Transverse electroreflectance in semi-insulating silicon and gallium arsenide

J. Phy.s. Chem. Solids

Pergamon

Press 1970. Vol. 3 1, pp. 227-246.

Printed in Great

Britain.

TRANSVERSE ELECTROREFLECTANCE SEMI-INSULATING SILICON AND GALLIUM ARSENIDE* RICHARD

A. FORMAN?,

Department

of Physics.

DAVID’E.

ASPNESt

Brown University,

and MANUEL

Providence.

IN

CARDONA$

R. I. 02912.

U.S.A.

(First received 8 Muy 1969: in revisedform 7 July 1969) Abstract-The

transverse electroreflectance (TER) spectrum of GaAs is measured over an energy range extending from the fundamental absorption edge to the lambda structure. The TER structure in the vicinity of 3.4 eV in Si is also presented. Quantitative measurements of the variation of the spectrum with field have been performed, utilizing the facts that the TER geometry permits the calculation of the value of the electric field. and that the field is essentially uniform in the region of measurement. A detailed investigation of the effect of collision broadening on the electroreflectance response in the one-electron approximation is given, which qualitatively correlates the dependence of peak amplitude. peak location. and number of subsidiary oscillations observed in electroreflectance with the magnitude of the field and collision broadening lifetime. A method of measuring the collision broadening lifetime is applied to the fundamental edge of the GaAs sample used in this experiment. A conformal mapping is presented which enables a calculation of the field at any position in the crystal for the common TER geometries.

1. INTRODUCTION

the p-n junction method of Handler and coworkers [5]. High-resistivity crystals allow electro-optic measurements to be made in the parallel [l 1, 121 or transverse[6,7, 101 electrotransmission or electroreflectance [8101 configurations, where the modulating field is sufficiently uniform and accurately determined to compare with theory. In this paper, we apply transverse electroreflectance (TER) methods to insulating samples of GaAs and Si, with the objective of obtaining the additional symmetry information available from TER, and also to demonstrate a method of obtaining collision broadening which requires quantitative measurements of the variation of the electro-optic response with field strength. A conformal mapping of the field distribution for typical TER gemoetries is given in the Appendix, which permits field magnitudes to be calculated.

effect has been used with great success for the determination and assignment of interband critical points in the band structure of a wide variety of solids[l-121. In most of the previously used techniques [l-5] the electric field is non-uniform and quantitative comparison between the experimental lineshapes and existing theories of the elctro-optic effect for a uniform field [ 13-l 71 is difficult. These methods, applicable to semiconductors, make use of a space charge region for the production of the electric field and include the field-effect configuration of Seraphin and co-workers [ 1.21, the semiconductor-electrolyte-interface technique of Cardona and co-workers [3,4], and THE

ELECTRO-OPTIC

*Supported bv the Army Research Office. Durham, .. and the National-Science Foundation. ton training assignment from the National Bureau of Standards. Permanent address: Solid State Physics Section, National Bureau of Standards, Washington. D.C. 20234, U.S.A. $Present address: Bell Telephone Laboratories. Murray Hill, N.J. 07974, U.S.A. $A. P. Sloan Foundation Research Fellow.

2. EXPERIMENTAL

The basic experimental configuration is similar to that reported previously [3,4, lo] and is shown in Fig. I. The source is a 450 W 217

R. A. FORMAN

728

/ hiqh volioge

Fig.

et al.

high-voltage square - woire generator

I. Diagram of the experimental apparatus. The sample is in the transverse electroreflectance configuration.

XB0450 Osram high-pressure Xenon arc, and the monochromator is a 3/4 m Spex Model 1700-l I. A 600 line/mm grating blazed at 500 nm or a 1200iinelmm grating blazed at 200 nm was used in conjunction with a Dumont 6911 photomultiplierwith S 1 response or an EMI 6256B photomultiplier with S13 response, depending upon the wavelength range of interest. Quartz optics were used throughout, and the light incident on the sample was polarized with a Polaroid HNB’P film polarizer. The effective light intensity I was held constant by controlling the gain of the photomultiplier in the usual manner[4] so that a constant DC current was maintained at the anode. The variation AZ of light intensity caused by the electric field modulation was phase sensitively detected with a Princeton Applied Research Corp. Model HR-8 lock-in amplifier and the output recorded as a function of wavelength. The GaAs single crystal samples [ 1S] used in the experiment were chromium doped and had a room temperature resistivity of 10y R-cm and a liquid nitrogen (LN) temperature resistivity well over 10’” &cm. High resistivity Si was produced by diffusing gold into a 100 single-crystal silicon slice &cm p-type 1 mm thick at 1260°C for 23 hr. and following

this with a rapid quench[l9,20]. All samples were lapped and polished to a thickness of approximately 0.5 mm in order to aid in achieving spatial homogeneity of the applied field (see Appendix). The exposed surface of the silicon crystal was of (110) orientation, so that all three major crystallographic axes were contained in the plane of the surface. Following a suitable light etch, a thick gold film was evaporated onto the surface to be studied. except for bare strips O-25 to 1.0 mm wide perpendicular to each of the major crystallographic axes. The modulating electric field was developed in the plane of the surface and parallel to the appropriate crystallographic axes by applying a high voltage across these bare strips. This geometry gave the usual TER advantages: [8-lo] EK response could be measured for polarizations of normally incident light both parallel and perpendicular to the electric field, no dielectrics were present on the sample surface, and the fieid was reasonably uniform and could be accurately determined (see Appendix). Limiting field strengths of the order of lO”V/cm were determined by electrical breakdown across the surface gap. In most cases. measurements were taken with the samples in LN to insure high resis-

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tivity and to avoid thermal modulation effects [21]. Nitrogen bubbling, a possible source of noise, was eliminated by slowly bubbling helium gas into the LN. The samples were glued to a quartz backing plate at one corner only to eliminate strain: the fine wires used to make electrical contact to the evaporated gold regions were too thin to either support or strain the samples. These wires were soldered to the gold films using Wood’s metal. The resistance of the samples was greater than 1OSfi, including possible photoconductivity[22] due to the incident light. The experiments were performed using great care to prevent moisture from entering the dewar, for any ice which contaminated the LN would eventually be attracted to the high-field region of the sample. A square-wave voltage was applied across the contact gap so as to produce a periodic modulating field. The generator[23] was constructed so that both peak-to-peak and DC values of the square wave were independently variable. A I kHz frequency was generally used. The use of square waves is essential for the interpretation of line shapes and deof electroreflectance on field pendence strength, as will be discussed below.

electro-optic functions [2, 141. In this section we investigate the effect of this broadening, and show that by measuring its field dependence the broadening energy Fc can be experimentally determined. Our treatment does not include exciton effects [ 171; analytic expressions are not yet available to describe the general electro-optic effect in the Enderlein formalism [ 171, and for typical broadening energies and electric fields the analytic results of the one-electron approximation [ 13, 141 should be adequate to describe moderate bandgap semiconductors such as Si and GaAs where interband transitions dominate the absorption structure. The Lorentizian boradened electro-optic functions F(q) and G(n) can be expressed analytically[l4]; if Tc is the collision broadening parameter (in energy units), then after Lorentzian convolution

where h0 is the characteristic energy and

electro-optic

3. THEORY

A. Determination of collision broadening The theory of the electro-optic or FranzKeldysh effect has been extensively treated in the literature [ 13- 171. In the absence of Coulomb effects in the effect of the electric field on the real and imaginary parts of the dielectric function in the vicinity of any critical point can be described analytically in the oneelectron approximation in terms of two electro-optic functions [ 13, 141, F(n) and G(q). Excited carriers have a finite lifetime due to collisions, which leads to a broadening of the electro-optic spectrum which can be expressed as a Lorentzian convolution of the unbroadened response [2,15] in terms of a collision broadening parameter I’,. This broadening rounds off the singularities of the

v=;+

e2E2fi2 no=

113

-

(lb)

1s2+ Eu2 IEz2 -=-E2 P,1 (lc) /-+ rPU, IJ.~ i

G!,

1

1

p!, is the interband reduced mass in the field direction [ 13, 141. Changes in the real and imaginary parts of the dielectric function are proportional to F(r], r,/fiO) or G(q, I’,/&?) with the absolute magnitude of the proportionality factor given by B

e = d

47r2e2)Z. P,,J" .$ m2?i

(2)

230

R. A. FORMAT

where 8 is field-independent and contains the optical selection rules in the product of the polarization vector E with the interband momentum matrix element P,,.. Changes in the real or imaginary part of the dielectric function for any critical point can be written in the form

tt ul.

whence

(Sal or

or

Explicit expressions may be found in Ref. [ 141. Determination of I‘, by measuring the dependence of the peak-to-peak (p-p) value of Ae upon E is contingent on the dependence of F(n,l’,/@i8) and G(n,r,/M) on (T&2?), as shown in Fig 2. We plot F(n, X) and G(q, x) for four values of x =r,.l&Iie. Figure 2 illustrates the rapid decrease in the p-p value of each function together with a rapid attenuation of the subsidiary oscillations; with sufficiently large broadening the structure is effectively reduced to a positive and negative pair of peaks. As we shall show, the dependence of this decrease upon the ratio of &? to T,. is su~ciently strong to enable a determination of f, independent of the actual p-p value of the electroreflectance response or type of critical point. provided I‘, is field-independent. The broadening energy I’, depends primarily upon decay processes such as phonon emission, recombination through traps. Auger electron excitation, etc.. which are reasonably independent of externally applied electric fields, hence I‘, should be field-independent to a good approximation. We examine the dependence of ~AE(w,E, 1‘,.)1,defined in equation (3). upon field for a fixed broadening energy I’,.. It is convenient to use I‘, as an energy unit, and to define a field unit E,, as:

where the entire field dependence is contained in the terms in brackets, which are written as a function of the dimensionless reduced field E/E, since

(;) = (!J?’

(6)

‘The experimental condition of variable field and constant broadening energy is contained in the bracketed terms, since the proportionality factor is field-independent. The dominant peaks of the bracketed terms in equation t-5) originate from the singularity T= 0 and from q = -0.5 of F(q, O), and from n = 0 and n 2 - I ’ 1 of G(q, 0). We plot the numerical peak-to-peak (p-p) value of the bracketed terms as a function of reduced field for the Lorentzian broadened F and G functions in Fig. 3, and the energy separation (in units of I‘,. = h&,) between these peaks in Fig. 4. Inspection shows that the theoretical El’:” dependence of the p-p variation which occurs in the absence of broadening changes substantially when broadening is present. The p-p variation is faster than El’:%even for very little broadening: it is approximately El’:’ for I’,. - 0.01 hH. Such small broadening (or large field) is difficult to realize even under the best conditions at the lowest threshold and certainly cannot be obtained at interband critical points above the fundamental edge. The energy separation between the peaks retains the unbroadened E”‘” functional dependence

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I F(B,I-~01

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I1

I G(B,l-=Ol

-

-

0.16 -

-0.16 -

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Fig. 2. The effect of Lorentzian broadening on the_ electro-optic functions F(r). r,/fi@) and G (-,J, I’,,%@) for various broadenjn~ energies r,. = g/r.

R. A. FORMAN

232

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I I111111

I I0

I E

et al.

I IllIll

I IO

I

I lllil 100

Fig. 3. The energy difference, in units of ho,, between the main positive peak and main negative peak for each function in equation (3). as a function of field for a fixed broadening energy I’,. The abscissa is in units of E,, the field at which I‘, = rWo

for fields such that /&?i > Te, but the field dependence vanishes when JtiHl 6 I‘,. These statements apply equally well to both F and G functions. Measurements of the electrooptic effect for higher transitions appear at best to have I‘, comparable to tie; Figs. 2-4 show that under these conditions the electroreflectance spectrum of any critical point should consist of only two peaks whose positions are nearly independent of the electrical field and whose rate of growth with field is at least linear. It can be shown that in the limit of 1~014 T’, that the peak to peak amplitude of the signal is proportional to the square of the field. The rapidly decreasing p-p value for I‘, Ifi@]implies the electroreflectance signal will not be observable experimentally if I‘, 3 lh@j

which means critical points of one or two small interband reduced mass components will be favored. Since different modulation methods such as thermal[21] and stress[24] modulation depend on other parameters, electroreflectance will not respond with similar intensity to the same critical points in a given crystal. The field dependence of the p-p values of F and G in Fig. 3 is sufliciently strong to permit a quantitative determination of I’, by measuring the field dependence of the p-p amplitude of an ER structure and fitting the experimental points to the curves of Fig. 3. If the field is known for each experimental point, the reduced field E& is obtained immediately as that field corresponding to an abscissa of 1 in Fig. 3, whence 1;. is given by equation (4). Although

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233

Fig. 4. The peak-to-peak amplitude of each function of equation (3), as a function of field for a fixed broadening energy Te. The abscissa is in units of E,, the field at which I’, = hO,,. To obtain AE, the ordinate is multiplied by the prefactor B0,“2/~“, as defined in the text.

equation (4) contains pII, TC varies only as (&‘13, hence piI need not be known to high accuracy to extract TC. The advantage of this method over direct curve fitting lies in the fact that the variation of the p-p amplitude with reduced field is practically identical for both F and G functions, hence it is not necessary to know the nature of the critical point giving rise to the ER structure nor to perform the KramersKronig analysis in order to separate AR/R into AC, and Aeq to obtain I‘,. We repeat again that excitonic effects are not included and the results should be applied only where interband effects predominate. The determination of TC by the above method assumes that the dependence of the amplitude upon field is accurately measured.

We stress here that an accurate dependence (using any ER method) can only be measured using square-wave modulation since the electro-optic response is nonlinear. The commonly used sine-wave modulation will give erroneous results. The system nonlinearities will in general distort the shape of the sine modulation, inducing both a shape and an amplitude component in the fundamental harmonic extracted by phase-sensitive detection. True amplitude dependence is obtained only by using a modulating waveform which cannot be distorted by system nonlinearities, i.e. a square wave. Errors introduced by using non-square-wave modulation will be less serious in the ‘linear’ response region discussed above, where TC - &I, but may be appreciable if rC < I/if3(.

234

R. A. FOR~IAN

B. Symmetry d~terminution by transverse electroreflcctunce TER measurements can be performed with light polarized both parallel and perpendicular to the applied electric held, permitting symmetry information to be obtained. The electro-optic effect depends on the interband reduced mass in the direction of the field, producing field-induced changes in the dielectric constant which are anisotropic. The selection rules for polarized light give characteristic changes in the observed response for certain high-symmetry field directions and critical point setsf25.261. We consider here only I’, A, and symmetries which permit the use of certain selection rules valid in many cases of interest. We assume that first-forbidden elt‘ects and effects arising from absence of inversion symmetry in crystals of zincblende structure are negligible. The electricfield-induced change Ae arising from a nondegenerate critical point at I‘ will be scalar under these assumptions, and will exhibit neither polarization nor field directional dependence (although degenerate critical points at I‘ may exhibit polarization dependence in electroreflectance 1271. In many cases of interest, the polarization dependence of the optical selection rules for direct transitions can be described by the functional form: wi = [l - (Ii * ;I,“]

(6)

where Wi is the weight given to a transition involving a critical point at kc in the B.Z. with the incident radiation polarized along ri. Equation 16) simply states that transitions are allowed if lit i li. The change in the electroreflectance spectrum with polarization vector can be calculated in principle, since the electric field gives rise to an ER signal at a critical point which, for nondegenerate critical points, depends entirely on the angle between ki and E. A set of equivalent critical points are first grouped according to the angle made by each with the held. then the weights

ef al.

Wi of equation (6) are summed over each subgroup. Table 1 shows groupings and weight sums for equivalent critical points of A and A symmetry for fields applied in the (001). f 1 1O), and f I 1 1) directions in a crystal with an exposed ( I TOI face. The total ER response is the sum of fractions having these weights regardless of the detailed shape of the ER response of any subgroup, The simplest identifications occur for A symmetry with (1 I I) fields. and for A symmetry with (100) fields. in which cases all critical points make the same angle with the field and no polarization dependence occurs. These simple results do not apply if there is appreciable mixing of the wavefunctions on equivalent critical points, or interband mixing by the electric field, and would be obscured if structures from more than one symmetry coincided in energy. 4. RESULTS

AND DISCUSSION

We report here results of measurements taken on Si and GaAs. These semiconductors are representative of the Group IV and I I I-V compounds, can be obtained in semi-insulating form, and their electro-optic response is given, at least qualitatively, by the oneelectron approximation. A. GillliL~l~~rsei~id~ Figure 5 shows the electroreflectance spectrum of the (001) face of GaAs. The experimental conditions are shown in the figure. Unpolarized light was used. for the data presented in this figure and the electric field was oriented in the (100) direction. All GaAs measurements were made on (100) faces since the only high resistivity samples available were t 1001 slabs. The electric field shown in Fig. S was determined by sample dimension measurements (see Appendix). At the longest wavelengths, there is a large signal which is not a true ER signal. but rather is due to absorption-modulated reflection from the back surface of the sample. A back-reflection signal of this type is present whenever any light scattered from the back surface re-

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I

I

-

a t

I

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R. A. FORMAN

et al.

Table 1. Electrorejlectance subgroups and optical selection rule weights for I’, A, and A critical sets for different jield orientations on the ( 110) face of a cubic crystal Critical point. symmetry

Number equivalent points

Angle with field

Field

Parallel

Perpendicular

(000) (l:lO, A,X

I

Any

6

00 I

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2 4 4 2 6

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1613 413 4 0 1613

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(111)

A.L

x

00 1 54.73”. 8

1IO III

35.27”. 4 90”. 4 0”. 2 70.53”. 6

emerges and is collected by the detector. It may be easily recognized by several characteristic features: its dependence on sample geometry, back surface roughness, and its similarity to electroabsorption. It has a characteristic shape much steeper on the highenergy side than the low-energy side due to the rapid rise of the absorption coefficient. The effect is similar to the internally reflected signals which occur in surface barrier methods under certain conditions[28]. It can be completely eliminated by cutting the back surface at an angle to the front. The remainder of the signals are true frontsurface electroreflectance signals and can be identified, as in previous work[4], with the assignments: 1.50 eV, the fundamental edge E,, at I‘: 1.8.5 eV, E, + A,, the transition from the conduction band minimum to the spinorbit-split valence band, also at I‘: 3.0 and 3.2 eV, E, and E, + A,, the main and spinorbit split transitions of A symmetry. The EO and E,, + A,, ER peaks show no polarization effects to within the accuracy of our measurements, although polarization dependence at EO has been detected in EA in similar materials [27]. No polarization dependence was

observed in the El and El + A, structures for (100) field directions, consistent with the interpretation of the transition having A symmetry. We estimate the amount of broadening at the E, fundamental edge in this sample in two ways: by the best fit of G(q. l’&@) to the ER spectrum in this region, and by the procedure outlined in Section 3. Figure 6 shows the ‘best fit’ to the LN ER structure at a field of 3 X 10J V/cm. Since the interband reduced mass at E, in GaAs is 0.37 m,[29]. the characteristic energy at E = 3 x lo4 V/cm is fi0 = 2 1 meV. The ‘best fit’ occurs for l’,/M = 0.9, hence I’, = 19meV. The method outlined in Section 3 can be used to determine the broadening by measuring the field dependence of the p-p amplitude and fitting the experimental points to the curve of Fig. 4. We show the evolution of the complete M0 structure with increasing field in Fig. 7. The main peak at 824 nm remains fixed, as expected, since this peak is the singularity which occurs at the critical point energy itself. The entire structure changes with increasing field in a manner qualitatively similar to that shown in Fig. 2. The fitting of the p-p

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!_

1’33

1.3”

ENERGY CeV)

Fig. 6. ‘Best fit’ of theoretical

broadened G-function to experimental the fundamental edge of GaAs.

amplitude of the structure to the theoretical curve of Fig. 4 is given in Fig. 8. The experimental points. from 600V to 2000 peak square wave applied across the 0.56mm gap, are shown as circles. The field unit E, corresponds to an applied voltage of 940 V, whence from the crystal dimensions and Appendix A, E, = 10.9 kV/cm and lC = 11 meV. The discrepancy between the two methods of estimating lC may be caused by the definition of best fit: it is expected that the procedure of measuring peak height variation is more accurate since it involves a range of fields. Although exciton effects [ 171 are not included, the fact that IC is much larger than the exciton binding energy Es = 3.3 meV in GaAs [30] means that exciton effects will tend to be washed out and the shape will be primarily determined by interband processes, justifying the use of the one-electron approximation. The broadening energies obtained in transverse ER at the fundamental edge are larger

curve

obtained

for

than those obtained in capacitor[l, 21 or electrolyte [3,4] configurations, which are estimated to be of the order of 2 or 3 meV even at room temperatures, since deep traps and recombination centers in the semi-insulating materials may make excited carrier lifetimes shorter. It should be pointed out that the presence of normal fields at the surface due to a space charge region can cause errors in the calculation of the true field in transverse electroreflectance by modulation away from a nonzero field. If a normal field E, exists at the surface, the net field obtained in transverse ER is E=E,+Et where E, is the applied transverse field. Neglect of E,, if present, can cause serious errors. However, the large concentration of deep traps in semi-insulating semiconductors makes space charge regions fairly short

23x

R. A. FORMAN

c’t trl

Fig. 7. The evolution of the fundamental ER spectrum of GaAs with increasing field. These curves illustrate typical signal-to-noise ratios obtained in the measurements.

(the Debye length for a semiconductor of dielectric constant 10 and ionized impurity concentration 10” cm-:’ at LN temperature is - 10 nm). Also, the surface states on real GaAs [3 11 and Si surfaces [32] pin the surface Fermi level near the center of the forbidden gap, close to the bulk Fermi level in semiinsulating material. Thus, the true field over most of the light penetration depth, at least near the fundamental edge where light penetration depths are of the order of microns, will be tangential field. Our results are therefore characteristic of the transverse field and not greatly influenced by a normal surface field. B. Silicon We report only ER results on the structure at 3.4eV, whose band structure assignment

has been the subject of much effort and controversy [2,4,33-381. A tracing of a typical experimental spectrum is shown in Fig. 9. The experimental details are given in the figure. All spectra of Si in the 3.4 eV region show the same characteristics regardless of field or polarization, except for differences in peak amplitudes. The structure is similar in shape to that observed by Seraphin[33], except the higher-energy part is more pronounced. While this spectrum has the qualitative appearance of structure arising from one equivalent critical point set, a main peak followed by a subsidiary oscillation, detailed measurements show a negligible energy dependence of all peak positions on field amplitude. Since the interpretation of this spectrum as a main peak with a subsidiary oscillation requires a two-thirds power dependence of

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* GoAs,E,. 77

239

K

Fig. 8. Fit of the evolution with field of the peak-to-peak amplitudes of spectra similar to those shown in Fig. 7, to the theoretical curve of Fig. 4. The field unit E, = IO.9 kV/cm corresponds to a simple voltage of 940 V. The correction factor corresponding to the contact spacing of 0,55mm on a O-50mm thick crystal is 065 isee Appendix A).

the subsidiary oscillation

position with fietd. it is obvious that the absence of such a shift requires two distinct critical points in the 3.4 eV region. The qualitative features are consistent with the earlier discussion of broadening, provided TYC 2 flil for the fields attained. Subsidiary oscitrations cannot be seen until the Ezf3 dependence of peak position is weIt established for the main peaks, which woutd require ]!$I/ 2 1‘,; since no subsidiary oscillations are observed, then I‘, 3 lfi8). The quantitative dependence of p-p amplitude on field, for various polarizations and field orientations. is also consistent with 1‘, 2 j&S/ although this variation was not sufficiently strong to provide a definite Iocation on the curve of Fig. 4. If a typical interband reduced mass of 0.1 m,

is assumed, 1; >i 20 meV, from the highest field applied. Interpretation of this structure as arising from at least two critical points is also consistent with the interpretation of Seraphin. [2,33] who observed small shifts of peak position with bias in the surface barrier con~~u~at~on and assigned an Ma and MI critical point set to the lower and higher energy peaks on the basis of these shifts. Although his fields were nonuniform and the surface magnitudes not known, it is probable that they were larger than those attained here, which made observation of the field shifts possible. A more definite assignment could be made by calculating AC, and AfZ separately from AR/R, f39j but the fractionai coefficients

240

R. A. FORMAN

F = 5X

3800

IO4

et ul.

v/cm

3700

3600

3500

3400

xc&, Fig. 9. A typical electroreflectance spectrum shown corresponds to an applied voltage

CYand /3 necessary for the separation are not known for Si at LN temperature, and vary too rapidly with energy in this energy range to permit the use of the room-temperature values[2]. A measurement of the relative reflectivity (Fig, 10) of high resistivity sihcon was performed at 87°K and indeed also shows the doublet structure found in the electroreflectance. Measurement of the LN temperature optical constants of Si are planned. We consider next the measurements for different field orientations and polarization directions on the exposed (170) face. The amplitude ratios of polarizations parallel and perpendicular to the field directions for the

for the 3.4 eV region in Si. The field of 3500 V across the contact gap.

dominant peaks are given in Table 2 together with the means square deviation for a series Tnble 2. Ratio of amplitude oftwo peaks in the 3.4 eV region of Si for pur~lle~ and perpendicular po~~ri~~tions. The resldts are averaged for measurements from 2000 to 3500V in all cases. All measurements were made with the Jield in and light incidentperpendicularly onto the (I i0) face Field direction

Peak

3.6 ev 3.4 eV

001 1.9to.1 2.0?0.?

110 1.67’-tO.l t-ot0.2

ill 2.7io.3 2~0~0~5

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241

A (8) 3700 I

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3600 I

3500

I

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I INTRINSIC

53

-

49

-

48

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T’87OK

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I

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3.40

3.50 ENERGY

Fig. IO. Reflectivity

of measurements at applied voltages between 2000 and 3500V. Typically, little variation was observed over this voltage range. The results shown in Table 2 can be compared with the theoretical variations of the high-symmetry critical points given in Table 1. We consider first the larger peak at 3.6 eV. A simple critical point along either A or A can be immediately ruled out because of the strong polarization dependence exhibited for fields in both (001) and (Ill) directions. Since simple equivalent critical point sets of lower symmetry should tend to be more isotropic than the high-symmetry A and A sets, it appears that this peak is not due to a simple equivalent critical point set, but is a result of a more complex system, as for example several nearly degenerate bands [34]. The calculations of Kane[36], Dresselhaus and Dresselhaus [37], and more recently Saravia and Brust[38], show this energy structure as probably arising from a complex set of critical points near f, and lying between A and C. A calculation of the electric-field

3.60

(eV)

of Si near 3.4 eV.

effect on the dielectric function for each proposed energy band structure could be performed in the one-electron approximation by the convolution formalism[ 161, but this formalism contains approximations which may not be valid in this region, and the amount of computer time required to do the calculations is at present prohibitive. Similar conclusions apply to the smaller structure occuring in the region of 3.4 eV. The difference between the two structures with respect to polarization dependence is sufficient to show that they arise from different regions of the B.Z. It appears quite certain that the B.Z. region giving rise to this structure is not located in the ( 1 I 1) direction as indicated by Saravia and Brust[38], since the polarization ratio for a (001) field is nearly 2. The agreement is not good enough with any model, however, to allow a definite choice to be made with any particular critical point. Finally, Ghosh [35] reports seeing structure at 3.25 eV, 3.42 eV, and 3.55 eV in electrolyte electroreflectance measurements. These mea-

242

R. A. FORMAN

surements are not inconsistent with the results presented here, since the maximum fields obtainable in transverse ER are generally an order of magnitude below those obtainable in the semiconductor space charge region. The stronger electric fields presumably present in Ghosh’s measurements could bring out additional structure. We have not been able to observe any separate structure in the 3.2-3.3 eV region. 5. CONCLUSION We have applied the TER method to the measurement of the electroreflectance spectrum of semi-insulating GaAs and the 3.4eV structure of Si at liquid nitrogen temperatures. The application of known. nearly uniform fields to a crystal enables quantitative electroreflectance measurements to be made. We have shown that these measurements can be used to estimate collision broadening energies by measuring the field dependence of the peak-to-peak variation of the ER structure. A detailed investigation of the effect of collision broadening[Z. 14, 151 on the quantitative behavior of the electrooptic functions in the one-electron approximation demonstrates that the generally observed higher interband ER spectra consisting of single oscillations with both peaks field-independent are to be expected whenever the broadening energy I‘, = /?$I, in contrast to the predications of the unbroadened theory. The same relation between I‘, and Iht)l constitutes an observability criterion, for unless TCsmaller than or of the order of I&@}, a critical point will not contribute to the ER spectrum. Our measurements of the field and polarization dependence of the structure of the 3-4eV peak in Si show conclusively that at least two regions of probably different symmetry give rise to this structure. The data obtained were not sufficient to enable a of the symmetries, unique determination but it appears that simple isolated sets of equivalent critical points are insufficient to explain the results. These results are in

e’f 01.

general agreement with piezo-electroreflectance experiments [34]. A~,1\,2o,l,/edsr,rrrrlrs-One of us t D. E. A.) wishes to thank Bell Telephone Laboratories, Murray Hill. New Jersey. for providing facilities for the preparation of the final manuscript. We would also like to thank Kerry Shaklee and Fred Pollak for many helpful discussion>. REFERENCES B. 0. and HESS R. B.. P/!ys. Rvr. I. SERAPHIN L.c,rt. 14. 13X ( 1965). B. 0. and BOTTKA N.. Phxs. Rw. ‘. SERAPHIN 145,678 (1966). K. L.. POLLAK F. H. and CARDONA 3. SHAKLEE M.. PII~s. Rec. Left. 15. XX3 I 196.0. M., SHAKIXE K. I. and POL.I.AK 4, CARDONA F. H.. Pfrys. Rec. 154,696 I 1967). A. and HARDLER P.. Plr_xy. RzL’. Lrtt. 5 FROVA 14. 17X (1965): FROVA A.. HANDLER P.. GERMAN0 F. A. and ASPNES D. E., Phy. Kc1.. 14s. 57.5 ( 1966). C.. Hclv. P//y\. Acta 39. 593 (1966): 6. G;I;HWILLER Soliti-St. C‘tlmmlltl. 5. 65 ( 1967). 7. PAINE E. G. S. and REES H. D.. Ph,v.s. Rev. Latt. 16.444 ( 19hh). 8. REHN V. and KYSER D. S.. Plr,xs. Rec. Lctt. 18. H4X ( 1967). report of this work is contained in 9. A preliminary Rrt//. Anr. p&s. sot. 12. 658 ( 1967). R. A. and CARDONA M.. in f/-c’/ IO. FORMAN Sr~lriL.Oltdtt(.tilz,~~ Compowtdx, p. 100. Benjamin. New York( lV67). I I. FRENCH B. T..P/qs. RPU. 174.991 (196X). Y., HANDLER P. and GERII!. HAMAKAWA MANO. P~Ix. Rec. 167. 703 (19681. K., P&Js. Rw. 130, 7204 I3 TH~RMA~~IN~AM (lV6.i); ASPNES D. E..Phys. Rcu. 147. 554(1966). D. E., Pfz~s. Rcr. 153, 97’ t 1967). 14. ASPNES R.. Phy\. Stutirs. So/i& 20. 7-93 1s. ENDERLEIN ( 1967): 23. I77 ( 1967). 0. E.. HANDLER P. and BLOSSEY 16. ASPNES D. F.. P11v.s. Rrr. 166. VII ( 1968): REES H. D.. J. Phy. Chrrrr. Solids 29. I43 ( 1968). R.. Phys. Sttrtus. So/id;. 26, 501) 17. ENDERi_EIN ( I VhXI. from Brll and Howell. Pasadena. C‘alif. 18 Obtained R. E., Standford Electronics LaboraIV AITCHISON tories Tech. Rep. No. 3 I I-:! ( I9hO). D. E.. Ph.D. Thesis. Univ. of lllinoi~ 70 ASPNES (1965) (unpublished). C-. N.. ./. tr& Phys. 37. ?iOlV (IV)hh): 71 BERGLUND A. G. and MATA’I AGUI E.. THOMPSON CARDONA M. Phys. RPC. In press. J. and Y.4COBY Y.. Phvs. RPC. 174. 937 22. SHAH ( 1968). generator was designed by L. 13. The square-wave National Bureau of Standards, MA RZETTA, Washington. D.C. W. E.. FRITSCHE H., GARFINKEL 74. ENGELER M. and TIEMANN J. J. Phys. Km. Latt. 14. IOh9

TRANSVERSE

75. 16. 27.

18. 19.

30. 31. 37. 33. 34. 35. 36. 37. 38. 39.

40. 41.

41.

43. 44.

ELECTROREFLECTANCE

(1965); GOBELI G. W. and KANE E. 0.. Ph_vs. Rev. Left. 15. 143 ( 196.5). R&SLER U. and BOTTKA N.. Solid-St. Commun. 5, 939 ( 1968). AYMERICH F. and BASSANI F.. Nltoco Cirn. 56B, 295 (1968). BAGAEV V. S.. BELOUSOVA TYa.. BEROZASHUILI Yu. N. and KELDYSH L. V., Proc. lnt. Corlf. on the Physics of Semiconductors. Vol. I, p. 384. Publishing House “Nauka”. Leningrad (1968). ZOOK J. D.. Phys. Rev. Letr. 20. 848 (1968). VREHEN Q. H. F.. J. Phys. Chem. Solids 29, I29 (I 968). quotes the following effective masses for GaAs at 77°K: M,.” = 0.067 me. tn,,, = 0.082 point the IN,. and nr,,,,* = 0.45 m,. At the critical light hole mass should dominate. whence p = 0.037 ,,I,,. STURGE M. D.. Phvs. Rec. 127. 768 ( 1962). FLlNN I.. SurJilcc Sci. 10. 32 (1968). ASPNES D. E. and HANDLER P.. Surface Sci. 3, ( 1966). SERAPHIN B. 0.. P/?ys. Rev. 140. Al716 (1965). CARDONA M. and POLLAK F. H.. Phys. Rro. 172.8 I6 ( 1968). GHOSH .A.. To be published. KANE E. 0.. Phvs. Rev. 146.558 ( 1966). DRESSELHAUS G. and DRESSELHAUS M. S.. P1z.w. Rec. 160. 649 ( 1967). SARAVIA L. R. and BRUST D.. Phvs. Rev. 171, 916(1968). EHRENREICH H. and PHILIPP H. R.. Phys. Rev. 129. 1550 (1963); CARDONA M. and GREENWAY, Phys. Rec. 133, A1685 (1964). OBERLANDER S. and WILHELM W. E.. Phvs. stut. sol. 12, 569 (I 965). CHURCHILL R. V.. Introduction to Complex Varicrbles und Applicutions. p. 17 1. McGraw-Hill, New York (1948). MILNE-THOMPSON L. M.. In Handbook of Mrtrhemtrtical Functions (Edited by M. Abramowitz and I. A. Stegun (U.S. Department of Commerce, National Bureau of Standards, Washington, D.C.. 1964). Appl. Mtrth. Series 55. pp. 567 ff. Ibid.,p. 591. ASPNES D. E. and FROVA A.. Solid-St. Commun. 7. I55 ( 1969). APPENDIX

Typical transverse electroreflectance geometries [6-IO] can be represented quite accurately in crosssection as idealized two-dimensional potential distributions having simple boundary conditions. These potential distributions can usually be expressed in closed form using Jacobi elliptic functions. [20,40]. In this Appendix, we obtain an analytic approximation for the potential distribution of the crystal in our geometry. Figure I O(a) shows a cross-section of our crystal taken perpendicular to both crystal surface and contact edges. and plotted in the W = u + iv plane. The crystal is assumed uniform in the direction perpendicular to the crosssection. We approximate the actual geometry by assuming the crystal and contacts continue as shown to infinity in both positive and negative u-directions. Errors are intro-

343

duced because the crystal is finite in the ~1 direction and nonuniform in the perpendicular direction. but the effects of these errors decrease roughly as e-a’XiL’, where x is the distance from the crystal boundary or nonuniformity, L is the sample thickness. (the characteristic length for the geometry) and a is of order unity. The truncation errors are small in the region of interest. the center of the gap between the contacts. To obtain the analytic approximation, we must further assume that the dielectric constant is isotropic. the contacts define equipotential lines. and the normal component of the electric field vanishes on all remaining crystal surfaces. The first assumption is satisfied for a cubic crystal. The second and third depend primarily upon the absence of surface states, which result in the presence of electric fields in the crystal even if no potential difference is applied between the contacts. Such surface fields are almost always present in crystals and represent the principal limitation upon the results calculated here. The vanishing of the normal field over the contact-free boundaries also requires that the crystal dielectric constant be much larger than the dielectric constant of the ambient. Unless the crystal dielectric constant is infinite. normal fields will be present at the contact-free boundaries. In the absence of surface charge, the normal field inside the crystal is related to the normal external field by the boundary condition E,,l = E amb./ cr. and amb. refer to the ecr. Eanthl. where the subscripts crystal and ambient respectively. and I to the normal component of the electric field at the boundary. Since the external normal field is a result of field fringing, it is expected to be much less than the tangential fields. where internal equals external. For the usual dielectric constants encountered in crystals and ambients. the normal field component inside the crystal due to fringing will be quite small. This is particularly true for the region of interest between the contacts to the crystal. Therefore, to the extent that normal field at free boundaries can be neglected. the potential distribution in the region centered between the contacts will be given fairly accurately by the potential distribution of the idealized geometry and boundary conditions of Fig. I I(a). The coordinate axes u and u are defined as shown in the complex )c-plane in order to utilize the symmetry of the idealized crystal. The potential distribution is completely described by either the upper or lower half of the crystal. and by choosing the upper and lower contacts to be at potentials + V/Z and- c’/2 the line u = 0 becomes an equipotential with @ = 0. We therefore need the potential distribution for only the c 2 0 half of the figure. where the boundary conditions are

I dV(w) -= d,r*

The

crystal

0: (0 =s II c d.u=O) O:(u = 0.0 zz

c

<

Qc)

i 0:

thickness

is d. the

spacing

between

the

R. A. FORMAN

244

et al.

W - PLANE

Z-PLANE

Fig. I I (a) The idealized sample geometry in the ut-plane. Contacts are shown as heavy lines, and dimensions are indicated. (b) The Zplane diagram of the rectangle conformally mapped onto the upper half of the idealized sample geometry. Equivalent points A, B, C, and Dare shown in both diagrams. contacts is s. and the voltage difference between contacts is V. The boundary conditions are defined over four straight line segments, which suggests that the idealized crystal represented by the region defined in Fig. I I(a) can be mapped onto the rectangle shown in Fig. I I(b), for which the potential distribution is simply 4(z) = yL’/21’,. Corresponding image points in Figs. I I(a) and 1I(b) are labeled A, B, C and D with B in Fig. I Ifa) receding to u -+ + m. We map the rectangle onto the idealized crystal using the conformal mapping W(z) = ~(x.y) -t iofx,~] defined in the z-plane of Fig. 11(b). By Schwartz-Christoffel transformation theory[41],

the straight line segments of the rectangle will be mapped onto the straight line segments of the idealized geometry if the derivative dWtz)/dz of W(z) possesses a symmetric distribution of poles and zeros about each of the infinite lines formed by extending the sides of the rectangle. Therefore, dW(z)/dz must be doubly periodic in the complex z-plane. In addition dW(z)/dz must be analytic everywhere except at those points where the map is not conformal, i.e. at points B and C where the angles are not preserved in the mapping, and al1 doubly periodic images of 3 and C. The required angutar changes are obtained if dMi(z)/dz has a pole at B and a zero at Cf4ll. This determines dW/dz to within a multiplicative constant.

TRANSVERSE

the idealized crystal potential distribution is

Including this constant gives:

y

245

ELECTROREFLECTANCE

=

@(u, 0) =

+(z)

V

Im

2K(1--m)

I

nlV,%i

dc)[l-msin20]-“‘(12)

and the electric field is W(Z)= u + iv = $ sin-’ [sn(z)]

(7b) E,+E,=

where the Jacobian elliptic functions[42] dn(z) and sn(z) are doubly periodic over the fundamental rectangle

4diyvm)

[*-msinZ(~(U+in))]~l’*

One of the objectives of transverse electroreflectance is to obtain a known, nearly uniform field in the region near the center of the gap between the contacts. Equation (13) can be used to calculate the field value at the center of the gap, and to estimate the width of a region where the field is uniform to within a given amount of this value. On the surface in the gap, w = d+ iv, and

K is the complete elliptic integral of the second kind and m is a parameter given by the location W, = d+is/2 of C in the w-plane. m is most easily obtained from the inverse transformation

PV E”=04dK(l-m)

1 -mcosh*

@a)

(

J

d0[ 1 - m sin2 @I-“*.

0

The integral representative larity at w,, where

(8b)

=wc

TV

(

)

.

(10)

Since the potential distribution on the rectangle is (II)

I

I

I

2>

(14)

V

c(m) represents a gap width/crystal thickness = s/d dependent correction to the simplest approximation, which sets the field equal to the voltage across the gap divided by the gap width. The correction e(m) is plotted as a function of s/d in Fig. 12. As can be seen, the simple approximation E = V/s is surprisingly accurate, with t(m) varying over the limited range 2/n = 0,636 at zero gap, to I at large gaps. Greatest uniformity of the tangential field near the center of the gap will result for largest ratios of gap width to crystal thickness. If s/d is large. m is small, and using[43]

= 0.

Using W, = d-t is/2, we have cash-2 ;

-“’

~EV.~=4&(]-m)2/(]-m)=5(m)~~

( >

m =

)I

giving the field minimum in the center of the gap as

of equation (8b) has a singu-

1 --m sin* 2d

g

u=d,Ocvsc

rrwlzd

E

(13)

I

I

I

I

I

,.o_____-__-__--_--------_--_-__-__

d

Fig. 12. The correction t(m) to the zero-order field approximation, with the asymptotic expansion which includes fringing at the edges of the contacts.

246

equation

R. A. FORMAN

( 13)

the index of refraction. Also, we suppose broadening ib present, in which case according to Fig. 3. the most rapid variation of E with E will be quadratic. at the peak locations when IhHI ,+ I’. Then equation ( IX) can be written

become5 in this limit

t 17) Each contact edge adds an effective length approximately equal to (2ln?/a)d to the gap width s for the purpose of calculating the tangential field at ;he center of the gap. Figure I?. shows that this asymptotic behavior is reached for a quite low s/d ratio. Surface uniformity can be estimated by setting I, = din equation ( 17) and calculating the range of P for which

IE,.(d+ir)l

s r). (E,,.(d.

et (I/.

O)/.

rl -)

I.

where 7 represents the allowable deviation of tangential field from the value at the center. The inequality is satisfied over the entire gap to within distance L = (d/n) In (q’/ + I) of the contact edges. As an example. IO per cent uniformity is achieved over the gap region which excludes the strips within L = 0.55d of the contact edges for any gap width greater than I. Id. Field uniformity is therefore achieved with quite reasonable gap/length ratios. It has recently been shown that field inhomogeneitie\ can strongly influence the ER lineshape in surface barrier ER. due to interference in the direction of propagation which is phase-sensitive and mixes the line-shapes of AC,, and he2 as well as averaging them. We prove here OUI previous statement that such inhomogeneities in transverse ER are negligible. using equation ( Ii) and the criterion given in Ref. [44] for the neglect of inhomogeneity in the propagation direction:

1.akmg the most Unfavorable cases, we suppose t is real so that h/c = 2~-r1. where A is the wavelength in air and II

which may be evaluated : @ u. We find equation

using equation is equivalent

( 19)

(13), noting

that

to

Since this effect is observable for a distance of the order of several wavelengths in from the surface, we set II = d. which neglect? terms of second order in h/d. and obtain.

But we have already restricted our measurements to regions where the abolute value quantity is not too large. Therefore. TER field inhomogeneity of the type discussed in Ref. [44] is negligible. We should emphasize that this concerns mixing of the real and imaginary parts of AC due to interference: spatial variation of E parallel to the surface can be included by performing a simple average of AC,. and ,IE? independently. This effect is also of minor importance in the presence of broadening. Gene