we have
g(E) =
N(E)
N0(E)
=
(
1 +
Ω0
16π2k2Ik
∫ ∞
0
dqq2Sei(k, q)2
Sq
×
{
(q2 + ∆k)
2
2kq
ln
|q2 − ∆k + 2kq|
|q2 − ∆k − 2kq| −
(q2 − ∆k)(q
2 + ∆k)
2
(q2 − ∆k)2 − 4k2q2
})−1
. (1.9)
This expression consequently follows the expression (1.7)–(1.8) if density of states
is assumed
N(E) =
Ω0k
2
π2
dk
dE
.
The calculations of density of states within this theory were presented in [7].
The pseudo-gap deviation of density of states in close range of Ferme level calculat-
ed in Relel-Schrodinger and Brilliuen-Vigner approximation of perturbation theory
was obvious and similar. Both were in good correlation with Zaiman’s and Mott’s
predictions for distribution of electrons in disordered metals. Variation approach
used in [7] gives a better agreement than the results of second order of perturbation
theory in terms of electron-ionic potential.
2. Perturbation theory
The well-known fact is that configuration averaging must be used in disordered
systems [1]. So, when expressing the electron Green’s function averaged over confi-
gurations as
G(E, k)conf =
1
E − ~2k2
2m
− Σ(E, k)conf
(2.1)
539
P.Yakibchuk
we achieve the following form for the mass operator in the third order of perturbation
theory [5]
Σ(E, k)conf = 〈k|W|k〉 +
1
N
∑
q 6=0
Sq
〈k|w|k + q〉〈k + q|w|k〉
E − ~2/2m(k + q)2
+
1
√
N
3
∑
q′ 6=0
∑
q 6=q′ 6=0
Sq,q′,q−q′
〈k|w|k + q′〉〈k + q′|w|k + q〉〈k + q|w|k〉
[E − ~2/2m(k + q′)2][E − ~2/2m(k + q)2]
+
1
N
∑
q 6=0
Sq
〈k|w|k + q〉〈k + q|w|k〉
[E − ~2/2m(k + q)2]2
〈k + q|w|k + q〉 (2.2)
and this can be easily shown by factoring the Green’s function into a row for pseudo-
potential. Here we take the following designations: W (r) =
∑
i w(r − Ri) – electron-
ion potential.
Picking out imaginary and real parts of Green’s function we get for real contri-
bution of the first and second order
Σ′(2)(E, k)conf = 〈k|w|k〉θ,ϕ − Ω0
4π2
m
~2
∫ ∞
0
dqq/kSq|〈k|w|k + q〉θ,ϕ|2
× ln
|E − ~
2/2m(k + q)2|
|E − ~2/2m(k − q)2| (2.3)
and for the imaginary one
Σ′′(2)(E, k)conf =
Ω0
4π2
m
~2
∫ ∞
0
dqq/kSq|〈k|w|k + q〉θ,ϕ|2
×
∫ 1
−1
dxδ
{
E − ~
2/2m(k2 + q2 + 2kqx)
}
=
Ω0
4π2
m
~2
∫ k+
√
2mE/~
k−
√
2mE/~
dqq/kSq|〈k|w|k + q〉θ,ϕ|2 . (2.4)
Now, consider the third order correction. We set the convolution approximation
for three-particle structure factor Sq,q′,q−q′ = SqSq′S|q−q′| [1]. The third order con-
tribution takes the form of 2 last items in (1.2); then for real part of this correction
we have
Σ′(3)(E, k)conf =
Ω2
0
(2π)4
m2
~4
∫ ∞
0
dqq/k
∫ ∞
0
dq′q′/kSqSq′S|q−q′|
× 〈k|w|k + q′〉θ,ϕ〈k + q′|w|k + q〉θ,ϕ〈k + q|w|k〉θ,ϕ ln
|E − ~
2/2m(k + q)2|
|E − ~2/2m(k − q)2|
× ln
|E − ~
2/2m(k + q′)2|
|E − ~2/2m(k − q′)2| +
2Ω0
4π2
m
~2
∫ ∞
0
dqq/kSq〈k + q|w|k + q〉θ,ϕ
× |〈k|w|k + q〉θ,ϕ|2
|E − ~2/2m(k + q)2||E − ~2/2m(k − q)2| . (2.5)
540
Electron structure of disordered metals
Finding an imaginary part, we had better consider those two items separately and
using the calculation method similar to the second order calculation for imaginary
contribution and taking into account the following equation for transformation of
the divisor
1
(E − ~2/2mq2)(E − ~2/2mq′2)
=
1
~2/2m(q2 − q′2)
{
1
E − ~2/2mq2
− 1
E − ~2/2mq′2
}
we get for the first item
Σ
′′(3)
I (E, k)conf =
Ω0
4π2
m
~2
∫ k+
√
2mE/~
k−
√
2mE/~
dqq/k
∫ ∞
0
dq′q′/kSqSq′S|q−q′|〈k|w|k + q′〉θ,ϕ
× 〈k + q′|w|k + q〉θ,ϕ〈k + q|w|k〉θ,ϕ ln
|E − ~
2/2m(k + q′)2|
|E − ~2/2m(k − q′)2| (2.6)
and the following expression for the second item
Σ
′′(3)
II (E, k)conf =
2Ω0
4π2
m
~2
∫ ∞
0
dqq/kSq〈k + q|w|k + q〉θ,ϕ|〈k|w|k + q〉θ,ϕ|2
×
{
δ(E − ~
2/2m(k + q)2) − δ(E − ~
2/2m(k − q)2)
}
=
Ω0
4π2
m
~2
Sq〈k + q|w|k + q〉θ,ϕ ×
√
2mE − k
k
√
2mE q=|k−
√
2mE|
− Ω0
4π2
m
~2
Sq〈k + q|w|k + q〉θ,ϕ ×
√
2mE + k
k
√
2mE q=k+
√
2mE
. (2.7)
Density of states of electrons in conduction band is defined as the imaginary part of
the electron Green’s function averaged over all possible configurations
N(E) =
1
π
Sp Im G(E − iε, k)conf
=
1
π
∑
k
Σ′′(E, k)conf
[E − ~2k2
2m
− Σ′(E, k)conf ]2 + Σ′′(E, k)2
conf
. (2.8)
Here
Σ′(E, k)conf = Σ′(2)(E, k)conf + Σ′(3)(E, k)conf ,
Σ′′(E, k)conf = Σ′′(2)(E, k)conf + Σ′′(3)I (E, k)conf + Σ′′(3)II (E, k)conf .
Proceeding to integral over impulse k and taking into account spin degeneracy fac-
tor 2, we finally get the following expression for density of states.
N(E) =
Ω0
π3
∫ ∞
0
Σ′′(E, k)confk
2dk
[E − ~2k2
2m
− Σ′(E, k)conf ]2 + Σ′′(E, k)2
conf
. (2.9)
541
P.Yakibchuk
Some computation results are presented below. Form factor of electron-ion in-
teraction is calculated for the following non-local model potential
W (0)(r) = −Z
r
+
∑
l
e
− r
Rl
(
Z
r
+ Al
)
Pl , (2.10)
here Al, Rl – are model parameters of potential, Pl – is the projecting operator.
Screened form factor of this potential takes the following form [5]
w(q) = − 4πZ
q2Ω0ε∗(q)
+ f(k, q) −
(
1 − ϕ(q)
ε∗(q)
)
4
πq3ε(q)
∫ kF
0
f(k, q) ln
∣
∣
∣
∣
k − 2q
k + 2q
∣
∣
∣
∣
kdk.
(2.11)
Here
f(k, q) =
l0
∑
l=0
4π(2l + 1)
Ω0
Pl(cos θ)
∫ ∞
0
e
− r
Rl
(
Z
r
+ Al
)
jl(kr)jl
(
|k2 + q2|r
)
r2dr
(2.12)
is non-local part of form factor and
ε(q)∗ = 1 − (1 − ϕ(q))(ε(q) − 1) (2.13)
is a dielectric permittivity accounting for the processes of electron exchange and
correlation. Here we used Hartry approximation for permittivity ε(q) and Heldart-
Vosko approximation for local field correction ϕ(q).
Figure 1. Density of states Cd-II.
542
Electron structure of disordered metals
Figure 2. Density of states Zn-II.
Figure 3. Density of states Al-III.
543
P.Yakibchuk
Figure 4. Density of states In-III.
Figure 5. Density of states Pb-IV.
544
Electron structure of disordered metals
So, in figures 1–6 we indicate the perturbation theory result with solid line and
free electron approximation result with dash line. We use Ashkroft-Lekner result of
Percus-Yevik approximation for structure factor in our calculations.
Evidently, electron density of states has a pseudo-gap behavior near Ferme level
range. In approximation not responsive to the imaginary part of mass operator the
second order gap takes place and in case of this theory the imaginary part of mass
operator spreads this gap and transforms it into the pseudo-gap deviation of density
of states. This happens due to the extinction of electron spectrum on structure
fluctuations.
We have also investigated the effects of the order of perturbation theory duri-
ng these calculations. For some metals (Cd,In,Pb) the third order contribution is
the most determinant and for the others (Zn,Al) the second order is already good
approximation for density of states.
The calculations within the non-local pseudopotential theory are more accurate
than the calculations within its local approximation (i. e., Ferme sphere approxima-
tion)and makes it possible to exactly define the shape and position of pseudo-gap
minimum on density of states distribution.
3. Conclusion
Our results show that in order to increase the accuracy of data on density of states
of disordered metals we must provide the non-local properties of pseudopotential and
the higher orders of perturbation theory in our calculations.
References
1. Zaiman J. Models of disorder. Mir, Moscow, 1982, 591 p. (in Russian).
2. Mott N., Davis E. Electronic processes in non-crystaline metals. Mir, Moscow, 1982,
386 p. (in Russian).
3. Vakarchuk S.A., Yakibchuk P.M. Variation calculation of energy spectrum and density
of states in liquid and amorphous metals. Ukr. Fiz. Jur., 1984, 29, No. 8, 1272–1274
(in Russian).
4. Vakarchuk S.A. Solution of Shrodinger equation for ground state of ionic subsystem
of metal in adiabatic appoximation. – In book: Mathematical methods and physic.-
mechanic. fields. Naukova Dumka, Kyiv, 1979, No. 10, 125–131 (in Russian).
5. Dutchak Ya.I., Vakarchuk S.A., Yakibchuk P.M. Accounting of the higher approxima-
tion of perturbation theory for liquid and amorphous metals. – In: Proc. Contributed
papers of III Un. Conf. “Laws of formation of structure in euthectics alloys”, Dni-
propetrovsk, 26–28 March, 1986, p. 52–54 (in Russian).
6. Chan T., Ballentine L.E. The energy distribution of electronic states in a liquid metal.
Phys. and Chem. Liquid., 1971, No. 2, 165–175.
7. Yakibchuk P.M., Vakarchuk S.O. Density of states of disordered metals. Preprint
ICMP–00–05, 2000, 16 p.
545
P.Yakibchuk
Електронна структура топологічно
невпорядкованих металів
П.Якібчук
Львівський Національний університет імені Івана Франка
Отримано 29 березня 2005 р., в остаточному вигляді –
16 червня 2005 р.
В даній роботі розглядаються два різних підходи до визначення
густини станів та енергетичного спектру електронів провідності у
невпорядкованих металах. Перший з них грунтується на варіацій-
ному принципі з використанням однопараметричної пробної хви-
льової функції електронів провідності. Для енергетичного спектру
отримано рівняння, яке має більш загальний вигляд у порівнянні з
результатами теорії збурень. Електрон-іонна взаємодія входить в
теорію через електрон-іонний структурний фактор, що дає змогу
застосувати теорію і в тих випадках, коли формфактор потенціалу
не є малою величиною і теорія збурень не може бути застосована.
Якщо формфактори екранованого потенціалу є малими, то із
виведеного варіаційного виразу в часткових випадках отримуються
відомі результати теорії збурень Релея-Шредінгера та Бріллюена-
Вігнера. Другий підхід пов’язаний з використанням методу функцій
Гріна та стандартної теорії збурень за псевдопотенціалом з ура-
хуванням членів до третього порядку включно. Для ряду металів
виконані чисельні розрахунки густини електронних станів. Виявлено
помітне відхилення відносно вільноелектронного наближення на
залежностях густини станів від енергії, що обумовлює появу
псевдощілини.
Ключові слова: густина станів, масовий оператор, формфактор,
структурний фактор, псевдощілина
PACS: 43.38.Kw
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