Exciton reflectance in wurtzite AgI

Solid State Communications,Vol. 15, pp. 1885—1888, 1974.

Pergamon Press.

Printed in Great Britain

EXCITON REFLECTANCE IN WURTZITE AgI M. Bettini, S. Suga, and R. Hanson* Max-Planck-Institut für Festkorperforschung, Stuttgart, Federal Republic of Germany

(Received 15 July 1974 by M Cardqna)

The reflectance spectra of the A and B exciton in f3-AgI were measured on different crystallographic planes. The n = 2 state of the A exciton and the longitudinal-transverse mixed modes of the A and B exciton ground states could be identified. Magnetic circular reflectance measurement (MCR) yielded the excitong-values.

modulator5 in the MCR measurements. We measured the MCR spectra in Faraday configuration with H II caxis only. The MCR signal was calibrated by inserting a pure circular polarizer into the optical path. The accuracy of the method was checked by comparing it with the directly measured Zeeman splitting of the B exciton in CdS.

IN THE LONG history of optical investigations of Ag!’ one main difficulty has been to obtain large and pure crystals of a single phase. Especially.the detailed exciton structures of the E 2 are very sensitive to impurities. We therefore 0-gap performed usual reflectance and MCR measurements on good ~3-AgIsingle crystals in order to elucidate the internal structure of the A and B excitons whose general features have been reported already.3

Figure 1 shows the reflectance structures in the A and B exciton region. The geometry is described by the angle of polarization 0, measured from the z—y, plane, and the angle of incidence 4). The measurements at 4) = 0 degree were performed on a cleaved face parallel to the c-axis, at 4) = 28 degree on a natural face. Besides the strong reflectance structures of the A and B exciton (AT, BT) we fmd the following structures:

The crystals were grown by slow (3—4 weeks) dilution of a saturated solution of Ag! in 67 per cent HI with a methanol overlayer similar to Lakatos and Lieser.4 Due to the difficulties in handling the soft ~3-Ag1, we used mostly natural faces of the hexagonal pyramid shaped samples. In this case the c-axis is indined by about 28 degree to the reflectance plane. We also used cleaved surfaces parallel or perpendicular to the c-axis which were not as good as good as the natural ones.

(a) a small reflectance dip at 3.003 eV (AT n = 2) which shows the same polarization dependence as the AT fl = 1 exciton. .

The reflectance measurements were performed with a usual experimental set-up, including a superconducting magnet up to 12 T and a photoelastic quartz *

(b) a reflectance structure on the high energy side of the BT exciton at 2.990 eV (BLM). It appears for polarization within the z—y plane (0 = 0 degree) and inclined incidence of the light to the main axis of the crystai uegree

On sabbatical leave from Physics Department, Arizona State University, Tempe, Az. Partially supported by the National Science Foundation.

~‘Y

1885

-?- U,

YU

1886

EXCITON REFLECTANCE IN WURTZITE AgI

(c) a reflectance structure at 2.950 eV (ALM) which is resolved as a distinct structure from the AT peak in the dashed curve (b). It shows the same polarization dependence as the BLM structure.

PHOTON ENERGY (eV)—* 292

r HH-

40

2.94

20

~

298

AT n~1 /

9~2O

300

~

AgI

~ ~a Lii

2.96

_________________ BT

8

k

A n~2 1

I’1HkV~BTmT 20

C

B~\~,/’

nrl

0

_______ _______

~

4240

4220

~

4200

4180

4160

4140

Table 1. Energies of the excitons determined by a Kramers—Kronig analysis, T = 4.5 K Exciton

Energy (eV)

KK-analysis of curve

AT

2.944

a

AL

2.952

a

BL(x,y) BT(x, y)

2.987 2.983

a

BL(z)

2.994

d

ALm BT(z) ATn=2 BLm

2.950 2.980 2.990 3.003

dC aC

The errors are for the strong and sharp reflectance structures ±1 meV and increase to ±3 meV for the weak and broad ones like AT n = 2.

30 LU

Vol. 15, No. 11/12

4120

WAVELENGTH (A)

FIG. 1. Reflectance spectra of AgI in the A and B exciton region. The geometry is defined by the angle of incidence 4) and the angle of polarization 0. In Table I we compiled the energy positions of the excitons after a Kramers—Kronig analysis. The AT, BT and ALm ,BLm modes were determined by the maximum of e. The positions of the longitudinal modes (AL, BL) were estimated from the maximum of the imaginary part of ~

The above described polarization behaviour of the peaks corresponds exactly to a longitudinaltransverse mixed mode due to the anisotropy 6 of the crystal proposed by Hopfield and Thomas. Their classical picture of the dipole—dipole interaction corresponds to the long range part of the electron—hole exchange interaction in the microscopic

7 In order to understand the longitudinaltreatment. transverse mixed mode more precisely, however, one should also consider the short range part o the exchange interaction. We have calculated the energies of the four components of the A and B excitoiis taking into account the difference in the dipole moments. As base functions for the A and B excitons, the hole functions i/h = 3/2, inJh = ±3/2) and 3/2, ±1/2) were used respectively. A mixing between B and C excitons was neglected because the spin—orbit splitting (830 meV) is much larger than the crystal field (39 meV). The result for the energy positions of the excitons is sketched in Fig. 2 as a function of the angle 4). If we include the exchnage interaction, the essential features are the energy splittings between FST

—

r

6 (A exciton) and FST ~‘2 (B exciton) and the splitting between the F1T(z) and I’5T(xy) states of the BT exciton. In Fig. 2 the highest energy mode of the A exciton and the two higher energy branches of the B exciton are the longitudinal-transverse mixed modes. They become purely longitudinal or transverse only for 4) = 0 and 90 degree. —

ALm, BLm

Concerning the A(6) exciton we can the use energy directlyposithe formula of reference and estimate tion of the mixed mode. It coincides exactly with the experimentally determined one. In the case of the B excition we have to regard

Vol. 15, No. 11/12

EXCITON REFLECTANCE IN WURTZITE AgI

1887

B- Exciton A- Exciton

r

11 (z)

r51(y) 0

r~y) ~

LU

~

~ r51(x) z r~

0

30 •(degree)

~~ 60

90

—.

r2

0

TT

r5~9j2~~

r~ 30 60 90 $(degree) —.

FIG. 2. A and B exciton energies as a function of the angle 0. ~T and I~L are the isotropic transverse and longitudinal exciton exchange energies. The dashed and solid lines in the figure of B exciton show the 4) dependence of the uncoupled and coupled oscillators respectively. two coupled oscillators I’1(z) and F~(y)which show an anticrossing effect (solid lines compared with the uncoupled modes shown by the dashed lines). Unfortunate tunately we could not determine a splitting of FiT FST beyond our experimental error. We therefore assumed a degeneracy of the FIT and F~3’states and evaluated the energy position of the mixed mode as 2.989 eV at 4) = 28 degree which is close to the experimental one (see Table 1).

PHOTON ENERGY (eV) —0. 2.97

From the polarization dependence we identify the 3.003 eV reflectance structure as n 2 state of the A exciton. Therefore, in the hydrogemc approximation, the exciton binding energy is 79 ±4meV. In our MCR measurement we also determined the diamagnetic en-

2.99

30

I

Agi H~ T~6K 12T

—

Finally we want to point out that the determined L—T splittings of the A, F~(8 meV), B, F~(4 meV) and B, F1 (14 meV) are qualitatively explicable by the ratio of the oscifiator strengths ~3(A,f’s) : ~3(B,F~): g3(B, r~) = 3 : 1: 4 of our simple estimation.

2.98

I

I

I

I

I I

t ~ C-,

b______________________

I

Lii —j

IL

Ui

ergy shift of the ATn = 2 state as E~ = 1.5 meV at 12 T. Using the dielectric constant e = 4.0,8 the reduced mass m = 0.093 me and Bohr radius a = 22.6 A (from the determined exciton binding energy), the diamagnetic shift is calculated as 4.9 meV according to reference 9. We can reproduce our experimental result, however, with a somewhat larger value of = 5.3. It is therefore suggested that the dielectric constant of ~3-AgIshould be checked carefully; in addition, the validity of the hydrogenic approximation should be examined, Figure 3 shows the result of our MCR measurement for then= lstateoftheBexcitonatl2T.

I

4180

4160

4140

(A) FIG. 3. Reflectance, MCR signal and derivative of the reflectance of the B exciton versus photon energy at 12 T. ~a: direct reflectance, b: MCR signal and c: derivative of the reflectance). The good coincidence of the b and c spectra confirms the rigid energy shift of the a~and a. spectra with respect to the magnetic field. From the Zeeman splitting E = gp~Hbetween a~spectra, 4

WAVELENGTH X

we evaluated the g-value: g(B,F5)

=

+ 1.05±0.1.

1888

EXCITON REFLECTANCE IN WURTZITE Ag!

Around the faint reflectance shoulder at 4165 A (corresponding to the I~peak of reference 2), we observed a considerable MCR signal. The coincidence of the b and c spectra also indicates that the 4, exciton has almost the same g-value as the BT exciton. Therefore the possibility of the non-degenerate F2 state is discarded and an assignment as a bound state of the B exciton to some impurity is suggested. The MCR signal of the A exciton is considerably smaller than that of the B exciton. We estimate the g-value: g(A, F~)= + 0.06 ±0.03. The AT n = 2 state show the reported diamagnetic shift but no MCR signal. Since the g-value of the Is and 2s transitionshould be the same,9 except for anisotropic contributions, the magnitude of the MCR signal can be estimated from the values of the ls transition. If one assigns the structure to the B exciton series, a MCR signal should be detectable. The absence of the signal is therefore an additional confirmation of the AT n = 2 assignment. In the quasi-cubic approximation the exciton gvalues are evaluated in a similar way as reported previously’°keeping in mind that Ch = Ce = 1 is assumed there.

Vol. 15, No. 11/12

g(A, F~)= 2(K + Ch)

—

2Ce

and g(B, F~)= ~(K + Ch) + 2Ce, -

whereK = (‘I’jl~i’1’),Ch,e = 2 (ah,e lSzh,e Iah,e) and ‘I’ transforms like ‘I’ = i(X + iY)/~.J2,ahe is the spin function of the hole and electron respectively. —

We can also start from the Luttinger Hamiltonian and estimate the g-values of the excitons as reported 2 If we neglect the amsotropy term and recently.” the contribution from the excited exciton states, the g-values are g(A, F~)= —6~—ge and g(B, E’~) =

—

2K +g~.

Luttingers parameters, thus obtained, are i~= and g~,= 0.78.

—

0.14

Acknowledgement We are very grateful to Dr. K.Cho for many helpful suggestions and a critical reading of the manuscript. —

REFERENCES 1.

GMELIN L.,Handbuch derAnorganischen Chemie, 61 Tl.B2, 186 (1972).

2.

BOHANDY J., MURPHY J.C., MOORJANI K. and FRALEY P.E.,Phys. Status Solidi(b) 49, K91 (1972).

3.

CARDONA M., Phys. Rev. 129, 69 (1963).

4.

LAKATOS E. and LIESERK.H.,ZPhys. ChemieNF48, 5228 (1966).

5.

SARI S.O.,Phys. Rev. Lett. 26, 1167 (1971).

6. 7.

HOPFIELD J.J. and THOMAS D.G.,J. Phys. Chem. Solids 12, 276 (1960). ONODERA Y. and TOYOZAWA Y.J. Phys. Soc. Japan 22, 833 (1967).

8.

COCHRANE G.,J. Phys. D :Appl. Phys. 7,748(1974).

9. 10. 1.1.

HOPFIELD J.J. and THOMAS D.G.,Phys. Rev. 122, 35 (1961). SUGA S., KODA T. and MITANI T. Phys. Status Solidi (b) 48, 753 (1971). ALTARELLI M. and LIPARI N.O.,Phys. Rev. B7, 3798 (1973);Phys. Rev. B8, Errata, 4046(1973).

12.

CHO K., SUGA S., DREYBRODT W. and WILLMANN F. (to be published).