Quantum wells with enhanced exciton effects and optical non-linearity

Materials Science and Engineering, B1 (1988) 255-258

255

Quantum Wells with Enhanced Exciton Effects and Optical Non-linearity EIICHI HANAMURA and NAOTO NAGAOSA Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113 (Japan)

MASAMI KUMAGAIand TOSHIHIDETAKAGAHARA N I T Basic Research Laboratories, Musashino-shi, Tokyo 180 (Japan)

(ReceivedJune 21, 1988)

Abstract Exciton effects are studied theoretically for a quantum well of a semiconductor sandwiched by barriers with a smaller dielectric constant and a larger energy gap. The exciton binding energy increases markedly so that the radiative decay rate of the exciton and the non-linear optical susceptibility are also shown to be enhanced. The optical properties of two-dimensional (2D) excitons in quantum wells have been the subject of extensive studies [1-4]. The following features were observed: (1) a large blue shift of the exciton level; (2) an enhancement (by a factor of up to 4) in the exciton binding energy. The exciton, however, in a GaAs-Gai_xAlxAs quantum well is thermally dissociated within 0.3 ps at room temperature [5]. This prevents the quick non-linear optical response because the electrons and holes thus thermally activated have a long lifetime of an order of a nanosecond. In this paper, we propose a new quantum well which is sandwiched by barriers with a smaller dielectric constant and a larger energy gap than the well. The electron-hole Coulomb attraction in the well works very effectively through the barriers with reduced screening. As a result, the exciton binding energy increases much more than in the G a A s - G a l _ xAlxAs wells when the well thickness is decreased. Then the exciton will become stable against thermal dissociation and at the same time will be shown to have a larger oscillator strength. These effects will also enhance the optical nonlinearity [6]. These points are discussed quantitatively in this paper. Let us consider a quantum well ( - l ~ < z ~ l and z < - l ) with a larger energy gap and a smaller dielectric constant e 2

0921-5107/88/$3.50

than the well. The Coulomb attraction working between an electron at (G, G) and a hole at (r h, Zh) is evaluated by the image charge method as follows: 2

e

co

E1 . . . .

qn

(1)

{(re-- rh) 2 -}- (Ze -- Zhn/ J

where q,, = q_ ,,

= (e~- e~2)t'l

(2)

kel + e2] and

= ]Zh + 2nl Zhn [-- Zh + 2nl

for even n for odd n

(3)

We can show that this expression is equivalent to that given by Keldysh [7] in terms of an integral over the wavevector. We consider the isotropic conduction and valence bands with effective masses m e and m h respectively, and we use the effective mass approximation to obtain the exciton state. Then the hamiltonian of an electron and a hole in the quantum well is described with respect to an energy gap Eg as 4'12

2

~2

H = - 2-~ VR - ~

.j~2

(1 Z

2E1 ,,0

.,~2

Vr 2 2m¢ 3Ze 2

e2 if- Veh "[- - -

02

qn

Iz~-Ze.I

2 m h CJZh2

1 ~-

02

Iz~z~.l

) (4)

where M = m e + m h and /U=memh/(me+ mh). R and r are the 2D coordinates in the quantum well respectively for the centre-of-mass and relative motions of the electron-hole pair. The last term

© ElsevierSequoia/Printedin The Netherlands

256

denotes the self-energies of an electron and a hole respectively due to interactions with their image charges where ze, is obtained from eqn. (3) by replacing Zh by Z~. This term should be taken into account in evaluating the 2D Bloch function. However, the difference between this treatment and the present treatment is neglected in the firstorder approximation for the infinite barrier potential. For such a thin quantum well, as 2l is less than or of the same order as the exciton Bohr radius a B ell~2/~e 2, we take the following trial function for the lowest exciton state [8, 9]: =-

cos/ 7]

".'(r, zo, zhl=

Z,,exp(r)

× cos/ /)

where the variational parameter a denotes the extent of electron-hole relative 2D motion in the well and is determined so as to minimize the expectation value of the hamiltonian (4) in the trial function (5): E = ( W I H [ ~ ) . Here the wavevector K for the centre-of-mass motion of the exciton in the 2D plane of the quantum well was set equal to zero. The variational parameter a which minimizes E is shown in Fig. 1 as a function of the ratio el/e 2 of the well dielectric constant to that of the barrier for several values of the well thickness 21laB. It should be noted that the effective exciton

i

i

i

Bohr radius a becomes less than the value a~{/2 m the 2D limit for 2l/aB = 0.4 and for c~/e: larger than 5. The exciton binding energy E~,~b is evaluated in terms of the wavefunction ~ (eqn~ 5)) with the fixed a as

I5~,~ h = E(a= o o ) -

E(a)

/ i

=-i~

-~V£+

VchN

where the second equality is justified for the trial wavefunction in eqn. (5). This exciton binding energy is plotted in Fig. 2 as a function of el/g 2 for several values of the well thickness 21laB. Here Ee×cb is normalized by the effective Rydberg energy R-= jAe4/2~2612 of the bulk exciton in the well material. An enhancement of exciton binding energy was qualitatively pointed out by Keldysh [7] for a limiting case. For example, let us consider the quantum well of GaAs with barriers of alkali halides, e.g. an NaCI crystal on both sides. NaC1 has a static dielectric constant e 2 of 5.6 and a band gap of 8.6 eV. The ratio el/e z of the dielectric constants is 12.9/5.6 = 2.3. Then the exciton binding energy in a well of the width 2l/aB = 0.4 ( 2 l = 4 0 A) is estimated from Fig. 2 to be 6.2 times that of the bulk exciton (5 meV), i.e. 31 meV. For a CdS well sandwiched by NaC1 barriers, the exciton binding energy is estimated to be 4.5 times the bulk energy, i.e. 135 meV for 2l = 12 A and el/e 2 = 1.5. These are larger than the lattice temperature. The barrier of NaC1 has a

r

i 2-

4 20i

g t.u

2

~or 0.4 0.1

1

10

L i

6

1 0

6~ /62 Fig. 1. Extent of electron-bole relative motion a in the welt plane. The value a is plotted as a function of ratio of the well dielectric constant e~ to the dielectric constant ez of the barriers. The 2D radius a is normalized by the bulk exciton Bohr radius a~. The numbers 2, 1 and 0.4 denote the well thicknesses 2 l] aB.

0L-----0.1

10

1

100

( , ,(~

Fig. 2. Binding energy Eexcb of the 2D exciton in the well. The value of Ee,cb is plotted as a function of the ratio el/e2. E¢,cb is normalized by th binding energy R of the bulk exciton in the well material. The well thickness 2l/a B is chosen to be 2, 1 and 0.4. The numbers in the figure denote these thicknesses.

257 band gap about five and four times larger than the wells of GaAs and CdS respectively so that our approximation of the infinite barrier may be justified. The combination of GaAs (a = 5.65 A) and NaCI (a=5.63 A) has lattice constants which coincide rather well although their crystal types are different. The combination of AlP (a = 5.467 A; e~=9.8) and CaF2 (a=5.463 A; e2 = 1.96; Eg= 10 eV) is also interesting, where el~e2=5 and an exciton binding energy which is 10 times larger is expected. As to the eigenenergy of the exciton, the enhanced binding energy is almost compensated by the self-interaction energies of the electron and the hole with their own image charges, which are described by the last term in eqn. (4). As a result, there is a blue shift in the exciton level because of the increase in the kinetic energies as a result of the confinement in the z direction. This is almost independent of el/e 2. The exciton in the quantum well loses translational symmetry in the direction perpendicular to the well. Therefore, it can radiatively decay in this direction. Furthermore, it keeps the 2D macroscopic transition dipole moments:

[11] and f/ALT= 1 meV for CdS [12]. The following material constants are used: for GaAs, aB = 100,~k, Eexcb=5 meV, e0=12 and 2 = 8 0 0 0 A ; for CdS, aB = 30 A, Eexcb= 30 meV, e0 = 8 and 2 5000 A. Bulk excitons behave as almost ideal bosons, i.e. harmonic oscillators. As long as this is the case, these excitons cannot contribute to nonlinear optical responses. When exciton-exciton interactions, i.e. anharmonic terms, decay and/or relaxations of excitons become sufficiently large, the third-order optical susceptibility )(3:, (co; - co, co, -co) becomes finite and is enhanced by the fourth power of the macroscopic transition dipole moments. In terms of the decay rate F0 = 27 =-1/T~, the relaxation constant F = 7 + 7' (7' is the rate of pure dephasing) and the exciton-exciton interaction energy ¢/~o~m,we can evaluate Z i3~ for such an incident frequency co nearly resonant at the exciton coo, as contributions from other levels are negligible [6]: =

X(3}((D; -- co, co, -- co)

Ip,,[ 4

.fi3(co_ co0 + iF)2 co- (°ll-iF

P,, ~

x \ sra~/

1

/~

+ co - co,,- col,,+ iF] V

(6)

;

where L 2 is the optically active area of the well and /~, is the band-to-band transition dipole moment. The radiative decay rate F~ of the exciton with the zero wavevector in the plane is evaluated as F~j-=27=24Jr(X/a)27~ [10], where h~o0 is the energy of the lowest exciton. ?/~= 41~c~]2/3h22 and 2 is the relevant wavelength corresponding to the energy of the lowest exciton hco0. The radiative decay rate of the 2D exciton is enhanced by a factor 243(2/a) 2. This comes from the coherent nature of the 2D exciton over the whole optically active region and the reduction in the effective Bohr radius a due to the enhanced exciton effect. The value F 0 is independent of the well thickness as long as l = a B'~ 2 < L. It is also independent of the barrier material as long as the band gap of the barrier is much larger than that of the well. It is estimated to be 0.36 meV (2.8 ps) for a GaAs well with a = aB/2 and 1.2 meV (0.6 ps) for a CdS well with a = aB/2. Here we estimated the transition dipole moment from the longitudinal and transverse splitting of the bulk exciton l~iALT=4ktc~=/eoaB3=O.1meV for GaAs

Here P~ is the transition dipole moment of the lowest exciton in the direction Ok of the incident radiation field: \~a~/

#~,(k

(7)

and V is the optically active volume of the quantum well 2/L 2. Let us consider the case in which the radiative decay (Fo---)2y is dominant over 7'. When we normalize energies by fifo, )(3: is expressed as

;¢i~,,_ IP°I~ 1~3F03V

(~(coZco"com~) /

Ftl ' Ftl

The dispersion of Z <3; is represented by G and the magnitude of Z <3; is determined by the prefactor ]P014/C/3F~3 V. Here G depends only on (co-co0)/F0 and coint/F0 and is independent of material constants as derived from eqn. (6). The real and imaginary parts of G are evaluated as a function of the off-resonance energy (co- coo)/Fo for several values of com]F0. The dependences of

258 Z i3) on the well thickness 21/a B and the ratio e~/e2 of the dielectric constants are represented through P0 and F 0. T h e real part of G(A) with A =- (~o - W o ) / F o shows a maximum of about unit)' at around A -- 1 and a minimum of about minus unity at around A = - 1, while the imaginary part of G(A) has a minimum of the order of - 2 at around A = 0. T h e coherent enhancement of the exciton transition dipole m o m e n t is limited by the lateral size of the optically active region of the well a n d / o r the spatial extent of optical coherency of the exciting light field, whichever is smaller. H e r e it is assumed to be of the order of the optical wavelength 2 so that L 2 is nearly equal to 22. T h e n the magnitude [Pol4/~3F~3V with V - L2(2l) ~ 22aB is estimated to b e Z '-~I= 9.6 esu for the GaAs quantum well 2 1 / a ~ = 0 . 4 ( 2 l = 4 0 A) with e l / e : = 2 . 3 and Xi3i= 1.6 esu for the CdS quantum well 2 l / a u = 0 . 4 ( 2 l = 12 A) with e l / e : = 1.5, both corresponding to the NaC1 barrier. T h e enhancement of X 3) under nearly resonant 2D exciton excitation arises for the following three reasons. First, the exciton is a coherent excitation over the whole optically active region of the quantum well so that the transition dipole moment has a macroscopic enhancement L / a - ~ 2 / a given by eqn. (7). Secondly, the exciton effect is enhanced by the welt structure sandwiched by the barriers with a smaller dielectric constant. As a result, as Figs. l and 2 show, the exciton binding energy becomes larger and the effective Bohr radius a is reduced. This also enhances the transition dipole moment by a factor L / a and Z 3' by ( L / a ) 4 together with the first effect. Thirdly, the large decay rate F0, i.e. the short decay time T t = 1/F(~ which is shorter than a picosecond, makes the 2D excitons deviate from ideal bosons and makes the nonlinear response, e.g. Z ~3), finite. At the same time, the fast decay guarantees a fast optical response in this time order, even under near-resonant excitation. Still there remain several problems. First, the enhanced optical non-linearity and fast response are limited to a lattice temperature much lower than 100 K in which the radiative decay dominates p h o n o n scattering. Secondly, we assumed

perfect well and barrier structures. Fluctuation.~ in thickness and defects will limit the coherent length of excitons and consequently the enhancement of Z'3( We can conclude that a quantum well sandwiched by barriers with a smaller dielectric constant has larger exciton effects. As a result, the transition dipole moment is enhanced so that both the radiative decay and X -~ become larger. At the same time, we can expect a faster response time and stability of the exciton against thermal dissociation because the exciton binding energy can be made much larger than the thermal energy k BY' at r o o m temperature.

Acknowledgments Two of the authors (E.H. and N.N.) thank the Ministry of Education, Science and Culture of Japan for financial support under Grant-in-Aid 6 3 6 0 4 5 1 6 for Scientific Research on Priority Areas, New Functionality Materials--Design, Preparation and Control.

References 1 A special issue on excitonic optical nonlinearity, J. Opt. Soc. Am., 2 (7)(1985). 2 C. Weisbuch, R. Dingle, A. C. Gossard and W. Wiegmann, Solid State Comrnun., 38 (1981) 709. 3 L. Schultheis, M. D. Sturge and J. Hegarty, Appl. Phys. Lett., 47(1985) 995. 4 L. Schultheis, A. Honold. J. Kuht. K. K6hter and C. W. Tu, Phys. Rev. B. 34 (1986)9027. 5 W. H. Knox, R. L. Fork, M. C. Downer. D. A. B. Miller. D. 8. Chemla, C. V. Shank. A C. Gossard and W. Wiegmann, Phys. Rev. Lett.. 54 (1985 1306. 6 E. Hanamura, Phys. Rev. B. 37 1988 1273. 7 L.V. Keldysh, Pis'ma Zh. Eksp. feor. Fiz., 29 19791716 (Sov. Phys.--JETP Lea.. 291t 979) 658 8 G. Bastard, E. E. Mendez. L L. Chang and L. Esaki. Phys. Rev. B, 26 (19821 1974:28 1983) 3241. 9 D. A. B. Miller, D. S. Chemla. T. C. Damen. A. C. Gossard, W. Wiegmann. T. H. Wood and C. A Burrus. Phys. Rev. B, 32 (198511043. 10 E. Hanamura, Phys. Rev. B. 38 1988 1228. 11 R. G. Ulbrick and C. Weisbuch. Phys. Rev. Lett.. 38 (1977) 865. 12 J. J. Hopfield and D. G. Thomas. Phys. Rev.. 122 (1961 35.