Phonon-assisted trapping by shallow impurity in quantum well

Sohd State Commumcatlons, Vol 87, No 1, pp 27-30, 1993 Pnnted m Great Bntam

PHONON-ASSISTED

TRAPPING

BY SHALLOW IMPURITY

Yuri M. Sio Department

of Ekctncal

003% 1098/93$6 00+ 00 Pergamon Press Ltd

and Computer Engineering,

IN QUANTUM

WELL

and Vladimir Mitin Wagne State University,

D&oat,

Michigan &?OS!, USA

(Received 19 March 1993 by A. A. Maradudin) We have considered the capture of electron in a quantum well by shallow donor impurity due to the interaction with an acoustical phonon. Wave functronsof the zinc lowest bound states have been found by variational method. We have calculated the ratter of transitioor between the free statea and bound rtater u well IU between different bound Btates. In the cue of phonon wavelengthmuch rmalla than the width of the well L the trannition

ratter are proportional

energy, n, is a phonon occupation

to

l-I (% + l/2 f l/2) Lw6, where c ia the transferred

number

I. Introduction

by a number of author6 to account for different forms of confining potential, higher-excited levels, radiative transitions, etc. (for reviews see e. g. [8]).

The processes of free carrier trapping by impurity centers are known to play an important role in transport and noise properties of semiconductor structures [l]. Two approaches exist for the description of capture by a shallow attractive center in bulk eemiconductors. First, proposed by Laz [2] and then improved and extended by Leningrad’s group (31, treats the capture of carrier as classical descend through the quasicontinous spectrum of highly-ezcrted bound states due to cascade emission of low-energy acoustical phonons. The generalization of this model to the two-dimensional case was performed by Karpus [4]. Since this model is based on the assumption of the high density of impurity bound states at the negative thermal energy, its applicability is limited to low temperatures and materials with comparatively large effective Rydberg energy [5]. In GaAs quantum wells the Rydberg energy is less then a hundred kelvins and, at room temperature, the case opposite to that of the first model is realized. Therefore we will consider only the processes with the participation of the ground and several Zotu-ezcitcd states assuming that they give the major contribution to transport and low-frequency noise. For bulk materials the calculations of acoustical phonon assisted trapping have been performed with the help of exact coulombic wave functions of the free and several lowest bound states [S]. To our best knowledge there exist no correspondent treatment of capture by impurities in quantum wells. fn this paper we study the processes of trapping and detrapping of the electron in a quantum well by a shallow donor impurity placed either inside or outside of the well. We calculate the rates of electron transitions between the free states and several lowest bound states, as well as transitions between the different bound states assisted by the emission and absorption of longtudinal DA-phonons. Impurity bound states in quantum well are calculated within the variational method proposed by Bastard for the ground state of impurity in the infinite rectangular well [7] and generalized subsequently

II. Model

and Calculations

Following Bastard [7] we consider the system of the infinite square quantum well at -L/2 < I < L/2 and a shallow donor impurity at (O,O,z,). In the effective mass approximation the unperturbed Hamiltonian of the system is given by

where

rn* ir an effective mass of electron; er. is a lattice

&electric constant; R = 4~’ + (z - 2,)s is a distance from the rmpurity and p = (z,y) is a polar radius; term &W(Z) is the quantum well potential. For the ground and several low-excited states of Hamiltonian ‘H we use the hydrogen-like trial wave functions +Ml containing variational parameter6 A-

. _

Here m = 0, &l, f2,. . . irr a magnetic quantum number due to the cyhndrical symmetry of the Hamiltonian; n = 1, . . . , Iml+ 1 is the main quantum number, X(E) is the transverse wave function of a lowest subband,

X(Z)

=

CO6 7

0(L/2 - It]),

(3)

where e(z) is a step-functron; dimensronless factors A& are chosen to normalize the wave functions to unity; functions ‘P-(z) are polinomials of the order n - ]rn] - 1 wrth the coefficients chosen to provide orthogonality of the wave functions &,,,, with the different quantum numbers. In fact, trial functions (2) are equal to the product of the transverse wave function x(z) of free electron in the quantum well and hydrogen-like functions corresponding 27

PHONON ASSISTED TRAPPING

28

to orbital quantum number 1 = Inal (states with other I are not bound for a sufficiently thin well). Minimization of the expectation value of the Hamiltonian ‘H for the trial wave functions &,,,,(r) with conditions of orthonormality gives the values of the variational lengths &,,,, and ionization energies e,,,,, of the impurity bound states

On Fig. 1 we plot the results of calculations for ionization energies of nine lowest bound states corresponding to n = 1,2,3 vs. the position of the impurity for the quantum well of width L = 150 A (note that e,,,,, does not depend on the sign of m) For the sake of conveniency the ionization energies are measured m kelvins. Electron free states are taken as unperturbed plane waves corresponding to the lowest subband, flk(r) = s-"'

=p(h)

x(z),

(4)

where S is a layer area, and k = (Ic,, ky) is in-plane electron wave vector We use the Fermi golden rule to calculate the rates of transitions due to the interaction with the longitudinal DA-phonon modes

Vol 87, No 1

and V are crystal density and volume; sign Uplus’ corresponds to the emission and %inus” to absorption of a phonon We consider all possible sorts of transitions. a) trapping with i z k and f E nm; ai) detrapping with i E nm and f E k, aaa)mterlevel transitions with a E nm and f z n’m’ In the case of trapping one should multiply the left-hand side of Eq (5) by the number of Impurities A&, in the plane z = z, to obtain the probability for an electron to be trapped to the state &,,, of any impurity. For the matrix element of transitron with allowance for Eqs. (2) and (4) we obtain after integration over the polar angle ‘p

X

~4 I-= 0

e-R’X-m Jtmdls- klp),

(6)

and a similar expression for the mterlevel tramutions (here J,,, 1s a Bessel function). In order to find the overall transition rates W,+f we have to perform the integration over all possible phonon wave vectors QW 8-f = (2*)-3 V JkQ

w$

(7)

Magnitude of Q is specified by the transition energy e:

(5)

x+,-G-Q)

Here Wszr is the probablhty of transitron between mitral (a) and final (j) st at es per unit time mediated by DAphonon with wave vector Q = (q,qJ; D is the deformation potential constant, WQ specifies phonon dispersion relationship; nQ is the occupancy number of phonons; pi

---

edge of

m=l m+

1=2 k

yyy-_e-_w-L_

---___---

____

_

_____

QL>2r.

___

0 0

Irtf~urity pori+ioA’&)

150

1 Ionization energies of impurity bound states nm (m kelvins) versus impurity position z,. Width of the well L = 150 A. Fig

(8)

In GaAs this is realistic approximatron for charactenstical lengths of the order of hundred angstroms or larger and c 1 2irAsJL N 10 + 20 K. In this case due to the rapidly oscillating factors exp(aqsz) and Jl,,,l(lq - klp) m Eq (6) the trapping and detrappmg rates decrease rapidly with the increase of Q Analysis of the integral in Eq (6) shows that the main contribution to tranaition rates (7) is gven by the %ansverse” phonons with Q M qz > q, k since the matrix element of transrtion decreases more rapidly with Ik - ql than with qz The same conclusion is valid for interlevel transitions. Substituting the expression (6) for matrix element into Eqs. (5) and (7) and performing integration over phonon wave vectors Q we find the following results for the trapping rate

m-0

____________

where e = e- + tLzka/2m’ or c = Jc,,,,, - e,,,,,,,l In the following we restrict ourselves by the cases of such energies of transition, c, that phonon wave vector Q. is (much) greater than other values of the same dimeneionality, in particular

W(k+nm)=n,

-

D’

PLJ,

(n. + 1) Jr,,,,, eL2 (Q.L/zr)’

’

(9)

detrapping rate dW(nm-+

k)

d(haka/2m*)

m* = a

Da E

nC 3lL~ (QeL/2r)4

(10) '

Vol 87. No

PHONON ASSISTED TRAPPING

1

and the rate of mterlevel transitions

W(nm

Ds

n’m’) =

4

pLJ

C

’ (11)

number;

is a sheet concentration

z = s,; formfactors

the value s, I

[e-*+s_

pb4+1

[ewzt S,,,dd

(s+)

+ e-x-s_

at

and 2,

= e” l-e-’

Pk(t/2)

t dt

Justified,

(Z-)] ,

= ]z. f L/2] (A,

(I? - r#“l

+

group

S ,,,,,_,,,,,,,,(u) = e” L-e-I

(t’ - u’)‘“‘+‘~”

Results

t dt ,

)

function8

S,,,,,(u)

III. Numerical

For the numerical evaluation of the transrtion rates we use the parameters of the GaAs: m* = 0 067m+~,s = 5 24 x 10’ cm/s, pr, = 5.36 g/cm3, EL = 12 5, D = 7.OeV. The width of the well L is equal to 150 A, both phonon and electron temperatures are equal to 300 K On Fig. 2 we presented inverse electron capture times 1/r& and capture rates c,,,,, = l/nimpr- versus the impurity positron z, for the case of the nondegenerate electrons and nunp= 101o/cmz One can see that the dependence of W& on z, for different states nm has the form analogous to that on Fig 1, 1 e for the given wrdth of the well L the behavior of trapping probabilities is specrfied mamly by ionization energies c,,,,,. Note that for the

(s_)]

(2,) + e-Z-Sm+,w

where z* = ]2z, f L] /A, X,,,,)/X,X~,,,,

of impurities

F are given by

= XrL

FM

x

l/T&t= a E-‘W-~.

for small Q tends to the sound velocity a, factor

n,,,w = N-/S

T from Eqs. (9) and (10) follows that l/rtr a In equilibrium the relation ~,&/n~ = dairIn,, holds. rof E <

(n, + l/2 f 1/2) ~,,,,,,,,,, eX nmAn’m’ Ls (Q.L/ZTr)’

Here n, is a phonon occupation dwg/dQ

29

of levels with n = 3 the condition therefore

estimation

the results for l/r&

(8) is hardly

provide only an

for the actual rates.

On Fig. 3 the rates of mterlevel transitions calculated

with the help of Eq. (11) are plotted versus the impurity are pohnommls

Note that in equilibrium the transition

probabllitres (9) and (10) are related by the detarled balance equation. The transition

energies less or of the order of 100 K

mvolved m our problem tor Q m an almost therefore

correspond

linear

part

to phonon wavevec-

of WQ dependence

we can use the equation

Q. =

value of deformation potential constant Wrth the help of this relation we obtam -

(11) the following dependence

on the transferred

energy

s/fis

posrtlon

Two groups

of transitions

are presented.

a)

between the levels with n = 1 and n = 2, aa) between the levels with n = 1 and n = 3 We plotted only the results for transitrons

from the upper to lower level; the rates for

mverse processes

differ by the factor exp(-s,,,,,/T)

which

[9],

and the

D at Q = 0 from Eqs.

of the transitron

and the width

(9) rates

of the well:

W .-f a c+ (n, + l/2 f l/2) L-’ (numerrcal calculations show that for reasonable values of L and z, formfactors 3 are of the order of unity) Thus the capture (detrapping) of the carriers occurs mamly from (to) the free states near the bottom of the lowest subband Note that the transrtron energy 1s in its turn specified by ionization energies c,,,,, which depend strongly on z. and I, (cf Fig. 1) Following [3] we introduce the integral transition times rLr and rdetr which appear m the balance equations for carrier concentration The electron capture time T:~ specifies the probabrbty (per unit time) for ‘average” electron in the lowest subband to be trapped to the level nm of any impurity at z = z, plane 1/r:

=

I

smf(E) W(h -, nm) dE / imf(W

dE

(12)

The electron stay time ~2 grves the rate of electron detrappmg from the bound state nm to any free electron state m the lowest subband, 1/r=

=

J

-dE 0

Fig. 2. Inverse capture times l/r&, (left axis) and capture rates c,,,,, = l/nww,r,,,,, (right axis) to impurity bound state nm (Indicated

[l - f(E)]

[dW(nm -+ k)/dE] . (13)

Here E = tr”8’/2m’, and f(E) is the distribution function of the electrons on the lowest subband In the case

position

z,

Width

near the curves) vs. rmpunty

of the well L = 150

A, electron

and

phonon temperatures T. = Tph = 300 K; for calculation we used the sheet impurity concentration nlmP = of l/rk 10lO/cms

30

PHONON-ASSISTED TRAPPING

Vol 87, No 1

Fig 3. Itate~ of transitiona W(nm + n’m’) between impurity bound rtateo nm and n’m’ with the emission of acoustical phonon vemur the impurity porition z,. Width of the well L = 150 A; lattice temperature Tph = 300 K.

Analyrir of Figr 1 - 3 qim the following qualitative picture of the carrier trapping. For atatea with n 2 3 the energy reparation between levela ia rmall beii only a fraction of effective Rydberg energy and also due to the lifting of the degeneracy in quantum number m. Thus, for temperature greater or of the order of 10 K the rtates with n 1 3 can be treated = quasicontinucnrpectrum, therefore the tranritioru between the level are fast and can be treated in the fashion of Rena. [3, 41. The levelr with n = 1 and n = 2 are mparatcd from the other (cf. Fig. l), this implies the existence of several distinct groups of tranritione on Figa. 2 and 3. The time of capture to the ground state of the impurity ir specified by the transitiona from levelr with n = 2, direct trapping from the free rtate+shave much lower probabilities. Note that for the capture of electron to the ground state of on-center impurity other mechaniirmrof trapping can dommate, namely intaaction with an optical phonon or Auger-processes. The effect of carrier trapping in quantum wellr on nonequilibrium tranaport and noire should be examined by solving the balance equation for the electron dirtribution function for the free and impurity bound states with the trmition rates given by Eqs. (9) - (11) and proper scattering mechanirmaincluded (cf. [S]).

is cloee to unity at room temperature. Other tranritiona have higher probabilities and are not plotted on Fig. 3.

Acknowledgmentr. Thir work was supported by the National Science Foundation and Army Research O&e.

References [l] A. M. Stoneham, Rep. Prog. Phys. 44, 1251 (1981); L. Reggiani, V. Mitin, Riu. ZVwv. Csm., 12, 1 (1989). [2] M. Lax, Phys. Rev. 119, 1502 (1960). [3] V N. Abakumov, et al. Nonradiative ncombmation in semiconductors (North-Holland, Oxford, 1991). [4] V. Karp~, Sov. Phys. Semicond. 19, 1000 (1985). [5] L. S. Darken, Phge Rev. Lett. 69, 2839 (1992). [6] R. A. Brown, S. Rodriguez, Phga. Rev. 153,890 (1967); F. Belesnay, G. Pataki, phys. stut. sol. @I

13,499 (1966); W. Pickin, Solid St& Ekchwr. 21, 1299 (1978). [7] G. Bartard, Phys. Rev. B24,4714 (1981). [8] B. V. Shanabrook, PI+ca 146B, 121 (1987); A. A. Reeder et al. J. Quantum Electron. 24,169O (1988); s. FraissOli, A. pwqudO, Phg= stipb T39,987 (1991). [Q] 0. Madelung ed., Semiconductors: group IV elements and III-V compounds (Springer-Verlag, Berlin, 1991).