QUANTUM WELL ENHANCEMENT OF NONPARABOLICITY EFFECTS ON THE EFFECTIVE ELECTRON MASS

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QUANTUM WELL ENHANCEMENT OF NONPARABOLICITY EFFECTS ON THE

EFFECTIVE ELECTRON MASS

U. Ekenberg

To cite this version:

U. Ekenberg. QUANTUM WELL ENHANCEMENT OF NONPARABOLICITY EFFECTS ON THE EFFECTIVE ELECTRON MASS. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-207-C5-210.

�10.1051/jphyscol:1987542�. �jpa-00226746�

QUANTUM WELL ENHANCEMENT OF NONPARAROLICITY EFFECTS ON THE EFFECTIVE ELECTRON MASS

U. EKENBERG

D e p a r t m e n t

of

T h e o r e t i c a l P h y s i c s , 1 K e b l e R o a d , G B - O x f o r d OX1 3 N P , G r e a t - B r i t a i n

Abstract An analysis of the nonparabolicity effects on the subband structure in a quantum well is presented. We derive simple analytical expressions for the perpendicular mass, which describes how the confinement energies are influenced, and the parallel mass, which gives the curvature at the bottom of a subband. The anisotropy of the bulk conduction band is included and is found to have a larger effect in quantum wells than in the bulk. We find that the nonparabolicity enhancement for the parallel mass is more than three times stronger than for the perpendicular mass. An expression is also derived for the Landau levels in a perpendicular magnetic field and the cyclotron mass is found to be equal to the parallel mass in the limit 8 4 . We finally compare our results with recent experiments.

The electron subband structure in a quantum well can often be determined to a reasonable accuracy by a simple particle-in-a-box calculation with the kinetic energy given by ti2k2/2m. For subbands fairly far from the bulk conduction band edge nonparabolicity corrections can be important. The subband structure for a quantum well is influenced by nonparabolicity effects in three different ways: 1 ) The confinement energies are shifted, 2) the curvature of the dispersion parallel to the layers E(kll) at the bottom of a subband corresponds to another effective mass than the bulk mass, and 3) the subband dispersions deviate from parabolicity.

While the nonparabolicity effects on the confinement energies have been considered by several authors, the E(kll) dispersion of the electron subbands has received less attention. We derive E(k,,) in a GaAs quantum well starting from an accurate expression for the bulk conduction band obtained from the 14-band calculation by Braun and R6ssler[l

1.

They have shown that if the spin-splitting due to the lack of inversion symmetry is neglected the conduction band dispersion expanded to fourth order in k is given by the expression

where m , = 0.0665 (in units of the free electron mass), a, =

-

1.969-10-~9 eV cm4 and 8 , =

-

2.306.10-29 eV cm4 for GaAs. The @,-term describes the anisotropy of the conduction band.

We assume that the layered structure is grown along the [001

]

direction, replace kz by -i dldz and add the potential V(z) of a finite quantum well. Since we have translational invariance parallel to the layers, kx and k remain good quantum numbers. When the kz4 term is included it is easily verified that the e~genfunctions still are of the form cos(Kz) and sin(Kz) (in the well) and Y exp(-XIZI) (in the barriers). Applying boundary conditions with continuity of the envelope wave function and its derivative divided by the effective bulk mass we obtain the usual transcendental equations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987542

JOURNAL DE PHYSIQUE

m X

t a n ~b =

1

( a v i . ~ ~ pi;; ; y j

m K ^{( 2 n )}

m X c o t Kb = -

m K ( o d d p a r i t y )

where b is half the well width and m, is the effective mass in AIxGal-x;\s. T h e difference compared to the parabolic case is that the relation between the energy E and the parameter I; is more complicated. In the general case we find

. .

m 201 +/3

w h e r e G = -

-

l 2 201

CY h

2 ^t

4 m , E 2m, 2 4 0 2 2

a n d G = - - - , 2 k l , + k + ~ k k CY'li4 CYti I 1 CY x Y

We have here introduced two parameters cr' and 0' defined by a '

= -

^{(2m1/h2)2 a.}

For GaAs at= 0.600 e v - l and P'= 0.702 e v - l . For simplicity we neglect the nonparabolicity effects in AlGaAs (which a r e not accurately known) and find

where V,, is the conduction band discontinuity. T h e equations (2a-b) can be solved numerically for different values of kx and ky and in this way we can determine the energy dispersion E(kll).

If k,=ky=O we find that (3) simplifies to

since a ' c < < l for most energies of interest. Here E is the confinement energy [ E I E(kll=O)].

From this relation it is appropriate to define an energy-dependent perpendicular mass

One can also derive a n approximation, which gives E(kll) a t the bottom of a subband in analytical form. We treat t h e kll-dependent terms in Eq.(l) in first-order perturbation theory and make some additional approximations, which should be reasonable as long as the penetration of the wave function into the barriers is small. (Details will be given elsewhere). We then find that the coefficient in front of kIl2 becomes

By equating this to ti2/2myl we define the energy-dependent parallel mass myl which is approximately given by

oscillator operators a and a t and eigenfunctions un defined by

a'a u n = n u n

When the Hamiltonian (1) operates on un it gives terms proportional to un with the exception of some terms proportional to un+4 and un-4, which, however, can be shown to be negligible.

After straightforward calculations we find

h e B ^{1 1} 1 tieB 2 2

'

2

E ( B ) = r

+ -

m [ I - ( 2 a

+@

) r ] ( n + $ ) - -

[-]

[ ( S n +Sn+J)a + ( n + n + l ) @ '

]

(12)

1 "'3

The cyclotron mass is defined by the relation

where AE(B) is the energy difference between the Landau levels with index n and n + l , which is what is measured in cyclotron resonance experiments. We thus find that in the limit B 4 the cyclotron mass is equal to the parallel mass. From (12) it is seen that the cyclotron mass increases with B. We should, however, expect further corrections to the B~ term due to confinement effects.

If the bulk expression (1) were expanded to sixth order, there would be terms proportional to k 2 k 1 F , which would give corrections to the coefficient of the. B~ term in Eq.(12).

We have calculated the dispersion E(kll) for a 50 A thick GaAs quantum well between A10.4Ga0.6As barriers taking the conduction band discontinuity to be 324 meV and using m,=0.1 for the effective mass in the barrier. In Fig. 1 we show ~ ( k ~ ? ) for the lowest subband in the

[ l o o ]

direction for three different cases: (a): the numerical calculation of Eq.(2a) together with Eqs.(3-6), (b): the analytical result with Eq. (10) for myl, and (c): the parabolic case. We see that

110- 1. Energy vs.

y3-

for the lowest.

subband in a 50 A

I 1 I I

.. -

GaAs quantum well

for three cages:

-

¹⁰⁰

^-

(a) Numerical solution of Eqs.

-

_A

(2)-(6),

e

(b) Analytical

2

approximation using Eq.(lO), and

(c) Without nonparabolicity effects.

-

I 9

0 1 2 3 4 5

~ , 2

^110-4A-2)

JOURNAL

DE

PHYSIQUE

Fig. 2. Parallel and perpendicular effective mass YS. well width.

0.070

Bulk

. - ^.-.

mass

I

0.065

1

^{I I}

0 50 100 150 200

Well width

(A)

the simple expression (102 is a quite good approximation to the exact curve (a). In Fig. 2 we show the parallel mass m,, and the perpendicular mass m i for the lowest subband as a function of well width in comparable approximations. Both the masses increase with decreasing well width, but the ratio between their enhancements remains almost constant somewhat larger than 3.

Recent interband magneto-optical experiments with an 80 A quantum well 131, for which this calculation gives mT, = 0.073, could indeed be quantitatively explained only if an 11% higher electron mass were used as input for a six-band model. Other recent experiments [ 4 ] with a 22

A

wide GaAs quantum well indicate that the cyclotron mass in the limit B 4 is smaller than according to Eq.(lO). One possible reason for this discrepancy is that for very thin quantum wells the wave function substantially penetrates into the barriers, and the neglect of nonparabolicity effects there can be serious.

In conclusion we have calculated the dispersion of the electron subbands in a quantum well and the Landau levels in a magnetic field perpendicular to the layers. Nonparabolicity effects have been taken into account using an expression with the conduction band dispersion in the bulk expanded to fourth order in k. The spinsplitting has been neglected but the anisotropy of the conduction band has been included and it is found that it gives a stronger effect in a quantum well than in the bulk. We have given some simple analytical expressions for the parallel and the perpendicular mass which explicitly depend on the experimentally accessible confinement energy. We find that the parallel mass is enhanced over the bulk mass about three times more than the perpendicular mass, which determines how nonparabolicity effects influence the confinement energies. (The relative shift of the confinement energy is in the present approximation about the same as the relative change in m i ) . The parallel mass is of importance for transport properties along the quantum well, including cyclotron resonance and in-plane exciton motion, and also determines the density of states.

Acknowledgements

-

I am grateful to Dr.R.J.Nicholas and Dr.J.Singleton for communicating results prior to publication and for many valuable discussions. I would also like to thank Dr.M.Altarelli for valuable comments. This work was financially supported by the European Research Office of the US Army.

1 M.Braun and U.ROssler, J.Phys.C

18,

3365 (1985).

2 J.M.Luttinger, Phys.Rev.

102,

1030 (1956).

3 F.Ancilotto, A.Fasolino and J.C.Maan, Superlattices and Microstructures

3,

187 (1987).

4 J.Singleton, R.J.Nicholas, D.C.Rogers and C.T.B.Foxon in Proceedings of the 7th International

I I

Conference on Electronic Properties of Two-Dimensional Systems (to be published).