Pis'ma v ZhETF, vol. 101, iss. 11, pp. 840-845
© 2015 June 10
Surface photocurrent in electron gas over liquid He subject to
quantizing magnetic field
L. I. Magarill, M. V. Entin^ Institute of Semiconductor Physics, SB of the RAS, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia Submitted 13 April 2015
The photogalvanic effect is studied in electron gas over the liquid He surface with the presence of quantizing magnetic field. The gas is affected by the weak alternating microwave electric field tilted towards the surface normal. Both linear and circular photogalvanic effects are studied. The current occurs via indirect phototransition with the participation of ripplons emission or absorption. The photogalvanic tensor has strong resonances at the microwave frequency ui approaching to the frequencies of transitions between size-quantized subbands. The resonances are symmetric or antisymmetric, depending on a tensor component. Other resonances appear at ui ~ nuic, where n being integer and uic is the cyclotron frequency. It is found that the latter resonances split to two peaks connected with emission or absorption of ripplons. The calculated photogalvanic coefficients are in accord with the experimental observed values.
DOI: 10.7868/S0370274X15110053
Introduction. The stationery surface photocurrent (in other words, surface photogalvanic effect, SPGE) appears along the border of isotropic homogeneous bounded medium under the action of tilted alternative electric field [1, 2]. This effect was also studied in size-quantized systems [3-6]. Recently, SPGE attracted attention [7] as a tentative source of the microwave-induced photoresponse oscillations in 2D electron gas over the liquid He surface (EGOHeS). The microwave-induced resistance oscillations (MIRO) were the subject of numerous publications (see, e.g., review [8]). In particular, this effect was studied theoretically [9-11] in relation to EGOHeS (the theory of EGOHeS, see, e.g., in [12]).
Surface photogalvanic effect can be considered as another source of the observed photoresponse oscillations manifesting itself without stationary in-plane electric field. The theory of SPGE in EGOHeS was developed in [13] for the case of no magnetic field.
The present paper is a continuation of [13] with accounting for a strong magnetic field B directed perpendicular to the He surface (x,y). The forced progressive in-plane electron motion in a quantizing magnetic field is a result of the transitions between size-quantized subbands with synchronous directed in-plane transitions between the Landau states. The mathematical reflection of this idea is the second order optical transition probability with the participation of scattering, in particular, ripplon-induced scattering. The translational mo-
tion results from the interference of transition amplitudes caused by out- and in-plane components of alternating electric field. The scattering leads to a change of the in-plane electron momentum with a shift of the orbit center.
We will mainly follow the conditions of the experiment [7] on MIRO. We consider the electron gas of low density 106cm~2) over the He3 or He4 surface. At such low density the electron gas is non-degenerate. The photon energy is chosen close to the distance between the ground and the first excited size-quantized electron states A. The magnetic field is assumed to be weak enough so that the cyclotron quantum is some times less than A.
The mentioned resonance works clS cl magnification factor for transitions via the intermediate state. The mechanism of SPGE can be illustrated as follows. The population of subbands changes in- or contra-phase with the normal component of the alternating field if the frequency exceeds or it is less than A/h and has ty/2 shift if huj = A. This fact, together with the phase shift of the in-plane field component, determines the current direction.
Problem formulation. The phenomenology of SPGE in a magnetic field B = bB = B(0,0,bz) is determined by the relation for the current density
«iRe
(E-n(nE))(nE*
lot 2
n, [E, E*
-^e-mail: entin@isp.nsc.ru
a3[(E-n(nE)),b](nE*
iO!4
[[n,[E,E*]],b], (1)
840
IfiicbMa b >K3TO tom 101 Bbin. 11-12 2015
where n is the outer normal to the quantum well, E(t) = Re(Ee_iWi) is the uniform microwave electric field (E = (ex,ey,ez) is its complex amplitude). Real parameters on are the functions of magnetic field value B; «i and «3 correspond to linear and «2 and a.4 - to circular photogalvanic effects, respectively. "Drift" components «i and «2 exist in the case of zero magnetic field, while the "Hall" components «3 and a.4 originate from the magnetic field action, change their signs with the magnetic field and vanish if B =0.
The current components oc «2, «4 can be treated similar to translational motion of a rotating wheel. Electromagnetic field spin flow ic[EE*] (c is speed of light) transfers its angular momentum to electrons as a moment of force. The friction converts this moment to the translational electron motion.
We will base on the same model of 2D EGOHeS as in [13]. Electrons are attracted to He via the dielectric image force and the normal static electric field which composes Coulomb-like states xi(z) with energies q. The cyclotron frequency is supposed to be much lower than the Bohr energy. The interaction of electrons with surface waves (ripplons) and the homogeneous alternating electric field leads to the stationary surface current with density j.
Eq. (1) for current contains additive contributions oc EZEX and oc EzEy. The system under consideration is axially-symmetric. Hence, to determine all components of the photogalvanic tensor, one can find only x component of the current.
To calculate the photocurrent, we will use the approach first suggested by Titeika [14]. According to this approach, if unpertubed electron states are localized, the current can be expressed via the transition probability between these states. For the x-component of the current density, one can write
3x = | - - f(ep,)], (2)
where is the probability of transitions (caused
by perturbation) between the electron states with quantum numbers /3 and /3', £¡3 and Xp are the energy and the center of localization, correspondingly, f(ep) is the Fermi function, S is the system area, e is the electron charge.
Let us choose the vector potential of magnetic field in the form of A = (0, Bx, 0) when the electron states are localized in the x-direction. Electron states are described by a set of quantum numbers /3 = (l,n,ky), I
is the number of size quantized level, n is the Landau number, ky is the «/-component of electron momentum:
|/3) = —^==eikyV(f>n f^Il) XI{Z),
V Sa \ a J
where 4>n(0 are dimensionless oscillator functions, X/3 = —bza2ky is the localization center (the cyclotron orbit center), a = \Jcj\e\B is the magnetic length (we set h = 1). These states have energies £¡3 = £„+q, where £n = ojc(n + 1/2) is the n-th Landau level (n = 0,1,...), cjc = \e\B/mc is the cyclotron frequency, q is the Z-th size quantization level (/ = 1, 2,...). If image attraction to liquid He prevails q = —l/(2mag/2),
Xi (z) = 2zexp(-z/aB)aB3/2, X2(z) = z(2 - z/aB) exp (-z/2aB)(2aB)-3/2, (3)
where ae = «/me2 is the effective Bohr radius, k = = 4«i(ki + k2)/(k2 — ki),«i,2 are dielectric constants of gaseous and liquid helium. Note, that the functions (3) can be used also in the presence of the normal static field, if to consider them as variational functions with fitting parameter ae-
In our case transition probability Wp^-p/ is determined by the interaction with a microwave field and ripplons with the Hamiltonian 7i;nt(i) = Wer + F{t). The Hamiltonian of electron-ripplon interaction Uer is
(W)er = S-1'2 J2 Jq(&±q + (4)
q
where
Vq(z) = -J— ff^qVq(zy, Vq(z) = ^[1 -qzK^qz)], maB3 y 2/3 zA
(5)
Jq = e*qr, r = (x, y), 5+, 6q are the operators of creation and destruction of ripplon with wave vector q and frequency <jjq = q3/2 y/ao/p, p is the liquid helium density, <to is the helium surface tension coefficient. Interaction of electron with microwave field is given by
1
F{t) = — Eve-iwi + h.c. = -Ue-iut + h.c.; (6) Zuj 1
? P ? P
U = —Ev = —(E||V|| + Ezvz), (7)
to to
where v is the operator of electron velocity. In the first order on the interaction Hamiltonian, the contributions of (W)er and F(t) to the transition probability are additive and do not produce the photocurrent. Hence, the transition amplitude should be searched in the second (mixed in (U)eT and F{t)) order. In this order we have
842
L. I. Magarm, M. V. Entin
W,
TT
2S
1±1
Çr-2-2
q,± v
X [ö(£ßtß, -UJT^q)\(Bq)ßß'\2 4 + ö(eßiß,+üJTuq)\(BCi)ßß,\2],
(8)
where
(ßq W=E
ßl
(l
qtßßjUßuß
T) -10J + l£ßuß' Ußßl(Iq)ßl,ß>
rj + iuj + ießuß
(ri = + o).
(9)
Here /q = jqvq(z), ep^i means ep—e'p, Ag is the ripplon equilibrium distribution function. Quantity (Bq)pp/ is determined by Eq. (9) with change uj —> —uj and U —>• —> U+. For matrix elements (/q)/3,/3' one can write the following expressions:
(Iq)ß,ß' - (jq)n,ky;n',k'(vq)liV,
(10)
where
(Jq)n,ky;n',k' — &X,X'-a2qyi
x tJlqœX+bzUç sin (2tp)/2]eibz(n'-n)tp y
I min (n, n')\ ■ " -
'Join' \ )
max (n, n')!
/2e-"«/2L|nr" ,.,(11)
mm ) ' v y
L™(u) is the generalized Laguerre polynomial, uq = = q2a2/2, is the polar angle of vector q (qx = q cos
= çsin^); {Vq)ij, = /0°° dzVq{z)xi{z)xi'{z)- We will consider the PGE at resonance conditions, when the microwave frequency is close to the distance between size-quantization subbands I = 1 and 2. The expressions for the necessary quantities (Vq)(Vq)2,2, and (Vq)it2 can be found, for example, in [13].
The matrix elements of operator U are
x Re
2e e2a2bT 2naujr
xE E f^n,i)[i-f(£n',i')]x
X Jn,n> (,nq + i ± i J qyv^i/ X
X j<5(£„,„' + il,l> -UJT^q) X
r/ - iuj + ehii> r] + iu} + eiui
■■II \( ibrip-n ^'+14,11'+ l — ï/nJn-l,r,
x«sig(n - n) e *VE--:—!---—
V <jj„ — <jj
- e-ib^E4
-1 - Vn + 1 Jn+l,r;
(w ->• -
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