Dependence of the indirect energy gap of silicon on hydrostatic pressure

Solid State Communications, Vol. 17, pp. 1021—1024, 1975.

Pergamon Press.

Printed in Great Britain

DEPENDENCE OF THE INDIRECT ENERGY GAP OF SILICON ON HYDROSTATIC PRESSURE B. Welber IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. and C.K. Kim, Manuel Cardona and Sergio Rodriguezt Max-Planck-Institut für Festkorperforschung, 7000 Stuttgart 1, Federal Republic of Germany (Received 17 June 1975 byM. Cardona) A diamond anvil optical cell is employed to measure the pressure dependence of the fundamental indirect r—x transition of crystalline Si. The result is E(eV) = (l.llO±0.002)—(l.41±0.06)x l03P where the pressure Pis in kbar. Between 115 and 126 a transformation takes place to a phase opaque to electromagnetic radiation ofwavelength between 0.3 and 2.0 pin. We believe this is the same phase transition reported by Minomura and Drickamer. A pseudopotential calculation assuming a relatively soft core is carried out and its results are in rather good agreement with experiment.

ACCORDING to an empirical rule, first stated by Paul,1 the pressure coefficients of the energies of electron states at a given point in the Brillouin zone aresimilarforallsemiconductorsofthegrouplVof the periodic table of the elements and for those of the zinc-blende family. They vary substantially, how. ever, from one high symmetry point to another. For example, the energy at the point X in the conduction band decreases relative to that of the top of the valence band r with increasing pressure. The magnitude of this change is small compared to the corresponding shifts of energy at other symmetry points. Paul and Warschauer2 have made measurements of the indirect optical absorption edge of Si as a function of pressure *

in the range from zero to 8 kbar and established that this energy difference E varies as dE = —1.5x 10-6 eV/bar. (1) Even though the minimum of the conduction band in Si is not situated at X but along the line A in the Brillouin zone, it is sufficiently near I to regard E as the difference in energy between the electron states at the points X of the conduction and 1’ of the valence band. De Alvarez and Cohen3 have recently made a theoretical calculation of the pressure coefficient of the I’—X energy gap of Si and find a value in close agreement with that of equation (1).

Supported in part by the National Science Foundation. On leave from Purdue University, West Lafayette,

Precise optical measurements at pressures in excess of 100 kbar have been made possible by recent developments in high pressure techniques. The authors have carried out a study of the direct energy gap of GaAs to pressuresup to 180 kbar and found that it varies non-linearly with both pressure and lattice

~ Alexander von Humboldt Senior US Scientist on t sabbatical leave from Department of Physics, Purdue University, West Lafayette, IN 47907, U.S.A. 1021

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DEPENDENCE OF THE INDIRECT ENERGY GAP OF SILICON ____________________________________________________________________

Vol. 17, No. 8

lJ~

_______________

l.4kbor

0.3 02 01

0.9

j /1 1 / /•I/ / J / /Y r /

~

FIG. 2. The shift in the Si r—x absorption edge vs hydrostatic pressure. An irreversible phase transition occursbetween ll5and l26kbar.Thesolidlineis

/.// ;~/ I / I I ~/ 1.0

12

.3

.4

!~w(eV)

FIG. 1. The effect of pressure on the

edge of crystalline Si.

r—x absorption

constant. The non-linearity as a function of lattice constant seemed to be related to the pressure of the d-core-electrons of Ga about 18 eV below the measured gap. Silicon has no such d-electrons. For this reason it was decided to carry out an investigation of the pressure dependence of the lowest energy gap of Si up to 108 kbar while keeping in mind the fact that this gap is indirect (1’ X) and, according to Van 6 Vechten, rather msensztive to the presence of core d-electrons. In this range the gap was found to vary linearly, both with pressure and with lattice constant. Beyond this pressure a transition to a metallic phase occurs. This takes place at a pressure which is substantially lower than that reported by Minomura and Drickamer.7 -~

.

(kbor)

..

1. METHOD OF MEASUREMENT The experimental set up, consisting of gasketed diamond anvils, is similar to that employed for GaAs and described in reference 5. The pressure transmitting fluid was a 4: 1 mixture of methanol and ethanol. Due to the longer wave length of the indirect energy gap of Si, the source of radiation and detection procedures are different from those used in reference 5. The former was a quartz—iodine tungsten lamp chopped at 400Hz. The transmitted radiation was collected by a 15x microscope objective of the reflecting type, fed to a monochromator with 16 A resolution, and detected by a cooled PbS cell whose signal was processed with

a least squares fit to the data and the dotted line represents the results of our calculation. For ease of comparison, the curves are shifted so that they coincide at P = 0. routine lock-in techniques. The pressure was measured using the ruby fluorescence method.4 The sample was a small slab 0.004 cm thick and across, polished on both sides, and with its plane surfaces perpendicular to a (111) axis. It was n-type with a resistivity 100 fl-cm. The measurements were made at room temperature. 2. RESULTS We show in Fig. 1 the variation of (ad)”2 as a function of photon energy h~for several pressures. Here a is the absorption coefficient and d the thickness of the sample. Since the shape of these curves proved to be relatively independent of pressure, we obtain the absorption edge by extrapolating the linear portion to ad = 0. To obtain these data it was necessary to

know the values of the transmission in the zero ab-

sorption region. These were obtamed, for each pressure, by averaging neighboring maxima and minima of a well developed set of interference fringes found below the gap. We note, in passing, that the transmission in this region was observed to increase monotonically with pressure in a manner consistent with the pressure dependent increase in the refractive index of the pressure transmitting fluid. In any case, the values of the intercepts of (&)1.12 proved to be relatively insensitive to small changes in the line of zero absorption. To achieve pressures above 90 kbar where the gasket material was beginning to yield under the fluid pressure, the system was first brought back to atmospheric

Vol. 17, No.8

DEPENDENCE OF THE INDIRECT ENERGY GAP OF SILICON

1023

liL

us

change in lattice constant. The points are obtained from those displayed in Fig. 2 using Mumaghan’s

1.10

Fig. 3 also shows the values yielded by the theoretical

1106.

t13

Os

~09

6 FIG. 3. The shift in the Si F—X transition as a function of lattice compression, 6 (ao a)/ao, where a0 is the lattice parameter at atmospheric pressure. The solid and dotted lines are, as in Fig. 2, the least squares fit to the experimental and the theoretical results, respectively. —

pressure and some fluid allowed to drain off. Recompression produced a smaller (collapsed) hole and permitted pressures. this Upon fashion the optical attaining data werehigher extended to 108Inkbar. further compression the sample became opaque between 115 and 126 kbar (transmission less than 0.01% at 0.9 eV). We presumed that a first order transformation to a metallic phase, identical to that 8 reported in reference 7 at 150 kbar, had occurred. Figure 2 displays the variation of the indirect energy gap with pressure. A least squares fit to the data yields the relation 3P, (2) E = (1.110 ±0.002)—(1.41 ±0.06) ~ lO~ where E is eV and Pm kbar. The coefficient ofP is in close agreement with the result of Paul and Warschauer2 given by equation (1). The data do not permit us to write a term proportional toP2 as was the case in reference 5.

calculation equation. Todescribed provide below a convenient which basis are likewise for comparison, displayed in Fig. 2 after appropriate use of the Murnaghan equation. The value ofE as a function of 6 can be

represented by the straight line, obtained through a least squares fit: E = (1.115±0.007)—(5.04±0.23)6, (3) where E is in eV. It is not possible from the statistical analysis of the data to add a meaningful quadratic term in 6 to equation (3). The calculation of the indirect gap was done with a Heine—Abarenkov potential~similar to that of

reference 5 in our work on GaAs. We obtain the energy of the top of the valence band as a function of lattice constant a by diagonalizing the pseudo-Hamiltonian 12 in a representation the partners. plane waves (000), (ill), (200),generatedby (220) and their This involves 27 plane waves and 3 r~ 5triplets. For the determination of the conduction band state at X we employed the plane waves (000), (O0~)which belong to the which irreducible representation X~; (ii(0~0), I), (Ti(~00), T) and (ill) generate X~+ 14; (020), (20~),(200), (02~),(0~),and (~)~) belonging to X~+ 12 + 13 + 14 and, fmally, (iii), (1 Ti), (Ti 1), (T Ti), (11 ~),(ii ~),(T 1 ~),and (T T ~)which generate 2X~+ X 3 + 14. The pseudopotential cut-offR0 at zero pressure13 follows from an independent fit’4 and is 1.027 atomic units. We expect that as a decrease with increasing pressure, this core radius will likewise decrease. The theoretical dotted lines displayed in Figs. 2 and 3 were found with the assumption R

=

R 0

!.;

(4)

a0

3. THEORY Comparison of the results discussed above With those of the theory, which are naturally obtained as functions of the lattice parameter, require the knowledge of the equation of state of Si.9 We use, for thisthe together with purpose, the Murnaghan equation work of McSkimin and Andreatch1°who measured the elastic bulk modulus B 0 and its pressure derivative B~at zero pressure. Figure 3 shows the dependence ofE on 6 = (a0 a)/a0, the negative of the relative —

these lines fit rather well the experimental results; that of Fig. 3 is linear in 6. The assumption of equation (4) represents a stronger (softer core) variation of R with 6 than that required to fit the pressure dependence of the direct gap of GaAs, a fact which may be related to the absence of d-eiectrons in the case of Si. Acknowledgements — We wish to acknowledge valuable conversations with R.W. Keyes, W.E. Howard, J.H. Van Vechten, and W. Holsapfel. G. Piermarini and S. Block have offered timely advice regarding high pressure techniques. P. Perry has helped with the data processing.

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DEPENDENCE OF THE INDIRECT ENERGY GAP OF SIUCON

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REFERENCES 1. PAULW.,J. Phys. Chem. Solids 8, 196 (1958). 2. PAUL W. & WARSCHAUER D.M.,J. Phys. Chem. Solids 5,102(1958); see also PAUL W., I. App!. Phys. 32, 2082 (1961). 3. DE ALVAREZ C.V., & COHEN M.L., Solid State Commun. 14,317(1974). 4.

BARNE11~J.D., BLOCK S. & PIERMARINI G.,Rev. Scz Insir. 44,1(1973). 5. WELBER B., CARDONA M., KIM C.K. & RODRIGUEZ S., (to be published).

6. VAN VECHTEN J.A.,Phys. Rev. 187, 1007(1969). 7. MINOMURA S. & DRICKAMER H.G.,J. Phys. Chem. Solids 23,451(1962); see also DRICKAMER H.G., Rev. Sc: Instr. 41, 1667 (1970). 8. We cannot explain the discrepancy in the values of the critical pressure. However, we note that in our measurements there was no detectable broadening of the R line of ruby at 126 kbar so that the pressure can be regarded as hydrostatic. Furthermore, in a private communication WEINSTEIN B.A. and PIERMARINI G. report a transition range of 116—125 kbar based on the disappearance of the Raman line of Si. We also remark that despite the fact that our measurements were carried out within 10% of the phase transition, no tendency toward metallization is apparent in our data. Possibly, this is to be expected if one supposes the transformation is of the first order. 9. MURNAGHAN F.D., Proc. Nat. Acad. Sci. 30, 244 (1944); see also reference 5. 10. McSKIMIN HJ. & ANDREATCHP.. Jr., I. App!. Phys. 35,3312(1964). The values ofB0 and B~for Si are B0 = 0~97x 106 bar,B~= 4.16. Similar results are obtained by BRIDGMAN P.W.,Proc. Am. Acad. Arts. Sci. 76,55(1948); VAIDYA S.N. & KENNEDY G.C.,J. Phys. Chem. Solids 33, 1377 (1972); ANDERSON O.L., J. Phys. Chent Solids 27,547(1966); and by ENDO S., YAMAKAWA K. & MITSUI T.,Japan I. AppL Phys. 12, 1251 (1973). They disagree with those of SENOO M., MII H., FUJISHIRO I. & FUJIKAWA K., Proc. 4th mt. Conf on High Pressure p. 240. Kyoto (1974). 11. COHEN M.L. & HEINE V., Solid State Physics (Edited by EHRENREICH H., SEITZ F. & TURNBULL D.) pp. 51—57. Academic Press, NY 24, 37 (1970). 12. We designate by (h1 h2h3) the plane wave exp [i(K + k) rJ where k is the reduced wave vector and K = (2ir/a) (h1 , h2 ,h3) a vector of the reciprocal lattice. 13. See equation (8) in reference 5 or equations (4.3) and (4.4) in p. 55 of reference 11. 14. This is obtained from Table XVIII (p. 235) in reference 11. Notice that qR

=

v/2.