Channels of electron-electron interactions in highly doped heterojunction Текст научной статьи по специальности «Физика»

CHANNELS OF ELECTRON-ELECTRON INTERACTIONS IN HIGHLY DOPED HETEROJUNCTION

V. A. Ambartsumyan1, E. A. Andryushchenko1, K. V. Bukhenskyy1, A. B. Dubois1, E. A. Dvoretskova1, T. V. Gordova2, N.I. Ivanova3, S.I. Kucheryavyy4, S.N. Mashnina2, A. S. Safoshkin1

1 Ryazan State Radioengineering University, Ryazan, Russia

2 Moscow State University of Economics, Statistics and Informatics (Ryazan filial), Russia

3 Moscow State University of Economics, Statistics and Informatics (Yaroslavl filial), Russia

4 Institute of Atomic Energy of National Research Nuclear University of Moscow Engineer-Physics University, Obninsk, Russia dubois.a.b@rsreu.ru

PACS 72.10.-d, 73.23.-b

Electron-electron interactions in a single highly doped heterojunction are considered, taking into account both intra- and intersubband transitions. 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 with the fine structure of the energy spectrum, and for the electron density spatial distribution. The theoretical dependences r^ (T,ns) provide a good description of the experimental times of Landau levels collisional broadening r|xp (T, ns ).

Keywords: electron-electron interactions, random phase approximation, Fourier analysis, Shubnikov-de Haas oscillations. Received: 3 May 2014 Revised: 4 June 2014

1. Introduction

Starting from the pioneering works [1] and up to the present time [2-20], electron-electron (e-e) interactions are the subject of ever growing interest because of their fundamental role in kinetic phenomena. Among others, one should note the hot electrons effects, quantum corrections to the conductivity, and damping (destruction) of Landau qunatiza-tion in bulk and two-dimensional semiconductors with degenerate electrons. Also known are anomalies in the low-temperature magnetotransport arising when 2D electrons fill several size- quantized subbands. In particular, the authors of [2] predicted non-monotonous behavior of kinetic coefficients as the density of the 2D electrons is changed and several size-quantized subbands in a 2D system are filled. Experimentally, the reduction of mobility with the growth of electron surface density ns was discovered in [3, 4]. Later [5-7], a complicated set of phenomena was discovered and studied in the AlxGa1-xAs/GaAs het-erostructure whose potential well contained two size-quantized (the main Em, and the first excited Ep) subbands. Most interesting among them are the amplitude and the frequency modulation of the transverse Shubnikov-de Haas (SdH) magnetoresistance [8], sharp bends in the magnetic field dependence of the oscillation amplitude ô (1/B) [9, 10] as well as the non-monotonous behavior of the Dingle temperature TD with the 2D electron density ns and

the temperature T [11, 12]. In spite of the variety of experimental details and approaches to their interpretation, the indicated effects were all shown to be caused by filling of the second excited size-quantized subband. The mechanism triggering the non-linear effects is the in-tersubband interaction. In [4, 5] following point was discussed; when the doping level of the AlxGai_xAs (Si) /GaAs heterojunction is high enough for ns to reach 8 ■ 10ncm_2, the quantum well contains two size-quantized subbands with energies Em and Ep electron densities nm and np, respectively, and the SdH oscillations of pxx exhibit frequencies Fm,p = (nh/e) nm,p with the periods Am,p (1/B) = F_1p. The main feature is the modulation of the main frequency Fm amplitude with the frequencyFp, the modulation depth growing with temperature and being more pronounced at the lower magnetic fields. The second feature is the development of oscillations at a different frequency Fm — Fp. These oscillations do not depend on temperature and transform to oscillations with a frequency Fm as the magnetic field is raised.

Assuming constant electron density and Fermi energy oscillations, Kadushkin [9, 10] explained the main features of the amplitude-frequency modulation of the SdH oscillations by the intersubband interaction. Authors of [2] derived an analytic expression for the amplitude of oscillations containing components at the frequencies Fm,p, Fm — Fp. They analyzed the experiments, taking into account the electron-phonon interaction. Formally, the Dingle temperature is related to the non- thermal collisional broadening time rq through the expression TD = h/2nkrq. Description of the SdH oscillations of pxx [8] based on the two subbands (m and p) model is in good agreement with the experimental results obtained in [12].

In the present paper, we report results from the study of e-e relaxation processes in a system of highly degenerate 2D electrons with finely structured energy spectrum and electron density spatial distribution. Expressions for the electron-electron intra-(rentra) and intersubband (r^67) interaction are derived and the matrix elements of the full screening potential Vtot (q, w) and the dielectric function for x (q, w) in the approximation far from the long wave limit are calculated. The oscillations in r®xp (T, ns) ~ r^ (T, ns) are shown to be related to the excitation of plasmons in the components of 2D electron system and the plasmon spectrum is studied.

2. Mechanism of Landau quantization destruction

One of the important points in the derivation of expressions for r6ntra and rint6r 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, w). In our problem, this corresponds to the following physical situation.

Let us consider 2D electrons, crystal lattice, and the source of perturbation as a thermodynamic system in equilibrium at the thermal bath temperature T. The electrons interact amongst themselves and with the crystal lattice with relaxation times t66 and reph, respectively. At lower temperatures and in a quantizing magnetic field B, the equilibrium energy and momentum distribution of 2D electrons is given by harmonic oscillators with the cyclotron frequency uc. The electron states are correlated and coherent because of the strong e-e interaction with the relaxation times hierarchy ree ^ rp ^ r£ where rp and r£ are the relaxation times of momentum p and energy e. The electron states on the cyclotron orbits are coherent because the lifetime on these orbits exceeds the mean free time (momentum relaxation time). Although the electron state initial phase is determined by the electron settling into the cyclotron orbit and has a random nature, the motion of electrons on cyclotron orbits is synchronized and it is precisely this point that allows one to apply the random phase approximation (RPA) to these magnetized electrons. It is worthwhile to note

that the reduced Wigner radius for the studied heterojunctions with ns ~ (0.5 ^ 2) ■ 1012cm 2,

V (q, u) is 0.6 ^ 1.2, so that the use of RPA is rather justified [1].

Electron collisions with the crystal lattice defects destroy the ground quantum state (cyclotron orbits), which is seen in the broadening of Landau levels, resulting in finite values for the kinetic coefficients' oscillation amplitudes.

In the absence of an external electric field, the equilibrium state of electrons is characterized by the temperature T0 which stipuleted the chaotization of the 2D electrons, which are in thermodynamic equilibrium with the crystal lattice (T, T0) in the quantizing magnetic field B. Note that the perturbations of the potential relief of the quantum well's 2D channel, caused by various defects, are time-independent: V (r,t). The electron system state (T,T0) stationary with respect to the electron-lattice interaction is controlled through the energy and momentum relaxation with the characteristic time T^j^.

The mechanism for the destruction of the quantum Landau state (T,T0)B and development of a new equilibrium stationary state (T,T^), which is not in equilibrium with respect to the initial equilibrium state (T,T0), can be described in the following way. The electric field E causes a drift of the 2D electron system along the heterojunction quantum well (in the xy-plane) so that the electrons "scan" the spatial distribution of the heterojunc-tion defects. In the reference frame associated with the 2D electrons, the observer sees the results of scanning the potential well defects as Vtot (q,u). Here, the electric field E acts only as the means of sweeping the external (with respect to 2D electrons) perturbations, converting them into a time-dependent potential which is expressed in the transition from

V (r, t) to the Fourier component Vtot (q, u).

In the 2D electron system perturbed by (T,T0) the collisions acquire a different nature and at T = const and B = const, the equilibrium broadening kT0 is affected by the perturbation caused by the external field E so that a new equilibrium state with the broadening kT0 is established. This new equilibrium state (T,T^), should be considered as a non-equilibrium one with respect to (T, TD0)B.

Fig. 1. The energy diagram of the conduction band Ec (z) for a hetero junction with two filled size-quantized subbands Em and Ep with electron densities nm and np = nn + nd; dp is the undoped spacer thickness, and N+, N-, A, A, 5x are the sources of perturbation of the 2D electron system

Our analysis of e-e interaction is based on the calculation of the conduction band energy structure Ec (z). We approximate the potential well of the heterojunction by a triangular profile [11] with the sharp bends at the size-quantized levels Em = Ec (dm) and

Ep = Ec (dp).

Fig. 2. Schematic model illustrating the channels through which the perturbation from the sources causing non-thermal collisional broadening of Landau levels passes. See text for details

Schematics of the e-e interactions in the 2D system is presented in Fig. 1, which also shows the typical channels through which the perturbation due to various sources, such as ionized donors N+, acceptors N_, growth islands of length A and height A, as well as the mole fraction variations Sx, affects the electron system. Here, nm, is the 2D electron density in the ground (main) size-quantized subband while nn and nd are the satellites of the perturbed (excited) subband component np = nn + nd located close to ("near") and far from ("distant") the heterointerface (in the following referred to as the n— and d—satellites). Of all the channels of Landau quantization damping which we consider, one should emphasize those three that reveal the major features. Since

^^^^ nn, nm, and the centers of gravity of (z)|2 at the levels Em and Ep, are spatially resolved, the perturbation is sensed (absorbed) during time r(1) by the set of 2D electrons. Then, through the intrasubband (rmm) and intersubband (r6mn, T^d, r^e) e-e interactions mediated by the (r66n) and (r^) intrasub-band interactions, the perturbation is extended to the entire 2D system. The path of this mechanism is shown in Fig.1 by the solid line:

-(i)

(Tmn^ rnd mn\ , Inn dd Tnd\ = (1) (1)

\ ee ) V ee i ee ) V ee i ee i ee ) ee ' V /

The major factor here is the interaction of nm, and nd electrons (m — d), while nn, is a passive element. The second scenario corresponds to the situation where perturbation is first sensed by the n—satellite. In that case, the time needed to destroy the quantization is formed in the chain shown in Fig.1 by the dashed line:

r

(2)

(rnn) , (rnd\ =b. (rmn rnd\ , i nn rdd rnd\ = (2) V ee ) V ee J V ee i ee ) V ee i ee i ee ) ee '

(2)

Here, the system behavior is governed by the intersubband "n — d", "n — m" and "m — d" interactions. The "n — d"- interaction is the major one in this channel and nm, plays the role of a passive element. The third version is formed in the chain where the "m — n" interaction dominates while nd is a passive element:

-(3)

^ (t6T) ^ (t6

_mn rmd

'ee , 'ee

frnn rdd rnd ^^ \ ' ee , ' ee , ' ee

t6

(3)

(3)

The corresponding path is shown by the dot-and-dash line. With respect to the nature of induced transitions the e-e interactions can be classified into three types: (1) interactions within a single subband limited to the transitions within the same subband; (2) intrasubband interaction exciting the intersubband transitions, and (3) intersubband interactions also resulting in intrasubband transitions.

To within the second order in the external potential Vtot (q, u) 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:

i = / du E^kif (W2

ij ™ k,m q — oo ' ^

x

S (Ej (k + q) + El (p — q) — E (k) — Ek (p)) fk fp (1 — fk+q) (1 — fp-q), (4)

k,p

where indices i,j,k,l run over the set consisting of symbols m (main component) and n,d (satellites of the np-component) which label the electron transition type; f - Fermi-Dirac function. Using the notations given in [12 - 20], (4) can be written as:

oo

Tj = / n'ch'ot^t) £ ^ I Vj(q'u) I »k <q'u) j u). <5)

— o ,m q

Matrix elements of the m — n, m — d intersubband interactions were calculated with the wave functions (z) as well as the ■n (z) and ■ (z) components of the wave function ■p (z) with appropriate boundary conditions.

Taking into account the parameters of the energy band diagram, the matrix elements of the full screening potential for the first and second type transitions are reduced to the form:

VtOtkl (q, u) = 0, g j2 3—?—Û , (6)

2diS (q3 + 2ne2q3Xik (q, u))

where S = L2 - square of 2D system.

For i = k and j = l, while for the third type:

yijki ( n) = Ej (1 - qdj) - Ej (1 + qdj) (7)

tot (q ' ) 2 (di - dj ) S (q3 + 2ne2q4Xik (q ,u))' ()

with i = j and k = l. In the form convenient for calculations, the relaxation times for the first and second type transitions are written as:

1 E2m*2S

j P-n (T), (8)

Tirdra 16n5 d2h4UiUj

while for the third type:

1 _ m*2S j (Ei - Em)) n (rr, (Ei - Em) (Eldm + Emdi) w ^ , TiT' = 8n6(di - dm)2«^ \ _IT"Q-n (T) ^fi W-n (TU (9)

for i = l. The polynomials in (8) and (9) are:

n /m\ Bl B2 B33 P-n(T ) ~T + t2 + T3 + "',

Q-n(T) t t 2 + t 3 — ''

W-n(T ) = Bf + B + ...

with the coefficients Bk defined by the Riernann zeta-function q(z,v). The non-monotonous behavior of r— 1 (T) is determined by the uniformly converging sums P-n (T), Q-n (T) and W-n (T) multiplied by the zeta- and gamma-functions [9]:

q (z, v)

1

tz-1e-vt

. -dt r(z)J 1 - e-t dt' 0

where z

dm

+dp

T2n

2kb T \ nm +np

The products of P-n (T), Q-n (T) and W-n (T) with q(z, v) are rather sensitive to the electron concentration in the size-quantized subbands. For example, for nm > 8 ■ 1011cm-2 (nd = 0.1nm, nn = 0.1nd, dpldm = 3.5) the factor q(z, v) in (9) results only in some smoothen-ing of the non-monotonous behavior while at nm > 8 ■ 1011cm-2 the curve t— 1 (T) does not contain any non-monotonous parts at all.

Calculations of t^ were performed within the outlined schematic model of the Landau quantization destruction, taking into account the paths corresponding to channels (1-3), including both intra- and intersubband transitions according to (8) and (9) employing the

Matthiessen rule t— 1 = ^(t-1)^ where the summation is performed over all intra- and

i

intersubband components of the schematic model presented in Fig. 2.

3. Electron "bottleneck"

Presented for comparison in Figs. 3a and 3b are the experimentally measured and calculated curves T®xp (T, ns) and rf^ (T, ns) for several heterostructure samples where 2D electrons are certainly known to fill only the lowest size- quantized subband (ns < 8 ■ 1011cm-2). Figures 4a and 4b show the experimental and calculated curves for the time of destruction of Landau quantization for two heterostructures with electron density sufficient for filling of the two size-quantized subbands (see [9, 10] for the details of the analysis of the experiment).

Fig. 3. Comparison of the experimental rexp (T) (nm,nd or nn) [2] (a) and calculated rf^ (T) (b) curves for different values of the m—subband electron density: nm, 1011cm-2: 1, 4 — 8.5; 2, 5 — 6.9; 3, 6 — 6.3

v

The energy and geometrical parameters were taken from the energy diagrams E (z) for the samples with appropriate electron densities. The first result is a quantitative "hit" of

the calculated times into the range of T in the studied temperature interval 2 ^ T ^ 12K for the real densities um ~ 10l2cm-2, nd = 0.1nm , nn = 0-01nm and, respectively, dp/dm = 3. Of all the considered versions of the model presented in Fig.2, scenario (3) is the most satisfactory one and the curve r^ (T,ns), plotted in Fig. 4b, was calculated exactly for this scenario. Further, it should be noted that at low temperatures (T < 5K), the Landau quantization damping is governed by nm-electrons. Numerical analysis of the expansion of the dielectric functions (6) reveals the appearance of non- monotonous parts in the curve tI^ (T,ns) at nm > 8 ■ 10llcm-2 and T > 5K (see (10-13)), allowing one to argue [9,10] that the typical oscillations of t^ (T) arise only after the electrons fill the second excited size-quantized subband and for the system response to the thermal perturbation at T > 5K. The third obtained result is the role of nn-satellite in the appearance of the oscillations regardless of which component of the 2D electron system senses the perturbation, as shown in Fig.2 by channel t(3).

Fig. 4. Comparison of the experimental T®xp (T) [2] (a) and calculated T^h (T) (b) curves for different values of the m—subband electron density: 1 - 9.1; 2 - 10.0

This was also directly seen in (9). Indeed, for nn = 0, only T%mma (T) and T%d[djtra (T) are different from zero and no oscillations arise in rfh (T). The bottleneck effect is explained in the following way; variation of temperature initiates the frequency scan of the external perturbation Vtot (q, u) towards higher u. The 2D electron system is transparent for Vtot (q, u) until the frequency uT of one of the components (nm nd, or nn) is reached. The lowest uT corresponds to the nn—satellite and it is this component or, to be more precise, its intrasubband relaxation that is the bottleneck for Vtot (q,u), perturbing the 2D electron system as a whole and finally destroying the cyclotron orbit quantization (damping of Landau quantization). This bottleneck effect in the e-e interactions illustrates the coincidence of the resonant frequencies for "m — n"and "d — n" channels (see Fig. 5 (curves 1 and 3) below in the range of low frequencies).

Thus, the experimentally observed features in rexp (T) at T < 5K are only related to the intrasubband e-e transitions, r®xp (T) « r^h (T) ~ Tmma (T). At higher temperatures, a mixed mechanism of the Landau quantization destruction is realized:

t,

exp

(_mn) — 1 __ra\

( ' ee ) V mn )

int ra\ 1 I f int er mn

1

jnd

Te.

1

int ra

1

+ T

int er

' nd

1

It should be noted that by varying the well parameters, one can obtain a satisfactory agreement with experimental results. This technique offers the possibility of recovering the actual potential profile from the superposition of the curves rqixp and rf^, measured for the

3

ee

nn

samples with different doping levels, and therefore, showing different variations of the form-factors dp/dm and Ep/Em. However, this matching of the calculated curves rf^ with the experimental ones r®xp is limited by a certain arbitrariness in the adjustable parameters dp/dm and Ep/Em (the potential well form-factors) since the curve E (z) cannot be derived with sufficient accuracy because of the uncertainties in ND, NA and the band discontinuity AErJAEg for the AlGaAs/GaAs heterostructure [7, 9, 10].

Fig. 5. Frequency dependence of the dielectric function for the interaction between the main size-quantized m—subband electrons with the n—satellite of the p—subband. nm, 1011cm-2: (1) — 8.5, (2) — 10, (3) — 11.5; dp/dm = 3.5

4. Spectrum of collective excitation

Bearing in mind the oscillations in r®xp and t^, it is natural to expect the resonant response of the components of a complex 2D electron system to the external perturbation Vtot (q, w) at the plasma oscillations' frequency. The 2D system responds to the spectrum of Vtot (q,w) by one of its nm, nn, nd components (or their combination), and in a timeree (Tintra, Tinter), the perturbation extends to the entire system resulting in the destruction of quantum states (cyclotron orbits), which is experimentally observed as the reduction of the 8(1/B)T oscillations' amplitude. The latter is formally equivalent to the rise of temperature T at which the measurements are taken. Therefore, the resonant response featuring the Landau quantization destruction corresponds to a minimum in the curves Texp and Tg^. We have performed a spectral analysis of the dispersion equations (6) for x (q,w) for various channels of e-e interactions according to the scheme shown in Fig. 2 and various relative values of the densities nm, nn, nd in the situation where two size-quantized subbands are occupied (nm > 8 ■ 1011cm-2). The plasma oscillation frequencies wT are found from the dispersion equation x (q,w) = 0, the minima in t^ (T) corresponding to the minima in Re x (q,w) and Im x (q,w) = 0 while the maxima in t^ (T) correspond to Re x (q,w) = 0 and maxima in Im x (q,^) [1, 19].

To calculate Re x (q, and Im x (q, w), the expression (6) should be take in the form of function x (w):

xik (w) = 1 ^ xik (q,w). (10)

q

Taking into account the characteristic scales q g {dm1, d-1} of the 2D electron system, the summation over q results in the expression:

Xik (w) = f (ni,w) + f (nk,w), (11)

where

The latter expression reduces to:

fi (ni,u) =

/— I m*wL n (m*wL\ ^ ^ (m*wL \ y2n-{——--ln _ + G - Ci

hyfni'K

hyfni'K

/— I m*wL f m*wL \ / m

V 2n - i + ln „ - G + Ci

hyfni'K

hyfni'K

2

hyfni'K j 1/2

hyfni'K j

l/2

(13)

where G is the Euler constant, f (nk,u) is given by (12) and (13) after the substitution of

x

ni by nk; Ci(x) = J dt - integral cosinus. The alternating sums over k in (12) prove to

be rapidly converging. Fig. 5 illustrates the partial contributions of various mechanisms to the Landau quantization destruction and the density-dependent singularities. For example, plotted in Fig. 5 are the frequency dependences Re x (q, w) and Im x (q, w) for nm ~ 10l2cm-2 for three intersubband transition channels. It is seen that the nm — nn and nn — nd interactions are dominat. Moreover, the resonant frequency is determined by the n—satellite density. Shown in Fig. 5 is the influence of the second size-quantized subband filling factor. An increase in the density nm, (and hence nn and nd) results in the resonant frequency shift to higher values while the discontinuity.in Re x (q,w) and Im x (q,w) is reduced which is consistent with the third scenario of the schematic model presented in Fig. 2 (domains "a" the real and "b" the imaginary parts of x (w)).

Similar analysis of the other channels for paths (1) and (2) of the model presented in Figs. 2 confirmed on the whole the trends presented in Fig. 5.

It should be noted here that in [12], the curve Tee (T) was obtained for a pair of coupled rectangular wells in the long wavelength limit at T = 0, which is incompatible with tile conditions of the experiments reported in [9]. In that case [12], the calculations of Tee (T) result in divergences which cannot be neglected when solving a particular problem. On the other hand, formulas of the type of (9) and (10) derived by us for the inter- and intrasubband e-e interactions allow one to obtain expressions for Te,ntra (T) and Tenter (T) given in [12] with the coefficients Bi in (8, 9) do not containing any divergencies.

2

5. Conclusions

In conclusion, it should be noted that a similar problem for 2D electrons seems to have been first considered in [12], and since then, numerous attempts have been undertaken [13-18] 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 amplitude magnetic field dependence make the description of Landau quantization damping, in terms of the Dingle temperature, rather problematic. Another point to be considered is the fact that in the magnetic field range where a strong amplitude-frequency modulation takes place, the p— subband electrons are in the state close to the quantum limit and one can only speak of the oscillation's period in a rather limited sense.

Acknowledgements

Authors thank Prof. Yu. E. Lozovik, L. A. Falkovskyy, G. S. Lukyanova and A. I. Novi-kov for fruitful discussions.

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