Photoelectric thresholds for tetrahedrally coordinated semiconductors

Solid State CommunicationsNol. 15, PP. 575—578, 1974.

Pergamon Press.

Printed in Great Britain

PHOTOELECTRIC THRESHOLDS FOR TETRAHEDRALLY COORDINATED SEMICONDUCTORS* S. Ciracit Applied Physics Department and Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, U.S.A. and B. Bell Department of Applied Physics, Stanford University, Stanford, California 94305, U.S.A. (Received 6 April 1974 by A.A. Maradudin)

The photoelectric threshold, F, is calculated for a variety of tetrahedrally coordinated semiconductors using the Bond Orbital Model. The variation of 4 with anion compo~ientand charge transfer is discussed. These results are compared with experiment and good agreement is found. The pressure dependence of ‘F is also discussed.

1. INTRODUCTION THE PHOTOELECTRIC threshold, 1, is the energy required to take an electron from the top of the valence band to the vacuum level. Itis a quantity that is of considerable interest to. experimentalists but, so far, it has not been possible to predict its value from standard band structure calculations. A recently introduced theory, the Bond Orbital Model1 (BOM) has been successful in studying many properties of tetrahedrally coordinated semiconductors. In this report we discuss its application to calculations of the photoelectric threshold for tetrahedrally coordinated solids from the 3rd, 4th, and 5th rows of the periodic table. It is shown that the photoelectric threshold can be calculated by BOM and there is good agreement between experiment and theory, both

quantitatively and in the systematics for a variety of materials. We also give the predicted threshold for a number of materials where it has not, as yet, been measured. 2. BOND ORBITAL MODEL In this section we will give only a brief sketch of the BOM in order to discuss its application to photoelectric thresholds. A more complete discussion of the model and the nature of the approximations involved is given in the papers of Harrison,1 çiraci,2 I-Iarrison and çiraci.3 The BOM is a tight-binding method in which the Bloch sumsare constructed in terms of 8 different sp3 hybrid orbitals of a unit cell: 1 X,~(k,r) =

~ exp(ik .R~)~(r—Rn)

where / = 1 to 4 denotes the direction of these orbitals belonging to either the cation (x c) or the anion (x = a) site. The expectation value of the crystalline hamiltonian is then calculated with respect to the Bloch wave functions formed from these X~~(k, r). We are led to the standard matrix equation:

Supported in part by National Science Foundation Grant GP 25945 and in part by the National Science Foundation through the Center for Materials Research at Standford, and DA ARO D3112473G1 1. *

tOn leave of absence from the Middle East Technical University, Ankara, Turkey.

575

576

TETRAHEDRALLY COORDINATED SEMICONDUCTORS (3C—Xj)a

=

0

Vol. 15, No.3

and

where ~JCisa (8 X 8) model hamiltonian, a simple generalIzation of the hamiltonian used by Weaire and Thorpe.4 It consists of four independent matrix elements, defined as follows:

1. (3) —~ ~ = V~X~ Here Xb is the binding energy of the bond orbital ~4,: =

1 V’b

(i)

~

(J~(J~J)

=

=

corresponding to hopping within the orbitals of the anion or cation

v

(ii)

2

=

—(p~l~chp~~>

is a standard matrix with well-known eigenvalues: ~va1ence

=

—

./14-+-v2 +

(singlet)

V1

E2 = — — 3V1 (triplet) Therefore, without going through the complete band calculation it is possible to say that the top of the valence band occurs at

corresponding to hopping from an anion orbital to the nearest cation orbital 1)] (iii) V3 == ±~ [(p~I I> — (I I p

______

/V~+V~+V 1.

(4)

=

(plus sign refers to cation lattice). This diagonal term reflects the tendency to transfer charge from the cation to the anion. The determination of these energy 1’2 Vparameters has been discussed previously. 1 is taken to be one-fourth the free atom s—p splitting. By expressing the dielectric constant in terms of V2 and comparing with experiment it is found that:’ 3 (1) V2

=

39.5d

where d is the bond-length (in Angstroms).

The energy eigenvalues and the properties of these valence bands are discussed in reference 5. We will now go on to discuss the location of this valence band edge for various semiconductors and its relation to the photoelectric threshold. 3. PHOTOELECTRIC THRESHOLD The photoelectric threshold, 1, is defined as the photon energy required to take an electron from the top of the valence band to the vacuum level. Thus

V 3 is related to the coulomb integrals for the anion and cation and can be determined from the experimental values for the dielectric constant of compound semiconductors. Note that V3 is measured relative to the intrinsic energy,E0 = ~(e~ + e~).The values ofr eg” are determined from the behaviour of V3 for different materials having the same cation (anion).

By applying Jacobi transformations to ~Cto separate the conduction and valence bands, we obtai~i,at k = 0

measurement of I~ gives direct information on the location of the valence band edge. The photoelectric yield, Y, defined as the number of photoelectrons emitted per photon, has been measured for many semiconductors. It is very small for hi’ ~ 1 and rises rapidly as hi’ exceeds 4~.Therefore the onset of photoemission might be considered to be a measure of 1. [A more complete discussion is given by Davison and Levine.] 6 Within the BUM, the edge of the valence band zero6f energy. The energy of this zero level with

~~va1ence

=

r XbV3 L-V1

A~

-—

—

—V1

L- V3 where V1

=

~

[(1

—

~)

V, V1

~,

V~+ (1 +

— —

V1

I

(2)

is given to by the equation (4). This is by: relative to intrinsic respect photoelectric threshold vacuum is isE0. given Therefore, thethe BUM

Xb

~)

______

X~] V~]

~BOM

=

E0 +

+ V~—

[(1 +~~)Vfl As in —~~)V~ Section 2,-~-l all quantities in equation (5) discussed are obtained independently of any photoelectric

Vol. 15, No.3

TETRAHEDRALLY COORDINATED SEMICONDUCTORS

577

Table 1. Comparison ofphotoelectric threshold from the BOM with experimentally determined values. 4~is related to the relaxation energy (see text). Intrinsic energy zero, E0. Experimental values are taken from (a) reference 8; (b) reference .9; (c) reference 7; (d) reference 10. Semiconductor

—E0 [eV]

GaSb° InSb° AlSb” Sia InAs° AlAs GaAsa AgI CuI InPb AlP GaP ZnTec MgTe CdTe” CdSec ZnSec MgSe CdSC ZnSC MgS CuBrd CuC1

7.92 8.00 8.35 7.84 8.37 8.17 7.85 8.80 8.70 8.85 8.45 8.87 8.80 8.60 9.57 9.50 9.27 9.80 9.80 9.45 953 10.27

I

J

j~~s

-

.

// cdsT/ 7

-

6

~

~ /

B-1”~

.

jnS.

AgI A2PTMGCP CdT...J’~ZnTe

—

A~A8 /

—

/

-

—

5.47 — —

5.66 — —

5.76 —

5.78 6.62 6.82 —

7.26 7.50 —

6.85 —

average hybrid energy, —E0, increases with increasing valence difference between the anion and cation. This increase dominates the decrease in “BOM because of changes in ~ and V~.Therefore: (1) *I~BoMincreases with increasing L~Z. .

a

the cation energy, e~.The latter does not change 4~BoM

I

I 6

7

8

[.v]

FIG. 1. Experimental photoelectnc threshold 4 vs

~I~o).: Experimental value exists. I: Predicted 4 for compounds not having experimentally determined 4’s (width of bar indicates uncertainty in value). —

-

From the expression for ~IBOM we can deduce some general trends for different compounds: The

.

~

(4s0M

4.76 4.77 5.20 5.07 5.31

The anion hybrid energy, h, is much larger than

GoSb/ .SI 5

4

[eV]

~exp

inP

InAs. TJGaA. AlSba xnSb,/

tto [eV]

information. Thus equation (5) constitutes a direct check of the theory.

/

Cud

—

4.57 4.38 5.22 5.13 4.92 5.26 :~0.2 5.32 5.84 ±0.2 6.07 ±0.2 5.83 5.99 ±0.2 6.15 ±0.2 5.98 5.69 ±0.2 5.57 6.51 6.84 7.11 ±0.2 7.18 7.79 7.90 ±0.2 7.01 ±0.2 8.64 ±0.1

8.65

I

8

~BOM

appreciably withupon decreasing valence. Thus depends mostly the anion atoms, i.e., (2) Compound semiconductors with the same anion constituent have approximately equal photoelectric thresholds. (This was noted experimentally 7

by Swank, who pointed out a correlation between 4~and Pauling s electronegativity of the anion constituent.)

578

TETRAHEDRALLY COORDINATED SEMICONDIJCTORS

In Fig. 1 weshow a comparison ~BOM as calculated by equation (5), and the experimentally measured ~7—1o Perfect agreement would correspond to all points being on a line of unit slope passing through the origin. From Fig. I we deduce that ~exp

=

~BOM

~

(6)

where F~is approximately the same for all compounds

Vol. 15. No. 3

percent. Thus we would expect D~to be approximately material independent.’1 It is also very simple within this model to discuss the pressure dependence of the photoelectric threshold. The only parameter that is likely to change with pressure is V 2. This is because V2 is related to the S bond length by:

and has the value 3.8 ±0.2 ev. Figure 1 also gives the values of ~ for materials that have not been measured.

=

constant d_s.

Thus 112V 4. DISCUSSION

~“BOM

From Fig. lit can be seen that to within cI~,the BUM is in good agreement with experiment, bearing out the two statements of Section 3. The factor ‘~O can be explained in a very simple way: As with many band structure calculations, the BUM does not take into account the gain in energy due to the polarization of the neighboring atoms when the photoelectron is polarizability removed. Thisand energy gainelectric is 2 where a F, the aL’ elemental semiconductors,a~’-d5,E’~d2. field. For Thus i~.E d, the bond length. For the materials considered here, d does not vary by more than a few

=

S

[(I —a~)

2 ~(V~— Vf~a~(l —c~)] —

~p 3B where B is the bulk modulus of the material. Froni pressure dependence of Hill,13 the refractive index isbyconCardona 12 and Gibbs and s = 3, which eta!., with the value used in equation (I Thus the sistent BUM photoelectric threshold increases with increasing pressure. ‘P 0 d, thus the experimental photoelectric threshold is expected to increase with increasing pressure. Acknowledgements We would like to acknowledge helpful discussions with Professors W.A. Harrison and WE. Spicer. ),.

—

REFERENCES I.

HARRISON W.A.,Phys. Rev. BlO, 4487 (1973).

2.

CIRACI S., Ph.D. Thesis (Stanford University, 1974).

3.

HARRISON W.A. and ~IRAC1S. (to be published).

4.

WEAIRE D. and THORPE M.F.,Phys. Rev. B4, 2508 (1971).

5.

~ERACIS. and PANTELIDES ST. (to be published).

6.

DAVISON S.G. and LEVINE J.D.,Solid State Phys. 25, p. 90, Academic Press, New York (1970).

7.

SWANK R.,Phys. Rev. 153, 844 (1967).

8.

GOBELIG.W. and ALLEN F.G.,Phys. Rev. 127, 141 (1962); 127, 150(1962); 137, A245 (1965).

9.

FISHER T.E.,PhysRev. 139, A1228 (1965); 142, 519 (1966).

10. 11.

FU S.L. (private communication). HARRISON W.A. (private communication).

12. 13.

CARDONA M., PAUL W. and BROOKS H.,J. Phys. Chem. Solids 8,204(1959). GIBBS D.F. and HILL G.J., Phil. Mug. 9, 367 (1964).