The Role of Electron-Electron Interactions in Moderately Doped
Nanostructures
A.B. DUBOIS, R.V. LYSOV
Mathematical Department
Ryazan Institute of Open Education
2 Bronnaya Str., Ryazan, 390010
RUSSIA
Abstract: - The report is aimed at clarifying the contribution of intrasubband and intersubband electron –
electron interactions to Landau quantization damping of transverse magnitoresistance oscillations.
Expressions are derived for the time of electron-electron interaction, matrix elements of the full screening
potential and dynamic dielectric function in a 2D electron system. The calculated dependences provide a
good description of the experimental times of Landau levels collisional broadening.
Key-Words: - Nanostructures, Dynamic dielectric function, 2D electron system
1 Introduction
  q,   in the approximation far from the long
Starting from the pioneering works [1] and up to
the present time [2-4] the electron interactions are
the subject of ever growing interest because of
their important role in kinetic phenomena. Also
known are the anomalies in the low-temperature
magnitotransport arising when 2D electrons fill
several size-quantized subbands. In particular,
authors of Ref. [5] predicted non-monotonous
behavior of kinetic coefficients as the density of
2D electrons is changed and several sizequantized subbands in a 2D system are filled.
When
the
doping
level
of
the
AlxGa1 x As( Si) / GaAs nanostructure is high
wave limit are calculated.
enough
for
concentration
ns
to
reach
2
8 10 cm , the quantum well contains two size11
quantized subbands. In present paper we report
the results of the study of electron-electron
processes in a system of highly degenerate 2D
electrons with the fine structure of energy
spectrum and the electron density spatial
distribution. Expressions for the times of
 eeintra and
electron-electron
intrasubband
intersubband  eeinter interaction are derived and
matrix elements of the full screening potential
Vtot  q,   and dynamic dielectric function for
2 Theory
Interactions
of
Electron-Electron
One of the important points in the derivation of
expression for  eeintra and  eeinter is the calculation of
the full screening potential matrix elements which,
within the perturbation theory approach, implies
the transformation of the potential V  r, t  into
Vtot  q,   . To within the second order in external
potential Vtot  q,   in the perturbation theory
expansion, the time required for the e-e interaction
to change the state k p into k  q p  q is
given by a well-known expression
1
 iee


 d

 V
ijkl
tot
(q,  )
2
jkl k,p,q
  Ek  k  q   El  p - q   Ei  k   Ei  p  
 fk  fp  1  fk q 1  fp-q 
(1)
Where indices i, j, k, l run over the set consisting
of symbols m (first subband of dimensional
quantization) and n, d (second subband of
dimensional quantization) which label the
electron transition type. When performing
summation over k and p we transform (1) to
1

d


  2kBT 
(2)
ijkl

V
q
,


q
,


q
,

  tot   ik   ik  
 ee

 ch
2
2
i , j ,k ,m q
Vtotijkl  q,   
Ej
2d m S  q  2 e2 q 4  ik  q,   
 ns ,i
 (T , nn , nm , ni , d m , di ) 


 2   q 
 2 ns ,i




q


 q
 q



2






n
2
n
s ,i
 s ,i 


m 

ns ,i q 
m 

ns ,i q 
2 1/ 2




2 1/ 2




 (3)

*2
E m S
P n T   T , ns 
16 d m2 4 ns2
(4)
while for the third type
1
 inter

2
in Eq. (5) are
e

2
1/ 2
2 k BTm*
0
 nm nd nn 


 nm  nd  nn 
1  et
t
dt
with zeta-function  (T , nm , ni , dm , di ) are rather
sensitive to the electron concentration in the sizequantized
subbands.
For
example,
for
11
2
nm  8 10 cm ... nd  0.1nm , … nn  0.1nd ,
d p / d m  3.5 the factor  (T , nn , nm , ni , dm , di )
in Eq. (5) results only in some smoothening of the
non-monotonous
behavior
while
at
11
2
1
nm  8 10 cm the curve  ee T  does not
3 Results and Discussion
(5)

W n T   T , ns  
8

where Q n T  , P n T  , W n T  - polynomials
3/ 2
t

4
  En  Em 2
Q n T   T , ns  

8

 En  Em 

dm
dm  dn
dm
 d d 
 m p 
 dm  d p 


contain any non-monotonous parts at all.
m*2 S
16 6  d n  d p 
 ee1 T  is
of
The products of Q n T  , P n T  and W n T 


  c.c.


In the form convenient for calculations the
relaxation times for intrasubband transitions are
written as
2
m
5
behavior
zeta-function
3
 ik  q,   

 (T , nm , ni , dm , di ) .
determined by the uniformly converging sums
Q n T  , Pn T  and Wn T  multiplied by the
and dielectric function
1
zeta-function
Non-monotonous
with matrix elements
intra
B1 B2 B3


 ...,
T T2 T3
B B B
Q n (T )  1  22  33  ...,
T T
T
B B
W n (T )  22  44  ...,
T
T
with the coefficients Bn defined by the Riemann
P n (T ) 
Presented for comparison in Figs. 1a and 1b are the
calculated and experimentally measured curves for
several nanostructure samples where 2D electrons
are certainly known to fill only the lowest size-

quantized subband ns  8 1011 cm 2

(see Refs.
[6,7] for the details of the analysis of the
experiment).
takes place the second – subband electrons are in
the state close to the quantum limit and one can
only speak of the oscillations period in a rather
limited sense.
Fig.1a
Fig.1b
Comparison
of
calculated
and
experimental (taken from [6]) curves for
different values of subband electron
density: ns ,1011 cm -2 : 1,4-8.5; 2,5-6.9;
3,6-6.3.
4 Conclusion
In conclusion, It should be noted that a similar
problem for 2D electrons seems to have been
considered in [8] and since then numerous
attempts have been undertaken [9-11] to study
this problem for 2D electron system where
several size-quantized subbands are filled at
T  0 in the long wavelength limit. However, the
plasma oscillation spectrum has not been obtained
in any of these works. Characteristic features of
2D electron systems, such as the amplitude –
frequency modulation, beatings, and sharp bends
in the oscillation, and sharp bends in the
oscillation amplitude magnetic field dependence
make the description of Landau quantization
damping in terms of the Dingle temperature
rather problematic. Another point to be pointed
out is the fact that in the magnetic field range
where a strong amplitude – frequency modulation
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