Exciton energy spectrum in GaAs in a magnetic field

Journal of Luminescence 12/13 (1976) 277—284 © North-Holland Publishing Company

EXCITON ENERGY SPECTRUM IN GaAs IN A MAGNETIC FIELD S.B. NAM

“,

D.C. REYNOLDS and C.W. LITTON

Air Force A vionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, USA

We have studied the energy spectra of the free exciton and a bound exciton—donor complex in GaAs in the magnetic field range from 12 kG to 40 kG; by photoluminescence (krypton laser—6471 A) and reflection. Calculations based on the perturbation and adiabatic schemes appear not to be applicable to present measurements. A phenomenological scheme proposed earlier accounts well for our data. We have also determined the following parameters; the effective masses of electron, m~= 0.066 ±0.001 and exciton, /20 0.048 ±0.002 via the binding energies of donor, 5.77 ±0.20 meV, and of exciton, 4.27 ±0.20 rneV, the effective Zeeman Splitting factors of electron, g~= —0.5 ±0.05 and of hole. = 1.0 ± 0.2, and the band gap of 1.5196 ±0.0005 eV, and the electron—hole exchange energy of 0.17 ±0.10 meV.

1. Introduction During the last several years there has been a great deal of interest as well as considerable effort extended in understanding the spectra of the free exciton and of exelton—impurity complexes in zinc blence type semi-conductors such as GaAs [1—4]. In particular, the magneto-optical studies in semiconductors allow one to determine various band parameters. The main objects of this paper are three fold: (1) to present the observations, by photoluminescence and reflection, of optical transitions of free excitons in the ground and excited states both in zero field and also in the field range from 12 kG to 40 kG, and those, by photoluminescence, of a bound exciton—donor complex; (2) to deduce the various band parameters such as the effective Zeeman splitting factors of electron and hole, and the effective masses of electron and exciton, and (3) to demonstrate that the phenomenological scheme [5] proposed for the exciton spectrum in magnetic fields accounts well for our measurements. In section 2 we discuss the experiment and in section 3 the theoretical description is given. In section 4 we discuss the comparison between calculations and measurements and summarize our findings.

*

Senior Research Fellow of National Academy of Sciences (USA) 277

—

National Research Council.

278

S.B. Nam et al.

/ Exciton

energy spectrum in GaAs in a magnetic field

2. Experiment In this section we discuss the measurements of Zeeman splittings of spectra of free excitons and a bound exciton—donor complex in GaAs. The ground states of the free excitons have been observed by reflection while those of a bound exciton— donor complex by photoluminescence. The epi-layers used in these experiments have very high mobilities (140 000— 250 000 cm2/V s at 77 K). The highest quality material is needed to observe the sharp line spectra (line width ~0.1 A). The samples were mounted on one end of a sample holder which was in turn placed in the tip of a glass helium dewar. The mounting was arranged so that the samples were immersed in liquid He. Provision was made for pumping on the liquid He and the temperature was measured by means of vapor-pressure thermometry, using an oil manometer. All of the experiments were conducted in the temperature range 1.2 K —2.1 K. The dewar tip was inserted in the air gap of a conventional dc electromagnet, the pole tips of which were separated by ~ in. The maximum field strength of this magnet was 45000 G. A krypton laser 6471 A was used to excite the luminescence. Reflection spectra were taken at near normal incidence using a zirconium lamp. Spectral analysis was made with a Bausch and Lomb 2-rn grating spectrograph. The spectrograph employed a large, high resolution diffraction grating producing a reciprocal dispersion of approximately 1.8 A/mm in first order. With this grating, the spectrographic aperture was about f/16. All of the spectra were photographically recorded on Kodak-type 1-N spectroscopic plates. The direct exciton (K = 0) is formed by an electron in the conduction band of symmetry F 6 (.1= ~)and a hole in the valence band of symmetry [‘8 (J’~).Excitons formed in this way will have the symmetries [‘3 + [‘4 + F~= ([‘6 X The j—j coupling scheme results in two excitons; one with total effective spin S = 1 and the other with effective spin S = 2. The S = 1 and S = 2 exciton states are split in zero field due to the electron-hole exchange energy. The S = 1 states are optically allowed in zero field, the S = 2 states are not. When a magnetic field is present, the S = 2 states may be optically allowed due to the mixing between the states of IS = 2, m5 = ±1,0) and IS = 1, m5 = ±1,0). The exciton ground states as observed in reflection are shown in fig. 1 a. This data was taken in the orientationKlH,EIH. Four transitions are observed down to relatively low magnetic field strengths. The lowest energy transition is associated with the 5 2 exciton, while the higher energy transition is associated with the S = 1 exciton which agrees with Willmann et al. [4] The n = 2 states of the free excitons are also shown in fig. Ia. In zero field the ground state energy is 1.5 1533 eV and the n = 2 excited state energy is 1.5185 eV. From these, we deduced the exciton binding energy of 4.27 ±0.20 meV and the effective exciton mass of j.t0 = 0.048 ±0.002. The line labelled A results from the IS = 1, m5 = 1) state and the line labelled B results from the IS = 2, ms = —1) state. The lines labelled C and D result from the lS=2,m5 =+l)and IS= l,m~—l), respectively states(fig. la). .

SB. Nam et al.

/ Exciton

energy spectrum in GaAs in a magnetic field

279

In an exciton—donor complex, the transitions have been associated with donor type complexes in which the initial state is that of an exciton bound to a neutral donor and the final state is an excited state of the electron on the donor. Transitions of this type in GaAs were first reported by Rossi et al. [6] They show four lines resulting from this complex. The upper state in this complex is a P312 hole state and .

the terminal states consist of excited states of the donor. This multiple degeneracy should result in very many lines in a magnetic field. Many of these transitions have been observed for the first time in this study as shown in fig. lb. In dealing with this type of complex one observes optical transitions from the upper state of the complex to the terminal state. In the upper state in inset in fig. lb, the electron spins are paired to give a bonding state and one is left with a degenerate hole state. In the terminal state the electron will split in a magnetic field according to the state it occupies. In zero magnetic field the n = 2 and n = 3 states of the ex1.52 33

I GaAs

1.5223

I

‘~

-

O~2’O3:4’O Fig. Ia. The energies of the ground states and the excited states of the free excitons in GaAs as a function of magnetic field strength.

280

SB. Nam et al. / Exciton energy spectrum in GaAs in a magnetic field

-6.00

GaAs

-

/

Excited State of Donor Complex -

—5.00

-

—4.00

-

/

/

~°°

0.00

~

~

~-----:

~!

-~

-

I

Fig. lb. Excited state transitions from a neutral donor bound exciton complex. The insert shows the configuration of this complex.

cited electron states are clearly observed. The energies of these states permit us to make a very accurate determination of the binding energy of the donor. The energies ofn = 2 and n = 3 states are 1.5100 eV and 1.5092 eV, respectively. From these data, we determined the binding energy of the donor of 5.77 ±0.20meV and the effective electron mass of m~’= 0.066 ±0.002. 3. Theory The exciton structure in wurtzite type semiconductors has been extensively studied within the hydrogenic model [7,8] However, in zinc blence type semicon.

SB. Nam et al.

/ Exciton

energy spectrum in GaAs in a magnetic field

281

ductors such as GaAs, the valence band is four fold degenerated and the description of the exciton is more complicated than in the former case. Starting from the Luttinger hamiltonian [9],the calculations of the exciton energy in the latter case have been carried out recently in the low and high field cases. In the low field regime, that is, the magnetic energy is less than the Coulomb energy, the calculations [10,11] are based on the perturbation scheme and the energy of the exciton for the state Ii) can be written in the form E~=Eo+GjB+DiB2,

(1)

where B is the dimensionless magnetic field, E 0 is the exciton energy in the absence of any magnetic field, and G~and D~are the linear and quadratic Zeeman factors, respectively, and they depend upon band parameters. On the other hand, in the high field regime where the magnetic energy is much greater than the Coulomb energy, one may obtain the Landau typ~spectra[12—141by the adiabatic scheme (2)

Et=LiB,

where L1 is of the order of = 0.01. However, the calculations based on either the perturbation scheme or the adiabatic scheme do not agree with present measurements which were carried out in the intermediate field regime, that is, the magnetic energy is of the order of the Coulomb energy. To understand the experiments, we developed a phenomenological scheme for the exciton energy spectra in magnetic fields, which reduces to the perturbation scheme and the adiabatic scheme at the two magnetic field extremes. For simplicity, we choose the exciton energy expression in the following form t

E.=E

0

2+II 3 I-4-JIiB2 GiB+DiB 1LiB

where parameters i~iare introduced. The above expression reduces to the eq. (1) in the low field regime and to the eq. (2) in the high field regime. With the value of (3~= 0.5, the calculated values from the above expression agree well with experimental data, which we discuss in the next section. The detailed calculations of the parameters i~iare in progress and will be reported elsewhere [15]. For the donor state, by solving the Schroedinger equation one may obtain the eigenvalues in the following form [5] E~= ~G~B’,

(4)

where G~is the energy of the donor in the absence of any magnetic field, and G~5~ for j> 1 are the appropriate numerical factors.

282

SB. Nam er a!.

/ Exciton

energy spectrum in GaAs in a magnetic fIeld

4. Discussion To compare calculated and experimental values we need-to-know various band parameters. From our experiments we determined the effective Zeeman splitting factors of the electron,g~,and the hole,~,as follows. Our experiment as well as others [4] indicated that the results are insensitive to the direction of applied magnetic field with respect to the crystal orientation. For simplicity we assumed such. The hamiltonian responsible for the magnetic field may be written as (5) HB g(S)SB +D(S)B2

,

where

S = J + a and J and a are the effective spin

respectively. The g-factors for the S g(S=2)_~+~g~,

=

operators

of the hole and electron,

I and S = 2 states are

The diamagnetic factor D(S) is independent of the magnetic quantum number m 5 and is irrelevant for our purpose since when one takes a difference between energies of the states IS = 1, ni~~ = ± I) or of the states IS = 2, ~ = ±1), then the term quadratic in B cancels out. Following the level assignment discussed in section 2, we determined gc and ~ to be g~—O.S0±0.05,

~1.0±0.2.

We also determined the electron—hole exchange energy of 0.17 ±0.10 meV by taking the limiting values of energy spectra at zero field. Our values of m~,p0,g~,and~ and the effective masses of hole, ~i = 0.823, and p2 = 0.148, and the dielectric constant e = 12.5 have been used to evaluate the spectra from eqs. (l)—(4). We have tried to fit our experimental data with values calculated from eqs. (1) and (2) with various values of the band parameters and found very poor agreement. However, with 13~= 0.5, calculated values from eq. (3) agree very well with the data as shown in figs. I a and lb. We summarized our findings in table I. Our value of the electron effective mass agrees with that obtained by the cyclotron resonance experiment, rn~’= 0.0665 [16] and also with that calculated by the self-consistent relativistic orthonormalized plane wave method at our laboratory [171, ni~’= 0.065. The value of the effective g-factor agree qualitatively with that calculated by one of us (Nam) by a new method [18], g~= —0.48. Our value for the

Table 1 Band parameters in GaAs

—0.50

±0.05

1.0

±0.2

ifl~

/20

0.66 ±0.01

0.048

±0.001

S. B. Nam et al.

/ Exciton energy spectrum in GaAs in a magnetic field

Luttinger parameter~agrees with the value K Lipari et al. [19].

=

1.1 + 0.1 of Willmann et al.

283

[41and

Acknowledgement Finally we thank C.M. Wolfe for supplying us the samples of GaAs. One of us (Nam) would like to thank the organizing committee for financial support and mak-

ing it possible for him to attend the conference. References [1] D.D. Sell, S.F. Stokowski, R. Dingle and J.V. Dilorenzo, Phys. Rev. B7 (1973) 4568. 12] R. Dingle, Phys. Rev. B8 (1973) 4627. 131 AM. White, P.J. Dean, K.M. Fairhurst, W. Bardsley and B. Day, J. Phys. C (Solid State Phys.) 7 (1974) L35. 141 F. Willmann, S. Suga, W. Dreybrodt and K. Cho, Solid State Commun. 14 (1974) 783. [51 D.C. Reynolds, C.W. Litton, T.C. Collins, S.B. Nain and C.M. Wolfe, Phys. Rev. B12 (1975) 5732. [6] J.A. Rossi, CM. Wolfe, G.E. Stillmen and J.O. Dimmock, Solid State Common. 8(1970) 2021. 171 J.J. Hopfield and D.G. Thomas, Phys. Rev. 122 (196 1)35 and others. [8] R.G. Wheeler and JO. Diinmock, Phys. Rev. 125 (1961) 1805. 19] J.M. Luttinger, Phys. Rev. 102 (1956) 1030. [10] M. Altarelli and NO. Lipari, Phys. Rev. B7 (1973) 3798; (E) B8 (1973) 4046. 1111 K. Cho, S. Suga, W. Dreybrodt and F. Wilmann, Phys. Rev. Bi 1(1975)1512. 1121 Ri. Elliot and R. Loudon, J. Phys. Chem. Solids 15 (1960) 196. [131 G.J. Rees, J. Phys. C4 (1971) 2822;C5 (1972) 549. [14] M. Altarelli and NO. Lipari, Phys. Rev. B9 (1974) 1733. [15] SB. Nam, to be published. 1161 J.M. Chamberline, P.E. Simmonds, R.A. Stradling and CC. Bradley, J. Phys. C4 (1971) L3 8. [17] S.B. Nam, T.C. Collins and R.N. Euwema, to be published. [181 SB. Nam, Int. J. Quant. Chem. 9S (1975) 551. [19] NO. Lipari, M. Altarelli and R. Dingle, Solid State Commun. 16 (1975) 1189.

Discussion P.J. Dean: My first comment is that I believe you have the wrong sign of the electron g value for GaAs in your tabular data. It is about —0.5, not +0.5. Second, my colleague Michel White and I have spent a lot of time trying to analyse the Zeeman splitting of both the principal donor and acceptor bound excitons in GaAs and InP. We concentrate on low magnetic fields, where the magnetic splittings are less than the.zero-field splittings. Despite much work on high quality crystals in the last three years, we have not yet emerged with a satisfactory, consistent parameter set in terms of the usual electron and hole magnetic spin hamiltonian for direct gap zinc blence semiconductors. The electrong value quoted above conies from analysis of the more tightly bound Sn acceptor cxciton, which gives a well-isolated single line at zero magnetic field. Finally,

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S.B. Nam eta!. /Exciton energy spectrum in GaAs in a magnetic field

I would like to point out that good evidence on the chemistry of shallow acceptors and donors in refined GaAs can be obtained, respectively, from edge luminescence and far infrared photoconductivity. Work mainly at RRE and fujitsu Laboratories has given the key to the acceptor analysis, or Oxford and Lincoln Lab, for the donor analysis. The chemical identifications of InP can be obtained in the same manner, but our understanding is less well developed so far. SB. Nam: Thank you very much for your comments. I think that the systems of the ionized donor—acceptor—exciton complexes are simple, but that of the neutral acceptor—exciton complex is complicated since an additional particle comes in. I hope that some progress will be advanced in the near future.