thermoelectric power of a quasi-two-dimensional quantum well in a corss electric field

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Page 1: Thermoelectric Power of a Quasi-Two-Dimensional Quantum Well in a Corss Electric Field

Short Notes K31

phys. stat . sol. (b) 164, K31 (1991)

Subject classification: 72.20 and 73.40

Institute of Metal Physics, Academy of Sciences of the USSR, Ural Division, Sverdlovsk’ )

Thermoe lec tr i c Power of a Quasi -Two-Dimensional Quantum W e l l

in a C r o s s Electr ic Field

BY V . V . KARYAGIN and 1.1. LYAPILIN

In a parallel electric field 2D-electrons as a result of heating acquire an energy sufficient to overcome the potential barrier and to go into a wide-gap material, thus leading to a negative differential conductivity / 1 / . Another situation arises in a strong electric field directed at right angles to the layer forming the quantum well (QW) . In this case the electrons do not warm up, but localize at one of the well walls. Hereby the dimensional quantization level energy is reduced / 2 , 3 / . In the case of the electric quantum limit the dimension quantization level drops considerably in a cross electric field and can even reach the QW bottom 1 2 1 . If hereby the ZD-electron concentration in the Q W is fixed, this will lead to a drop of the chemical potential in the well relative to the chemical potential in a wide-gap semiconductor. This stimulates the tunnelling of electrons from impurity donor levels of a highly alloyed wide-gap material via the interface to the 2D-channel. The electron concentration in the Q W ceases to be constant and increases with growing electric field.

Here we shall examine the differential thermo-e. in. f . and the electron thermal conductivity coefficient of a single Q W in a strong electric field in the case of an electric quantum limit. The calculation of the thermoelectric power will be made for the case when the temperature gradient VT is parallel to the layer plane, assuming that 2D-electrons are scattered on a 6-like potential.

Thus, we consider the Q W model having infinitely high walls in a cross electric field = (O,O,E) . The eigenfunction and eigenvalue corresponding to the bottom level of dimension quantization according to the varational method / 2 / are

(3) 6 = min c o p + F2/n2 f @(1/2F + PI(@ 2 2 +TI ) - ( l i2)cthp)I ,

’) 18 S . Kovalevskaya s t r . , SU-620219 Sverdlovsk GSP-170, U S S R .

Page 2: Thermoelectric Power of a Quasi-Two-Dimensional Quantum Well in a Corss Electric Field

K32 physica status solidi (b) 164

2 2 2 E = TI .h 12m Lz ; @ ( e l E L z / z o ,

where = (k , k ) and i? = ( x , y ) are the wave vector and radius vector in the x v x-y plane, R = L L L Lz the QW width, E~ the bottom level

energy in absence of an electric field, p is a variational parameter. Ho is the free

electron Hamiltonian in an electric field.

is the Q W volume, X - Y

In a strong electric field @ >> 1 ( p >> 1) ( 2 ) yields

For further analysis, it is convenient to represent the coefficients of the

thermo-e.m.f. Q and the electron thermal conductivity K in terms of the correlation

function I 4 I : Q = T - ~ ( J , , J ~ ) ~ = O ( ( J ~ , J ~ ) + w = o ) -1 ,

Here J1,J2 are two-dimensonal operators for charge flux and heat flux,

respectively,

[ i~ is the chemical potential) . The Hamiltonian for the interaction between 2D-electrons with three-dimensional

impurities reads

For further analysis we consider the correlation functions

t w = o t , Xij = ( J i , J . ) * Jiv = ( l l ih ) J 0 '

L.. = ( J iv , J . ) 11 J V

The subscript "0" of the correlation function ( A , B ) o means that the function is

calculated in the zero approximation for the interaction of band charge carriers and scatterers.

Using the equality

Page 3: Thermoelectric Power of a Quasi-Two-Dimensional Quantum Well in a Corss Electric Field

Short Notes

we have for the coefficient L.. '1

n.. 2 L.. = c.. C q ;(E; - v ) 'IF E+ 1 - F(E~~))&(EC+~ - E$ , '1 '1 if k ( k(

2-n.. 3nhN1G e "(G2+n2)

nll = 0, n12 = nZ1 = 1, nZ2 = 2 .

F( EG) is the 2D-electron distribution function.

Changing from summation to integration in (8) , we have

n. .+2 (LxL )' m3(kT) 'I

L.. = c.. B.. 'I 11 8 TI '+i6 '1 '

n.. Bij = 1 xrx - (p'/kT)] ch-'[x/Z - (11' /2kT)]dx; p'= p - 6 .

Similary for X. . we obtained 'I

n. .+2 exp(2-ni.)LxL (kT) 'I

x.. = B.. . 'I 87 fi2 11

K33

(8)

Using (9), (10) we have

2 2 2 B +n 3NiG m

1.. = co A(P) ; co = - , . A(P) = P cth P , 'I Z L P ~ 4 p +TI

where Zo is the 2D-electron relaxation frequency in absence of an electric field (with F = 0, + 0, A ( @ ) + 1 ) . If p >> 1 we have

A ( P ) = p / 4 = 0.49 @'I3 . (11)

Thus, the expression for the thermoelectric power becomes

Bz2; b = - K =-= J 3 2 ( k 3 3 ( 1 - b )

-

kT2Zo 8nh2Zo BllB22

In case of a degenerate electronic gas the kinetic coefficients Q and K are

3 physica (b)

Page 4: Thermoelectric Power of a Quasi-Two-Dimensional Quantum Well in a Corss Electric Field

K34 physica status solidi (b) 164

n2k2T Ns , K = k n k T m

6 e h2 Ns & = 9

3m C o

where Ns is the 2D-electron concentration in the QW. The expression (14) is in

agreement with the results of /5 / in absence of an electric field. In order to calculate the chemical potential we shall consider a QW the energy

structure of which is determined by the expressions / 6 /

v+ L with z € [- -2 = L ~ , - L]

2 2 ’

Here V ( z ) is a potential limiting the motion of electrons along the z-axis; N D , LD, E are donor volume concentration, depletion layer thickness, and dielectric donstant, respectively. E is the donor ionization energy. D

The electroneutrality equation can be written as

2NDLD = Ns (15)

while the condition of equilibrium between 2D-electron gas and donor states in a wide-gap semiconductor is as follows:

LzkT m(kT) 2 2 In (K) + p + E~ ,

2 v = - ln(K) + 8 n a N D h b 4a

K = 11 + exp(p’/kT)]; Vb = V(-Lz/2 - 0) - V(-Lz /2 + 0) and ~1 is the Bohr radius. Equation (16) enables one to find the chemical potential p in the QW in an

electric field with the exchange of electrons between 2D-channel and donor states in a wide- gap material held in view.

Fig. 1. Energy diagrams of a Q W in a cross electric field

Page 5: Thermoelectric Power of a Quasi-Two-Dimensional Quantum Well in a Corss Electric Field

S h o r t Notes K35

E - I n case of a degenerate electronic gas

we obtain from (16)

T h e r e s u l t s of numerical analysis of 6 , Q , and K ca r r i ed o u t f o r pa rame te r s : Vb /k = 4600 K, Lz = 5 nm, m = 0.067 m0 ( w h e r e mo is the f r e e e l ec t ron m a s s ) , T = 4.2 K , E = 12, are p r e s e n t e d in Fig. 2. T h e dec rease in thermoelectr ic power with growing electr ic f ie ld , as o u r analysis has shown , is due b o t h t o t h e rise of

the ZD-charge carrier concentrat ion and t o the effect of the electric f ie ld on the conduct ion electron relaxat ion f r e q u e n c y .

Re fe rences /1/ M.I. ELINSON and V.A. PETROV, Mikroelektronika 16, 522 (1987). / 2 / G. BASTARD, E.E. MENDEZ, L .L . CHANG, and L. ESAKI,

Phys. Rev . B 28, 3241 (1983). / 3 / I . J . FUTZ, J. appl. Phys. 61, 2273 (1987). 141 1.1. LYAPILIN and V.V. KARYAGIN,

Fiz. tekh. Poluprov. g, 295 (1983). / 5 / S . S . KUBAKADDI, B .G . MILIMANI, and V.M. JALI,

phys. stat. sol . ( b ) 139, 267 (1987); 137, 683 (1986). / 6 / V.M. GOMES, A .S . CHAVES, J . R . LEITE, anq J . M . WORBOCK,

Phys. Rev . B 35 3984 (1987).

(Received July 23, 1990; in r ev i sed form January 22, 1991)