Photogalvanic effect in asymmetric lateral superlattice

Physica E 9 (2001) 652–658

www.elsevier.nl/locate/physe

Photogalvanic eect in asymmetric lateral superlattice Lev I. Magarill Institute of Semiconductor Physics, Russian Academy of Sciences, prospekt Lavrenteva 13, 630090, Novosibirsk, Russia Received 2 May 2000; received in revised form 7 September 2000; accepted 27 October 2000

Abstract It is shown that asymmetry of a modulating potential causes appearance of photogalvanic eect in lateral superlattices. The dependence of photovoltage on magnetic eld exhibits Weiss-type oscillations. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx Keywords: Lateral superlattices; Photogalvanic eect; Weiss oscillations

1. Introduction Systems with broken space or time symmetry have long attracted physicists’ attention in connection with the problem of reversibility in statistical mechanics. The classical example of such a system is a ratchet. Its behavior under the in uence of thermal uctuations is the subject of one of the chapters in the book by Feynman et al. [1]. One of the examples of a phenomenon due to the broken space symmetry (medium with no inversion symmetry) is the so-called photogalvanic eect (PGE), the appearance of a steady-state current in a homogeneous system under the in uence of an alternating electric eld of an electromagnetic wave in the absence of any stationary driving force [2– 4]. Similarly, the presence of two coherent electromagnetic waves with frequencies ! and 2! breaks the time symmetry that also leads to the direct current (the coherent photogalvanic eect (CPGE)) [5,6]. PGE and CPGE appear in the lowest non-vanishing orders in the amplitude of an external electromagnetic eld (the second order for PGE and third for CPGE). Theoretically, PGE and CPGE were investigated on the basis of the Boltzmann kinetic equation or by means of Kubo-type formula or by using the diagrammatic technique. E-mail address: [email protected] (L.I. Magarill). 1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 2 9 0 - 3

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In the last few years, systems with broken space or time symmetry have been studied on the model of a classical [7; 8] and quantum [9] ratchet in an eort to explain such biological phenomena as movement of muscles and active transport through cellular membranes. The approach used in Refs. [7–9] is based either on the one-dimensional dynamic [8] or on the Langevin equation [7,9]. PGE was the subject of intensive investigations (both theoretical and experimental) in dierent materials and in various physical situations (see, e.g., review articles [10,11] and Section 10:5 in the book [12]). Asymmetry of the electron scattering and/or asymmetry of their energy spectrum are the main mechanisms which are responsible for the PGE. Recently, experiments on the study of transport and optical properties of a periodic array of triangle antidots have been carried out [13,14]. Measurements have demonstrated that illumination of the system by far-infrared radiation leads to the stationary lateral photoemf in the absence of an external bias. This experiment makes the theoretical investigation of quadratic responses in modulated two-dimensional electron systems to be of topical interest. In the present paper we consider PGE in a 2D system modulated by asymmetric periodic lateral potential V (r). Due to asymmetry of the potential (V (r) 6= V (−r)) such a lateral superlattice has no inversion center in the plane of the system that makes possible the PGE. 2. Derivation of the expression for PGE current We consider PGE for the case of the weakly modulated 2D electron gas (the amplitude of a lateral potential is small as compared with the Fermi energy), subject to a magnetic eld perpendicular to the plane of the system under normal incidence of the electromagnetic wave of the frequency !. To determine a stationary quadratic current the approach used in work [15] is generalized to the case of the lateral potential of arbitrary form and for the second order on an alternative electric eld. Derivation is based on the system of equations following from the kinetic Boltzmann equation (the situation of zero temperature is considered): ˆ ! = −evuE ! ; (−i! + L)    9! e ∗ ˆ Re E! h − u! : L = 2mv 9’

(1) (2)

Here we have introduced the notations: (t) = Re(! e−i!t ) and  are the linear p and the stationary quadratic non-equilibrium corrections to the equilibrium distribution function, v(r) = vF 1 − V (r)=EF ; (vF is the Fermi velocity, EF is the Fermi energy), E(t) = Re(E! e−i!t ); E! is the complex amplitude of the electric eld of the electromagnetic wave, u = (cos ’; sin ’) is the unit vector of the electron velocity direction, h = 9u=9’. ˆ is given by The operator L !   Z 2 9v 1 d’ 9 9 ˆ = vu + h + !c + 1− ; (3) L 9r 9r 9’  2 0 where !c = eB=mc is the cyclotron frequency,  is the relaxation time (as in Ref. [15],  is assumed to be constant). It is suggested that the electron scattering is symmetric, and PGE arises due to asymmetry of the lateral potential. For the PGE current we have 1 {I + [b; I ]}; (4) jPGE = 1 + 2   9V Ne  ; (5) I= m 9r

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where b = B=B; = !c ; N = m=˜2 is the density of states of an unmodulated 2D system. Angular brackets in Eq. (5) denote the following operation: Z 2 Z d’ dr h: : :i; h: : :i = (· · ·) hh: : :ii =

2 0 ( = ax ay is the “volume” of the lateral superlattice elementary cell, ax; y are periods of the superlattice along the axes x; y). Thus, the problem is to nd the quantity I . Instead of the functions ! and , we introduce other functions F! and  as follows: ! = !(0)

v + F! ; vF

 =  − (0) ;

(6) (7)

where !(0) ; (0) are linear and quadratic corrections for an unmodulated 2D gas. The introduced functions satisfy the equations ˆ ! = − C! 9V ; (−i! + L)F m 9r

(8)

  h 9F! 9v 9(0) u e ∗ ˆ Re E! [u(uC ! ) − h(hC ! ) + : − F! ] − h L = 2m v 9’ v 9r 9’

(9)

Here, ! = !c ! ; ! = =(1 − i!), and C! = e! (E! + ! [b; E! )=1 + 2! : We solve the system of equations (8) and (9) carrying out a systematic expansion in the parameter  = V0 =EF 1 (V0 is the amplitude of a modulating potential). It is obvious in advance that expansion of the value in question I begins from the third order in  because the potential does not “reveal” its asymmetry in the second P order. Let us represent the potential (and the required functions) as the Fourier series of the form: V (r) = g6=0 Vg eigr , where g = 2(rx =ax ; ry =ay ); rx; y are integers. Then we should solve equations of the type: # " Z 2 −1 d’ 9 − X!; g ≡ Gˆ !; g X!; g = A!; g (10) 1 − i! + ilgu + 9’ 2 0 (X!; g is the Fourier component of an unknown function, l = vF  is the free path length). Using the integral representation of the operator Gˆ !; g (see Ref. [21]), one can write the solution of Eq. (10) in the form hRˆ !; g A!; g i ˆ R!; g : X!; g = Gˆ !; g A!; g ≡ Rˆ !; g A!; g + 1 − hRˆ !; g i Here Rˆ !; g is the integral operator Z 1 ’ K!; g (’; ’0 ) e A!; g ; Rˆ !; g A!; g =

−∞

(11)

(12)

K!; g = −(’ − ’0 )= ! − igRc (sin(’) − sin(’0 )) (the angles are counted from the direction of g); Rc = vF =!c is the cyclotron radius. After some rather cumbersome transformations using Eq. (11) we nd the function  in the second order in . Substituting the found solution into Eq. (5), we obtain the following expression

L.I. Magarill / Physica E 9 (2001) 652–658

for I : I=

655

 g0C ! Ns e2 3 2 l2 P Im(Vg∗ Vg−g 0 Vg 0 ) Re hRˆ 0; g (Dˆ g; g 0 Rˆ 0; g 0 Pˆ ! + Pˆ ! Rˆ !; g Dˆ g; g 0 )Rˆ !; g 0 i g 8m2 1 − hRˆ 0; g i 1 − hRˆ !; g 0 i g;g 0 #! hRˆ !; g Dˆ g; g 0 Rˆ !; g 0 ihRˆ 0; g Pˆ ! Rˆ !; g i hRˆ 0; g Dˆ g; g 0 Rˆ 0; g 0 ihRˆ 0; g 0 Pˆ ! Rˆ !; g 0 i + ; (13) + 1 − hRˆ 0; g 0 i 1 − hRˆ !; g i

where Ns = NEF is the areal electron density, Dˆ g; g 0 = ug 0 + h(g − g 0 )9=9’; Pˆ ! = E!∗ (h9=9’ − u): Quantities of the type h: : :i in Eq. (13) can be transformed to expressions containing series of Bessel functions Jn (gRc ). Finally, we obtain I=

Ns e2 3 2 l2 P gIm(Vg∗ Vg−g 0 Vg 0 ) H (g; 0) Re((g 0 C ! ) H (g 0 ; !) 8m2 0 g;g ×[W (g 0 ) S(g; g 0 ; 0) H (g 0 ; 0) + W (g) S(g; g 0 ; !) H (g; !) + Q1 (g; g 0 ) + Q2 (g; g 0 )]);

(14)

where H (g; !) =

1 ; 1 − hRˆ !; g i

hRˆ !; g i = 

P Jn2 (gRc ) : 2 2 2 n  +n

(15)

(Here and below the summation is over all integers.)      Jn∓2 (gRc ) Jn (gRc ) nJn∓1 (gRc ) gRc 1 P Jn (gRc ) P ∗ + − ; (±)(E! g∓ ) W (g) = 2g n 1 + in ± 2  + i(n ∓ 2) ( + in )  + i(n ∓ 1) S(g; g 0 ; !) = − Q1 (g; g 0 ) =

Q2 (g; g 0 ) =

1 P Jn (gRc )Jk (g0 Rc )ein Z(g; g 0 ; k − n; n); 2 n; k ( + in )( + ik )

(16) (17)

Jn (gRc )ein 1 P Z(g; g 0 ; k − n; n) 4g0 n; k (1 + in )(1 + ik )    0   P Jk±2 (g0 Rc ) Jk (g0 Rc ) kJk±1 (g0 Rc ) g Rc ∗ 0 + − ; (±)(E! g± ) × 2  + i(k ± 2) ( + ik )  + i(k ± 1) ± Jn (gRc )ein 1 P Z(g; g 0 ; k − n; n) 4g n; k ( + in )( + ik )      P Jk∓2 (gRc ) Jk (gRc ) (k ∓ 1)Jk∓1 (gRc ) gRc + − : (±)(E!∗ g± ) × 2 1 + i(k ∓ 2) (1 + ik )  + i(k ∓ 1) ±

(18)

(19)

Here the following denotations have been introduced: P Z(g; g 0 ; j; n) = (±)[g0 (F(g; g 0 ; j ± 1; n) − ge∓i F(g; g 0 ; j ± 1; n ∓ 1)]; ±



 

9 F(g; g 0 ; j; n) = i − n 9

g − g0 e−i |g − g 0 |

j

! Jj (|g − g 0 |Rc ) ;

g± = g ± i[b; g];  = 1 − i!;  is the angle between vectors g and g 0 . Eqs. (14) – (19) are appropriate for numerical calculations.

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Fig. 1. Magnetic eld dependence of the PGE voltage for the electromagnetic wave linearly polarized along axis x at ! = 0:4; U0 = 23 eE!2 lL=EF (L is the length of a sample along x).

3. Application to 1D superlattice In Figs. 1 and 2 some examples of magnetic eld dependencies of PGE voltage U for 1D lateral superlattice with the simplest asymmetric potential V (x) = V0 [cos (2x=a) + cos (4x=a + )] ( 6= ) are shown. The curves have been obtained by numerical calculations on the basis of expressions (14) – (19). (For a 1D superlattice they can be somewhat simpli ed.) The following parameters of the system have been used: vF = 2:7 × 107 cm=s (Ns = 4 × 1011 cm−2 ),  = 7:6 ps ( = 2 × 105 cm2 V=s), m = 0:067 m0 , a = 180 nm, = =2. As one should expect, PGE voltages as a function of the magnetic eld demonstrate an oscillatory behavior. Oscillations occur due to commensurability between the cyclotron diameter and superlattice period. Oscillations of this kind have been discovered in the late 80th under measurements of the magnetic eld dependence of static magnetoresistance in samples with spatially modulated electron gas [16,17]. At the present time they are referred to as the Weiss oscillations. The Weiss oscillations were investigated theoretically by many authors. In particular, the theory of magnetoresistance oscillations of a weakly modulated 2D electron gas was built classically by using the Boltzmann kinetic equation by Beenakker [15] and quantum-mechanically from the Kubo formula by Gerhardts et al. [18,19]. In [18,19] it was shown that in the quasiclassical limit the band contribution to conductivity dominates and the result derived from the Kubo formula reduces to the one obtained in [15]. As it was mentioned above, in the derivation of the expression for PGE current we followed the approach used by Beenakker [15] extending it to the consideration of quadratic responses. Thus, we have restricted ourselves to the case of quasiclassical approximation. At small frequencies (Fig. 1), the minima positions (they are denoted by marks) of PGE voltage practically coincide with those of the static magnetoresistance. The latter follow from the condition [15 –17]: 2vF =!c a = (NW − 14 );

NW = 1; 2; 3 : : : :

(20)

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Fig. 2. The same as in Fig. 1 but at ! = 35; marks indicate the minima of the dynamic Weiss oscillations in accordance with Eq. (21).

With increasing frequency the peaks connected with cyclotron resonance (CR) harmonics appear (their appearance is caused by the modulating potential), and the Weiss oscillation period changes. Numerical calculations and theoretical analyses carried out in [20] have demonstrated that minima of the Weiss oscillations in dynamic regime are approximately given by   ! vF  (21) = (NdW − 14 ); 2 a!c !0 √ where !0 = 2vF =a; (x) = 1 − x2 − x arctan (1=x2 − 1) ((0) = 1); NdW = 1; 2; 3 : : : : At frequencies satisfying the condition 1 ! !0  there occur distinct beats: the envelope function associated with CR harmonics modulates the Weiss oscillations. Finally, at higher frequencies when 1 ! . !0  (for the chosen parameters !0  = 72:3) CR harmonics are modulated by the envelope function with minima obeying the relation (21) (Fig. 2). Here we have a manifestation of the so-called dynamic Weiss oscillations. In conclusion, we have shown that in lateral superlattices with the asymmetric modulating potential one can observe the photogalvanic eect. Magnetic eld dependencies of this eect demonstrate the Weiss-type oscillations typical for lateral superlattices.

Acknowledgements The author thanks M.V. Entin for highly useful discussions. This work was supported by the Russian Fund for Basic Researches (Grant 99-02-17127), by the State Program of Russian Federation “Physics of Solid State Nanostructures” and by NWO.

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