ALLOY FLUCTUATIONS IN THE PRESENCE OF STRAIN IN CdTe / (Cd, MnTe) SUPERLATTICE SYSTEMS

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ALLOY FLUCTUATIONS IN THE PRESENCE OF STRAIN IN CdTe / (Cd, MnTe) SUPERLATTICE

SYSTEMS

S. Jackson

To cite this version:

S. Jackson. ALLOY FLUCTUATIONS IN THE PRESENCE OF STRAIN IN CdTe / (Cd, MnTe) SUPERLATTICE SYSTEMS. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-349-C5-352.

�10.1051/jphyscol:1987575�. �jpa-00226779�

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Colloque C5, supplement au nOll, Tome 48, novembre 1987

ALLOY FLUCTUATIOEiS IN THE PRESENCE OF STRAIN IN CdTe/(Cd,MnTe) SUPERLATTICE SYSTEMS

S.A. JACKSON

AT and T Bell Laboratories, Murray Hill, NJ 07974, U.S.A.

ABSTRACT

The effect of strain on the valence band offset, bandgap and hole mass has been included in a calculation of the density of states arising from alloy fluctuations in the CdTe/(Cd,Mn)Te superlattice system for two growth directions. The consequences of this for the formation of interface states will be discussed.

In this paper we consider the effect of alloy disorder on optical properties of (111) and (001) oriented CdTe/(Cd,Mn)Te superlattices (SLS). First, we optimize the density of states (DOS) of an interface exciton in a QW localized by the random potential resulting from changes in the conduction and valence-band offsets due t o alloy disorder in the barriers. This is done w.r.t. trial wavefunction range parameters for motion both perpendicular and parallel t o the layers t o find the broadening of the exciton linewidth due to alloy fluctuations, which gives the characteristic localization energy for (001) and (111) QWs. We compare our results with estimates of binding energies of holes localized by local fluctuations in interface strain given by Zhang et al. [I] and Chang et al. 121. Secondly, we estimate the effect of the random strain field, due to fluctuations in the interface strain, on the band offsets via an effective Debye-Waller factor.

Previous derivations of exciton line broadening in random alloy semiconductors [3] have used the optimal fluctuation technique of Halperin and Lax [4], Zittarz and Langer [5], and Lifshitz [6]. One solves the self-consistent equation for the wavefunction of a particle in a random potential ( p = Lagrange parameter),

where the random potential

is a sum of contributions from each site and mattering type, and

W ( r

- - -

r l ) =

N 5

^{v ( r}

- - - ^-

^R)^{v ( r l}

^-

^R)^d3R ⁽³⁾

(N

= mean impurity concentration). In equation (2)

q, =

S ~ ~ = Q I A

N

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987575

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C5-350 JOURNAL DE PHYSIQUE

where a! = dE,/dx or dE,/dx for t h e conduction or Equation (1) results from optimizing t h e DOS = poexp

t h e wave function + ( r ) , where

-

K = t h e expectation value of t h e kinetic energy, E is t h e energy of the s t a t e in question and S is

For t h e combined electron-hole problem, if we assume t h a t t h e exciton equation, due to t h e Coulomb interaction, has been solved for t h e relative wavefunction and t h e exciton binding energy is larger t h a n typical energies due t o alloy fluctuations, then for t h e center-of-mass (CM) wavefunction, t h e self-consistent equation is of t h e same form as in equation (1), b u t where t h e coordinate r becomes t h e CM

-

me r

+

^{mh r}

-

^h

coordinate R

-

= ^me- e

+

^mn and t h e CM random potential a n d Q W potential result from averaging t h e full potential over t h e relative wavefunction, e.g.

where r is t h e relative coordinate (i=e, h; and i#j).

-

d ~ b u i k Then in equation (4), a = -

+

^---dE'train (note: = 1.5eV for this system).

d x dx dx dx

Under strain, t h e valence band degeneracy is lifted and a n anisotropic hole mass results, leading t o a n anisotropic (w.r.t. mass) form of equation (1). If we have a Q W grown o n a CdMnTe buffer layer, t h e CdTe layer is strained such t h a t the heavy hole band rises a n d for a (111) Q W (in units of t h e bare electron mass), m l l

=

m, = 0.77, ml = 0.143, while for an (001) QW, m, = 0.53, ml = 0.137; for t h e CdMnTe layers we use for t h e heavy hole mass, mhh = 0.5. Since t h e Q W makes t h e z-direction special, averaging over t h e relative coordinate suggests a wavefunction similar t o t h a t used by W u e t al (71, i.e.

We integrate t h e random potential over t h e z-wavefunction t o get

Then leads t o an effectively 2D random potential problem where it is known [8]

t h a t t h e DOS has t h e form

(4)

-

with Eefr = E- K, (X 47r), and

-

_li2

K, =

-I

^dz

/ I +

expectation values of QW potentials

2% (11)

The value of Eaff which maximizes the DOS is Eerf = -3/2

Q/x,

which gives E = K,

-

3/2 Q/X. If now this energy is minimized wrt u for Z, = L/2 (exciton localized !t the interface), we find for a (001) QW, the z-localization distzpce is u-' = 35A and E = 1.3 meV, while for a (111) QW, we find u-l = 28A and E = 2 meV for x = .25. This confirms the result of Chang et al. [2] that a disorder induced interface well can lead to binding of a heavy hole at the interface with different binding energies for the different SL orientations.

Next we take a different approach. As pointed out by Siggia [9] in his work on k.p theory in semiconductor alloys, the bulk bandgap gets renormalized due t o a rescaling of the pseudopotential form factors by a Debye-Waller factor, e x p ( - - ~ ' < u 2 > ) , where G is an appropriate reciprocal lattice vector and < u 2 > 1

6

is the mean square deviation of atoms in the semiconductor from their lattice positions. In the QW, the lattice mismatch strains eT(x) and exy(x) give rise t o an effect2e ~andomization of the atom positions in C d T s d u e to alloy fluctuations, since e--tet=T+d7, w h e r e 2 i s the strain tensor, where

e

is the position of an atom in the unstrained lattice, while? -is the position in the strained lattice. Alloy

de..

fluctuations mean that e i i = e ( p ) + A ~ x where e{p) is the usual strain tensor.

d x

Since the change in e,j is random, this gives rise t o fluctuations,

ge,

in lattice positions in the strained layers, which leads t o an effective Debye-Waller factor

++ -++

e- w = <e- iG.Ut

>

⁼

f

~ ( A X ) P ( A X ) ~ - ' ~ ~ '

(12) where 4 Ul = Ax

[g ^-71

and where P(Ax) is the distribution function for alloy fluctuations. If we take, as suggeste.d by Goede et al.

[lo],

a Gaussian form for P(Ax), where the mean fluctuation, A, is given by equation (4), we obtain a Debye-Waller factor e-c2A2 where

for a (001) QW and

for a (111) QW, where ele2e3, the number of unit cells in the volume of interest is --a&, the exciton volume. Then !1&2&3=-.

v

For ~ 1 . 2 5 we find for a (001) QW that the bandgap shrinks by a factor .997, and for a (111) QW, it shrinks by .918.

n

This gives a change in bandgap, due to randomization of lattice positions, of AEg=Eg-e-WEg. If we assume that this change is equally shared by a change in conduction band offset and valence band offset, i.e. AE,=AE,=-AEg, 1 we have

,

2

A E , = ~ ,Zg.

.

.-e expect the greatest effect on valence band states. If we assume 2

that this change in valence band offset extends over the same z-range as found in

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C5-35 2 JOURNAL DE PHYSIQUE

the DOS discussion, we have a potential well of depth Uo=,WEg. 1

-

This gives a binding energy for heavy holes of EB=2.4 meV for a (001) QW and EB=12.8 meV for a (111) QW, comparing favorably with experiment [1,2].

REFERENCES

[I] X.-C. Zhang, S.-K. Chang, A. V. Nurmikko, L. A. Kolodziejski, R. L.

Gunshor and S. Datta, Phys. Rev. B 31, 4056 (1985).

[2] S.-K. Chang, A. V. Nurmikko, L. A. Kolodziejski and R. L. Gunshor, Phys.

Rev. B 33, 2589 (1986) (and references cited therein).

[3] S. D. Baranovskii and A. L. Efros, Sov. Phys. Semicond. 13, 1328 (1978).

[4] B. I. Halperin and M. Lax, Phys. Rev. 148, 722 (1966).

[5] J. Zittarz and J. S. Langer, Phys. Rev. 148, 741 (1966).

[6] I. M. Lifshitz, Sov. Phys. JETP 17, 1159 (1963).

[7] J-W. Wu, A. V. Nurmikko and J. J. Quinn, Phys. Rev. B 34, 1080 (1986).

[8] D. J. Thouless and M. E. Elzain, J. Phys. C. 11, 3425 (1978).

[Q] Eric D. Siggia, Phys. Rev. B 10, 5147 (1974).

[lo]