Electric field effects on the direct optical transition to the 1s exciton in cuprous oxide

Solid State Communications,

Vol. 7, pp. 1129—1130,1969.

Pergamon Press.

Printed in Great Britain

ELECTRIC FIELD EFFECTS ON THE DIRECT OPTICAL TRANSITION TO THE is EXCITON IN CUPROUS OXIDE H.!. Ralph Electrical Engineering Department, McMaster University, Hamilton, Ontario, Canada (Received 6 December 1968 by B.N. Brockhouse)

It is shown that the results of recent measurements made by Brahms and Cardona on the electro optical absorption due to the is exciton in the yellow series of cupwus oxide are in accordance with Elliott’s theory of optical absorption.

RECENTLY Brahms et aL’ have made absorption measurements on the is exciton in yellow series of cuprous oxide in the presence of an electric field. The theory of optical absorption has been reviewed thoroughly by Elliott.2 The no phonon absorption coefficient in insulators is proportional to a quantity K given by

~I~2 S(hv’),

(1)

where 10> and g > are respectively the electron ground and excited states, q is the wave vector of the radiation whose polarisation is ~ ~ is the momentum operator and S(hi.i) is the density of excited states with energy hi.’ above the ground state energy. The excited state g > may be expanded in terms of the Bloch states 1k > which have an electron in the conduction band state with wave vector k and an electron removed from the valence band state with wave vector k. =

~ A(k) 1k>.

(2)

Near the absorption edge the matrix element in (1) can be expanded as ~ A(k) k cI EP lv> + ~ k.< clMIv >/m k + i + ...],

equation. The first term of equation (3) gives ~ ~) and is the dipole allowed term and is responsible for the exciton absorption in most materials. The second term giving c~’(0) is called ‘forbidden’ by Elliott and is responsible for the exciton series in cuprous oxide where the first term is zero. The relevant feature of the forbidden term is that the is exciton line is absent because it has çY (r) zero at the origin. (The prime indicates differentiation with respect to r). Weak absorption by the is exciton is seen, however, due to the third term giving q~(O) and is called quadrapole allowed. The application of an electric field changes the function 4 (r) so that ç5’ (r) has a finite value at the origin for non zero field; meaning that transitions to the is state can occur via the second term in (3). The field strength is conveniently expressed in terms of a dimensionless quantity f which is unity when the potential energy change of an electron moving through one exciton Bohr radius is one exciton Rydberg.

+

(3)

where Ic > and Iv > are the conduction and valence band states at the respective band edges. The integration is taken over the unit cell. M is defined in (5). The Fourier transform of A(k), ~ (r), is the solution to the effective mass

Putting the radius at about 10 A and the binding energy at 0.2 eV for cuprous oxide f is unity for a field of 2 x 106 V/cm. The experiments of Brahms et al. have been for values of about 0.05. 1129

f less than

1130

THE is EXCITON IN CUPROUS OXIDE

Ralph3 has studied this problem by means of a numerical solution of the effective mass equation and the forbidden absorption by the is exciton has been plotted. No field broadening is exhibited for f less than 0. 1 and the line shifting in this range is correctly given by second order perturbation theory. The maximum shift expected in Brahms et al. experiment is then about .0005 eV which is their stated resolution. The forbidden absorption strength is given by K =

>/m12.

~ ~(0).?i
(4)

In first order perturbation theory the Is function receives an admixture of r states in an amount proportional to the field strength so that = + !c~.The derivative of ~ at the origin is due to the presence of the P state so that the absorption in 4 goes like ~2• This will be the case whenever the forbidden absorption exceeds the quadrapole allowed absorption which should occur at fairly low field strengths. The operator M is given by p~i>
(5)

where the sum is taken over all the bands at ‘ except c and v. It is shown by Ralph3 that the modified is state has a finite slope only in the direction parallel to the field assumed in the z direction. For polarisations perpendicular and parallel to the field the matrix elements controlling the relative line strengths are ~‘

Vol. 7, No. 16

To summarise for fields less than 100kV/cm: (a) absorption increases with increasing field (b) the line strength increases with the square of the field strength (c) the strengths with parallel and perpendicular polarisations are different but neither are expected to be zero (d) no broadening and a shift up to only 0.0005 eV are expected. All these points are in agreement with the measurements of Brahms and Cardona within experimental error. There remains the problem of the small dip and peak to the high energy side of the is lIne. It is possible that this line is due to the interaction of the exciton with the background as Brahms et at. suggest. The way in which this can come about has been discussed by Phillips.4 It is not clear however, why they do not appear for all directions of propagation of the radiation. Also it has been pointed out that (5) predicts that the absorption coefficient is independant of the direction of propagation in isotropic crystals. Unfortunately it is not possible to test this prediction against the published data of Brahms and Cardona because their plots are of Al/I which depends on the crystal dimensions. In conclusion it can be said that Elliotts theory of optical absorption by Wannier excitons would appear to explain the current low field observations on the behavior of this is line quite well.

[/(E~ E~)+ /(E~— E~)I and —

+

1 + (E~ E~~’ U —

Acknowledgements author would like to acknowledge financialThe support from the National Research Council of Canada. —

~‘[ 1(E~— E~y respectively.

1. 2. 3. 4.

REFERENCES BRAHMS S. and CARDONA M., Solid State Commun. 6, 733 (1968). ELLIOTT R.J., Polarons and Excitons (Edited by KUPER C.G. and WHITFIELD G.D.) Oliver & Boyd (1963). RALPH H.!., J. phys. Chem. 1, 378 (1968). PHILLIPS J.C., Solid St. Phys. iS, 55 (1966). On montre que les résultats des récentes mesures prisent par Brahms et Cardona de l’ectroabsorption de l’exciton is de la série jaune de l’oxyde de cuivre sont expliqués par la théorie de l’absorption optical d’Elliott.