Electron energy bands and the optical properties of potassium iodide and rubidium iodide

Pergamon Press 1970. Vol. 3 1, pp. 265-274.

J. Phys. Chem. So/ids

Printed in Great Britain

ELECTRON ENERGY BANDS AND THE OPTICAL PROPERTIES OF POTASSIUM IODIDE AND RUBIDIUM IODIDE* A. BARRY KUNZt Department of Physics. Lehigh University. Bethlehem. Penna. 18015. U.S.A. (Received

12June 1969)

Abstract-Using

the relativistic mixed basis (M.B.) technique and the non-relativistic O.P.W. technique, electron energy bands are obtained for KI and Rbl under a variety of conditions. It is argued that the previous calculations and interpretations offered for the KI optical absorption spectrum are not in agreement with recent experimental results. The present calculation produces results which eliminate the difficulties associated with previous theoretical models, and the results of this calculation are in good agreement with experiment. It is found that the L, and X, conduction levels play a substantial role in understanding the experimental results. It is also shown that a stringent ‘muffin tin’ potential is a poor approximation to the KI potential. The Rbl results are similar to the KI results and a similar experimental interpretation is advanced. 1. INTRODUCTION

years the properties of Kl have been the subject of many theoretical and experimental papers. In this present paper we use electron energy band theory to investigate certain optical and transport properties of KI and RbI. The first attempt to obtain a band structure for Kl was made by Phillips[l]. In this work it was assumed that the conduction band was free electron like. The valence band was modeled after that which Howland obtained for KC1[2]. The optical absorption spectrum of KI has four sharp peaks in the low energy region. These peaks at 77°K are at O-430 Ry, 0.496 Ry. 0.507 Ry and O-526 Ry[3]. Phillips interpreted the first two peaks as being due to exciton transitions from the spin-orbit split 1-5~ valence band to the s-like conduction band at the point r in the first Brillouin zone. The second two peaks were interpreted as being due to exciton transitions from the upper two spin-orbit split 1-5~ valence states to the s-like conduction band at L. IN

RECENT

*Work supported in part by the U.S. Air Force Office of Scientific Research, Contract 1276-67. tPresent address: Department of Physics, University of Illinois. Urbana, Ill. 61803. U.S.A.

This model has been shown to be far too simple[4, 51. Firstly, the free electron model places the d-like conduction bands above the s-like conduction bands, whereas by a more detailed calculation the two conduction bands are seen to overlap[4]. Secondly, the exciton transition at the point L would be at a point where the energy separation of the valence and conduction bands is a relative maximum and one need not expect to see sharp exciton lines in this case. Onodera found that at the point X the lowest lying state is a d-like X3 state and the separation of valence and conduction band is a relative minimum just as it is at r and one may expect to find bound excitons here[4]. Onodera and Toyozawa[S] have developed a very elegant theory of these excitons. It is noted that Onodera identifies the first and fourth prominent absorption peaks as being due to the r exciton and the second and third peaks as being due to the X exciton. There are several experimental difficulties with the model of Onodera. Firstly, by a very careful analysis of the absorption spectrum of pure NaI, Kl and RbI crystals and mixed NaI-KI and KI-RbI crystals, Nakamura has shown that the first two peaks are due to transitions at the same point in the Brillouin 265

266

A. BARRY

zone]6 71. The first three absorption peaks are seen to have a similar temperature dependence [S]. Previously, the author has shown that the temperature dependence is explained simply by changes in the band structure due to the change in average lattice parameter as the temperature is changed[9]. It seems unlikely that transitions to different symmetry conductions bands should have the same dependence upon lattice parameter. In the case of CsI, Onodera has found that the first few exciton transitions are to s and d like conduction bands [ IO]. Lynch and Brothers have measured the pressure dependence of the first few excitons in CsI and find that the sign of the change in energy with pressure of the first two excitons is opposite [ 1 I]. It seems likely that transitions to s and d bands in both Kl and CsI would have a similar dependence on lattice parameter. In this paper we present a calculation of the energy bands in KI which avoids the difhculties of the Phillips model[l] and the Onodera calculation[4]. It is shown that the greatest part of the difference in the calculation of Onodera and the present one is due to a very stringent ‘muffin-tin’ approximation to the KI potential which is made by Onodera. We have found a similar difficulty in applying a ‘muhintin’ potential to NaCl[lZ]. In Section 2. we describe the techniques used in this calculation and discuss the various approximations made. In Section 3 the results of the various calculations are presented for KI in its normal lattice and in a distorted lattice. Results for RbI are also presented here. In Section 4 we compute the effective mass of an electron in the KI conduction band and compute the corresponding polaron mass. The results are compared to experiment in Section 5. 2. DETAILS OF THE ENERGY BAND CALCULATION

The energy bands in KI have been computed using both the non-relativistic O.P.W. technique [ 13,141 and the relativistic M.B. .(mixed basis) formalism[lSj. In the case of RbI

KUNZ

only the non-relativistic O.P.W. method is used. For KI we use a value of 13.236 Bohr units for our normal lattice constant. In the case of RbI we use the value of 13.8 Bohr units for the lattice constant. In the case of KI we also recompute the band structure using a lattice constant of 13*000 Bohr units in order to study the dependence of band structure on temperature or hydrostatic pressure. For these calculations we chose our usual model of potential[l4, IS]. This is a localized version of the Hartree-Fock potential for a valence electron located on a halogen ion. Localization is accomplished using the Slater exchange[l6] and imposing a Latter tail[ 171 so that the potential contributions have the desired asymptotic behavior. By using a localized potential we may not use the eigenvalues of the resulting Schrodinger equation as being the energy needed to remove that electron from the crystal since Koopman’s theorem is no longer valid [ 1S]. Thus we might regard the potential chosen as a type of pseudo-potential where it is hoped that the errors introduced by neglecting Koopman’s theorem compensate for the errors introduced by neglecting correlation effects. The ultimate test of such an approximation is the comparison with experiment. The normal lattice KI calculation was repeated using a ‘mufin-tin’ potential formed from the previously described potential using the values for sphere radii and height of the constant term in the potential given by Onodera et ~1.[4]. As will be seen, this ‘muffin-tin’ potential causes considerable distortion of the band structure. It is useful at this time to define the types of calculations used. The one electron crystal hamiltonian is given as H = &+

V(r)-t-II,tHd+Hs_o.

(1)

In equation ( If p is the momentum of an electron, m is the electron mass, I/(r) is the lattice potential, H, is the relativistic variation of

El ECTRON

ENERGY

mass with velocity term, Hd is the Darwin or non-physical term and H,_, is the spin orbit operator. In the non-relativistic O.P.W. calculation H,, Hd and HsTo are set equal to zero. In the M.B. calculation Hs_o is included using degenerate state perturbation theory 1191. The states which are included as core states are the K+. Is, 2s, 2p, 3s, 3p, the Rb+ Is, 2s, 2p, 3s, 3p, 3d, 4s, 4p. and the I- Is, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d. In the case of the O.P.W. calculation the one electron wave function is given as

and for the M.B. calculation we find Y

(3) In equations (2) and (3) @’ is the nth eigenstate which transforms according to the ath row of the yth irreducible representation of the point group of the wave vector k. The Q’s are properly symmetrized combinations of Bloch sums formed from the core orbitals. The S’s are properly symmetrized combinations of plane waves which have a common kinetic energy. The a‘s and b’s are variational parameters and are determined by simultaneously diagonalizing the overlap and hamiltonian matrix formed using the plane waves and the core orbitals or the orthogonalized plane waves as an expansion basis. In this calculation about 400 plane waves are used and the convergence of the energy levels is estimated to be better than O*OOlRy. 3. RESULTS OF THE ENERGY CALCULATIONS

BAND

The electronic energy levels of Kl were evaluated for the normal lattice constant at

BANDS

267

ten points in the first Brillouin zone by the M.B. method. These points are I, X, L, K, the midpoint of A, three equally spaced points along A and two points along 2. The X-ray energy levels obtained from this calculation are given in Table 1. In Table 2 the energies of selected points of high symmetry for the valence and conduction band are given. Spinorbit effects in the conduction band are neglected due to their small size[4]. The energy bands are shown in Fig. 1. It is seen that spin-orbit effects split the predominantly 5p I- valence band into two non-overlapping bands. At the center of the zone the splitting is 0.083 Ry. Neglecting Table 1. The X-ray levels for the KI crystal are given in column two for the normal lattice parameter (a = 13.236 Bohr units) and for a lattice parameter of 13*000 Bohr units in column three. In column one the atomic state of origin for the X-ray levels is given. In column four the levels obtainedfor the normal lattice parameter using a ‘mufin tin’ potential are given. Results are in Rydbergs One 1-1s 1-2s

Two

Three

Four

-2433. -2433. -2433. -3796 -379.6 -379.6 IEp, j = f -359.6 -359.6 -359.6 IEp. j = f -339.8 -339.8 -339.8 1-3s -77.62 -77.64 -77.62 I-3p. j = 4 -69.02 -69.03 -69.02 I-3p. j = 9 -65.24 -65.25 -65.24 lk3d. j = 2 -50.5 1 -50.52 -50.51 1_3d,j=$ -49.61 -49.62 -49.61 1-4s -14.02 -14.04 -14.02 lF4p,j=: -10.87 - 10.89 -10.87 I-4p. j = 3 -10.13 -10.15 -10.13 I-4d, j = + -4.603 -4.612 -4.603 [-4d, j = 2 -4.468 -4.476 -4.468 K+ls -236.7 -236.7 236.7 K+3S -27.36 -27.35 -27.36 KC2p, j = 2 -22.27 -22.25 -22.27 K+2p, j = 3 -22.05 -22.03 -22.05 K+3s -2.962 -2.952 -2.962 K+3pj = + -1.731 - 1.722 - 1.73 I K+3p. j = 4 -1.710 - I .701 -1.710

268

A. BARRY

Table 2. The energies of points of high symmete in the jkst ~riilo~ii~l zone of the KI crystal are given. Resrdts are presented for the valence and condl~ction bands. In column one the symmetry of the stute is given using the notation of 1241. In the second column the results obtuined using the relutivistic mixed basis method und the model potentiul described in [14, 151 are given. In the third ~oi~~rnn results obtained wing the relativistic mixed basis inet~lod und the ~rn~i~~ltin’ poter~tiul described in the text are given. In cohrntn fotrr, non-relativistic O.P. W. results are given and in column jive, rest&s for K1 obtained by Onodera et al. [4] using u ‘IW_@F~ tirl’ potential are given. For the results of columns tbvo to jive the lattice parameter is 13.236 Bohr rudii. In rolrrmn six, results ure presented wing a lattice ~~~r~iF~eterof 1JGOO Bohr units and the sume ~nethod us wus used for the results of column ti-rlo. Spin-orbit efects ure neglected for the conduction band. Ail energies ure in Rydbergs _.

One

Two

Three --Ssl- valence -

-1.1 X, L,

-1.525 -1.514 -1,519

J’JI’,,) 1’; x,(x;) x,(x:) X, k&ft;r

-0.675 -0.758 -0.707 -0.727 -0.805 -0.731 -0.703

i_$ 2 _

-0.809 -0.764 -.

1 ;1

-0.269 0.012 0.112 -0.058 -0,166 0.269 0,009 0.219 -0.206 -0.036 -0.009 0.155

1% 1.1, X1 X:3 X; X; X, L, L:, L; L,,

band -1.369 -1.357 -1.360

-0.663 -0.744 -0.673 -0.689 -0.756 -o&-j0 -O+i88

-06.59 -0,598 -0.691 -0.678 -0615 -0.719 -0.631 -0~710 -0.735 -0619 -0.671 -0.639

-1-47 -

-1.536 - i ,524 - I *52Y

-0.699 -0.793 -0.715 -0.737 -0.815 -0.741I -0.71

-0.718 -0.x19

conduction -0.080 0.160 0.159 0.019 0.102 0.462 0.263 0.232 0.026 0.061 0.147 0.766

Six _-.l--_......__

-1510 -1.503 -I.506

_.. 5pI- valence band

Lower

Five

Four

bands

-0.236 0.013 0.115 -0.048 -0.164 0.280 0.029 0.223 -0.194 - 0.033 0.008 0.157

-0.144 -0.042 -0.057 -0.063 -0.097 0,406 0.225 -0.022 0.005 0.020 0.027 0.067

-0.259 0.015 0.115 -0.049 -0.174 0.295 0,028 0.225 -0,205 -0.044 0.009 0.147

KUNZ

spin-orbit splitting the valence band is O+Yl76 Ry wide. The width of the spin-orbit split valence band is 0.134 Ry. It is noted that using the present I- potential, the author obtains a value of 0.066 Ry for the spin-orbit splitting of the free ion I- 5p state. The experimental value is 0*06S Ry and the agreement is excellent. it is found that the band gap is at I’ and is from a p-like r; state to the s-like T1 state, and has a value of 0.406 Ry. It is noted that there are other low lying conduction band minima at the points L and X. The minimum at X involves the d-like state X,. In general, it is noted that there is very poor agreement between this calculation and that of Onodera et al. 147with respect to the details of the conduction band. The most obvious differences are that the order of the d-like states at I‘ (r12 and I’,,,) are reversed in this calculation from those given by Onodera. It is observed that the order of the d-states obtained here is the same as this author and other authors have obtained for other alkali halides [ 14,15,20,23]. There are two things one can do to try to understand the differences between the present calculation and that of Onodera et al. [4]. Firstly, the energy bands were recomputed at I‘, X. L and the midpoint of A using the non-relativistic O.P.W. method. These results are presented in Table 2. It is seen that there is excellent correlation between the authors’ results using the two methods. The chief quantitative differences may be attributed to relativistic effects. Certainly the differences between the relativistic and non-relativistic results are similar to those found by the author for NaI. As a second test, the band structure of KI was computed making a ‘muffin-tin’ approximation to the author’s potential. As a matter of convenience, the author used the values given by Onodera[4] for the ‘muffin-tin’ constants. The radius of the K+ sphere was 3.275 Bohr, the radius of the I- sphere was 3.343 Bohr and the potential outside the ‘muffin-tin’ spheres was -0.145 Ry. The

ELECTRON

i

ENERGY

I

J-

I.0

BANDS

269

-

Fig. 1. The electron energy bands of KI are shown along lines of high symmetry in the first Brillouin zone. The lattice parameter is 13.236 Bohr radii. The B.S.W. notation is used[24]. The calculation was performed by the relativistic M.B. method.

calculation using the ‘muffin-tin’ potential was made at I, X, L, K and is presented in Tables 1 and 2. It is seen that the results are greatly distorted by the use of a ‘muffin-tin’ potential. However, the results of Onodcra are not obtained. The author believes this is due to remaining differences in the potential used in the two catculations. It does not seem likely that this is due to errors in the present calculation since the author, using his present program and a ‘mufhn-tin’ potential, was able to reproduce the NaCl results obtained by Clark and Kliewer[23] who used a ‘muffin-tin’ potential and the A.P.W. method. In addition, the author does not believe this technique of including relativistic effects is at fault since the size of the relativistic effects obtained by the author agree well with those of Onodera. Finally, using a lattice constant of 13-000 Bohr, the band structure of KI was evaluated at the points r, X, L, K using the usual potential. Qualitatively, these results are very similar to those obtained using the normal lattice constant. These results are presented in Tables I. 2. These results will be discussed in Section 5. In the case of Rbi a non-relativistic O.P.W.

calcutation was performed at the points I, X, L and the midpoint of A. In a quaIitative and quantitative sense the conduction bands of RbI are similar to those of KI. The results of the RbI calculation are presented in Table 3 and Fig. 2. Neglecting spin-orbit effects the valence band is found to be 0.063 Ry wide. The top of the valence band is at II5 and the bottom of the conduction band is at I’,. In these discussions the group theoretical notation of Bouckaert et al. is used[24]. The band gap is found to be 0.412 Ry. It is seen that there is a low lying minimum in the conduction band at Xs. The L, conduction level is also very low lying and by comparison of the Rbl results with those of KI. the author speculates that Lz is also a minimum. In presenting all of the above results a halogen ion site is used as the origin. 4. EFFECTIVE

MASS AND POLARON CONDUCTION ELECTRONS

MASS OF

Using the results of the energy band calculation the author has been able to compute the bare effective mass of an electron in the I’% conduction state. At points of high symmetry such as I. X. and L in the first Brillouin

170

A. BARRY

KUNZ

Table 3. The energies of points of high symmetry in the first Brillouin zone of the Rbl crystal ure given. These calculations were made using the non-relativistic O.P. W. method as described in [24]. The lattice parameter used was 13.8 Bohr units. All energies are in Rydbergs 5sl- valence band f-1

-1.35

X,

-1.34

5plL valence

I’,,

-0.646

Lower

I’, I‘;:,

-0.234 -0.002

I’,,

0.99

I.;5 I:’

0.504 0.086

zone the effective

X; .Y’

L,

-1.34

L; L’

-0.658 -709

band

-0.659 -0.690

conduction X, x:, X; X X:

-0.028 -0.162 0.268 0.11 0.203

.Y, X, X5

0.650 0.271 0.280

states L, L; L:, L,

L, L:, L;

mass of an electron

-0.175 -0.020 0.001 0,171 Fig. 2. The electron energy bands of Rbl are shown along lines of high symmetry in the first Brillouin zone. A lattice constant of 13.8 Bohr radii was used. The calculation was performed using the O.P.W. method. The BSW notation is used [24].

0.269 0.239 0.320

is given

by WI

(4) At the point rI this mass mij is found to be a scalar. The author has computed the effective mass about I’, by several different techniques. Parabolic fits of the computed band structure were made using the point rI as one of the points fit and several of the points along A and Z for the other point. In addition tight binding fits of E(k) about I‘, were made using the methods of Slater and Koster [26]. These tight binding fits seemed well converged and were employed in equation (4). The results of the several calculations were very consistent and effective mass of 0.44 free electron masses was obtained for the r, conduction electron using the tight binding method

which the authors believe to be the most reliable. The deviation from this value using the other technique was found to be +O.OO to -0.04 electron masses. In transport measurements the bare effective mass of the electron is not measured but a polaron mass is. Therefore, it is useful to compute the polaron mass from the bare effective mass. To do this, we employ the Langreth interpolation formula to the Feynman polaron mass [27]. Thus, one finds the polaron mass to be

(5)

Here m is the electron bare effective mass, e is the electron charge, w1 is the frequency of a longitudinal optical phonon. es is the static

ELECTRON

ENERGY

BANDS

271

two phonon absorption experiments of Hopfield, Warlock and Park[32]. Theoretical attempts to interpret the experimental data has been made by Phillips[l] and Onodera et a1.[4,5]. Onodera has demonstrated that the interpretations of Phillips are in error due to Phillips’ neglect of the d-like conduction levels. In the interpretation of Onodera, the first sharp peak is due to transition from the I; valence band to the exciton level associated with the I1 conduction level. m,=m(l+$ (6) He interprets the second sharp peak as being due to transition from X; to the d-like exciton state associated with X,. The third peak is and is found to be O-63 electron masses if one said to be due to an exciton transition from X; uses 044 for the mass of the bare electron. to X, and the fourth is said to be due to the 5. COMPARISON OF THEORY AND EXPERIMENT exciton transition from I; to Il. The interpretations of Onodera et al. are Experimentally, KI has received much attention. The optical absorption spectrum of not in agreement with recent experimental thin films of KI has been measured by Eby evidence. Firstly, the analysis of the mixed crystal KI-RbI data161 and of the KI-NaI et al. [31 Fischer and Hilsch [83 and Teegarden and Baldini 1291. Studies have also been made data171 indicates that the first two peaks on the reflection spectrum of bulk crystals of involve final states of the same symmetry. Hence, since the first peak is generally agreed KI by Roessler and Walker[30] and by Baldini to be due to a I‘; to I1 transition, the second and Bosacchi [3 Xl. In addition there is optical absorption data on mixed KI-RbI crystals by peak should be due to the I‘; to 1, transition. The analysis of the mixed crystal data by Nakamura[6] and on mixed ICI-NaI crystals by Nakamura and Nakai[7]. Basically, there Nakamura is supported by the temperature dependence studies of the sharp peaks in the is excellent agreement among the various Kl absorption spectrum by Fischer and authors on the features of the optical absorpHilsch [8]. The first two peaks are seen to have tion spectrum of KI. a similar temperature dependence in this work. In the analysis that follows the energy values used are those of Teegarden and It is also noted that the third peak has a temBaldini obtained at 10°K. The onset of opti- perature dependence similar to that of the first cal absorption begins at about 0.426 Ry with two. A shoulder on the high energy side of the fourth peak is found to have the opposite tema strong narrow peak identified as the gamma perature dependence of the first three peaks. exciton formed by a transition from the I; valence state. There is a shoulder in the ab- That is for the first three peaks the absorption energy shifts to lower energy as the temperasorption at about 0,458 Ry which is interpreted ture is raised. as being due to the band to band transition It is possible to explain most of the experifrom I; to Ii. There are other sharp absorpmentally observed features of the KI absorption peaks to 0.500, 0.514 and 0.534 Ry for tion spectrum in terms of the present calculawhich various interpretations have been protion. In Fig. 3 we present a chart of the MO posed. In addition there are certain broad peaks in the absorption spectrum at O-570, type absorption edges associated with the low lying conduction band minima at 11, L1 and X,. 0611 and O-661 Ry. The position of the band to band transition has been confirmed by the We also show the first few absorption peaks

dielectric constant. and E, is the optical dielectric constant. For KI, the value of a is 260 and one finds the polaron mass to be 0.72 electron masses when the value of O-44 is used for the bare effective mass. It is also possible to compute the polaron mass using the intermediate coupling theory result of Lee, Low and Pines [28]. In this case, the polaron mass is given by

212

A. BARRY

0.6C )-

xpx 3-

A+-

L-;L,--

x;*x- 3

c+* L,-

xpx 3-

r,-+r;L_*L,-

-

Fig. 3.The theoretical band edges are shown on the left hand side for KI. In the center the experimental peaks and the band to band transition is given. On the right hand side an empirically adjusted set of theoretical band edges are given. The notation of[24]. and the experimental data of[29] are used.

and the band edge obtained from the Kl optical absorption spectrum. Finally. we show a revised set of M0 type edges which we obtain by forcing the calculated band gap to agree with experiment. In doing this, we adjust the structure rigidly. It is noted that the error in the band gap is small (10 per cent). Previous calculations have shown that such readjustments are reasonable [ 141. We see that the first four sharp absorption peaks in Kl may be due to excitons associated with the I’; + I’,. I‘; -+ I‘,. L& + L,,and L; --;, L, band to band transitions. It is also seen that the shoulder on the high energy side of the fourth peak may be due to an exciton associated with the X; + Xn band to band transition. There is also good agreement

KUNZ

between the higher broad absorption peaks and the low lying M, edges shown in Fig. 3. From calculating the change in band edges with change in lattice parameter we find that the band edges at r, and L, will shift to lower energy as the temperature is raised. We also find that the edges associated with X3 will go to higher energy as the temperature is raised. We also find that the edges associated with X3 will go to higher energy as the temperature is raised. Thus, we see that the interpretation of the Kl absorption spectrum based upon the present calculation is in agreement with the experiments of Nakamura[l6], Nakamura and Nakai [7], and Fischer and Hilsch[8]. Hodby, Borders, Brown and Foner[33] have measured the mass of a polaron in the conduction band of Kl. They find the polaron mass to be 0.67-tO.05 electron masses, a value which is in excellent agreement with the value of 0.72 electron masses obtained from this calculation. Finally, using the results of Bridgman[34]. we have computed the hydro-static pressure dependence of the Kl band edges. They are as follows: I‘, ---, T‘, IQ -+ T‘, LY.5- + L, L;(upper) -+ L, x; + x:, X;(upper) -+ X:,

dE/dP= 5=0X 10wfiRy/bar dE/dP= 6.4X 10d6 Rylbar dE/dP= 1.6X lo-” Ry/bar dE/dP= 1.6 X 10V6 Ry/bar aE/aP==- lO-7 Ry/bar aE/aP- - 10e7 Ry/bar.

The author is not aware of any experimental studies of the pressure dependence of the band edges in Kl with which to compare these results. Recently Frohlich and Staginnus and also Park and Stafford have measured the two photon spectra of Kl and Csl[3_5]. These studies have included the polarization dependence of optical absorption in the region of the band edges. In the case of Csl a marked change in polarization dependence of the absorption occurs when one shifts from the absorption associated with s-like band edges

ELECTRON

ENERGY

to d-like band edges. This is in agreement with the results of Lynch and Brothers [I 11. In the case of KI as measured by Friihlich and Staginnus there was no observed strong change in absorption with polarization throughout the exciton region. This tends to confirm the assignments proposed in this paper. The less extensive results of Park and Stafford are not in agreement with the more recent work of Frohlich and Staginnus. Finally, we consider the soft X-ray absorption measurements of Fujita, Gahwiller and Brown[36]. In these experiments the soft X-ray absorption from the 4dII states to the p-like conduction bands are measured. The onset of this absorption is at about 3.94 Ry. There are sharp peaks observed at about 3.98, 4.10, 4.32 and 4.45 Ry. In addition there are smaller peaks at 4.07, 4.17, 4.19 and 4.23 Ry. These may be due either to exciton or band to band transitions. The splitting between the first two sharp peaks is 0.12 Ry. This corresponds well with the computed spinorbit splitting of 0.13 Ry of the 4dlI state. The p absorption band has a total width of about 0.47 Ry. The computed width of the p-like levels associated with the lowest II5 conduction level is about 0.50 Ry. In the case of RbI, the interpretation of the optical spectrum is conducted along the line indicated by the KI results. In this case, there are no other calculations available for comparison. In Fig. 4 we show the band edges of RbI associated with 11, L, and X,. To make this chart spin-orbit splittings have been incorporated in the RbI valence band by degenerate state perturbation theory. We also show the band edge and the exciton transitions obtained from experiment[20,37]. Finally. we show a revised set of edges obtained by having the band gap agree with experiment. It is seen that a similar experimental interpretation to that of KI is possible for RbI. It is clear that the first exciton peak is associated with the I; -+ I1 transition. The second and third peaks are associated with the L; + L1 and the IQ + 1, transitions respectively

273

BANDS RbI

+hory

experiment

adjusted theory

0.525 -

i edge

-

ply

-

0.425 -

J

0.375 ’

Fig. 4. The theoretical band edges are shown on the left hand side for Rbl. In the center the experimental peaks and the band to band transition is given. On the right the empirically adjusted set of theoretical band edges are given. The notation of Ref.[24] is used and the experimental data of [24] and [351 are used.

although it is possible that if one includes relativistic shifts the I‘; + I1 edge would lie lower. However the fourth peak seems associated with the X; + X, transition. Here as in KI low lying minima at I’, L, and X in the conduction band are essential in an interpretation of the RbI absorption spectrum. Recent Stark measurements by Menes on RbI tend to confirm our assignments[381. However, this would indicate that the spinorbit splitting in the 1’ exciton is smaller than for the I band edge. It is also possible that some small structure between the third and fourth peak is the I’; exciton. 6. CONCLUSIONS

We have seen that by using the relativistic mixed basis technique and a rather simple model potential results for the KI band structure may be obtained which are in good agreement with a wide range of current experimental evidence. A new interpretation of the optical absorption spectrum of KI is proposed

274

A. BARRY

which avoids the weak points of the experimental interpretations of Phillips [ l] and Onodera er al.[4,5]. In agreement with the results of Onodera[4,5] we show that the dlike conduction bands play a significant role in the fundamental optical absorption of KI. We have also shown, in agreement with Phillips[l]. that the L, conduction edge is essential in a satisfactory interpretation. We find that this L, level is a minimum whereas Phillips found it to be a maximum. We have also arrived at an interpretation of the RbI spectrum which is similar to that advanced for the KI spectrum. It has also been seen that the use of a stringent ‘muffin tin’ potential such as used by Onodera[4,5] produces a considerable distortion of the energy bands. It is suggested that calculations of the conduction band density of states, and the interband density of states should be performed. An effort should be made using the theory of Page et al.1391 to test the connection of the L bands of the F-center with transitions from the F-center ground state to the conduction band. as the results of the present calculation are not in agreement with the simple L-band theory proposed by Baldini and Bosacchi [3 11. Aclino,lled~ements-The author wishes to thank Professor W. B. Fowler for several stimulating discussions. The author also thanks Mr. Y. Onoderd for several discussions relating to his calculation. He thanks Dr. Jozef Devreese for a stimulating discussion relating to polaron theory. He also thanks Dr. D. Frohlich and Professor F. C. Brown for permission to quote from their unpublished works. Finally. the ethcient computing facilities of the Courant Institute are gratefully acknowledged.

REFERENCES PHILLIPS J. C.. Phys. Rev. 136, Al705 (1964). HOWLAND L. P.,Phvs. Reu. 109. 1927 (19.58) EBY J. E.. TEEGARDEN K. J. and DUTTON D. B..Phys. Rev.116. 1099(1959). ONODERA Y.. OKAZAKI M. and INUI T.. J. phys. Sot. Japnn 21.2219 ( 1966).

KUNZ 5. ONODERA Y. and TOYOZAWA Y., J. phys. Sot. Japan22.833 (1967). 6. NAKAMURA K.. J. phys. Sot. Japan 22. 511 (1967). K. and NAKAI Y., J. phys. Sec. 7. NAKAMURA Jupan 23.455 (1967). 8. FISCHER F. and HILSCH R., Nachrichten Akod. Wiss. Giitingen, Math. Physik. Klasse 141 (1959). 9. BARRY KUNZ A.. Phys. Rev. 151.620 (1966). Y ., J. phys. Sot. Japan 25,465 (1968). 10. ONODERA D.. and Brothers. Phys. Rer. Left. 21. 689 I1 LYNCH (1968). I2 KUNZ A. B.. FOWLER W. B. and SCHNEIDER P. M., Phys. Left. 28A, 553 ( 1969). I3 HERRING C.. P~_Ys. Rev. 5’7. I I69 ( 1940). 14. KUNZ BARRY A., Phys. Rev. 175.1147 (1968) I5 KUNZ BARRY A., Phys. Rev. 180.934 ( 1969). I6 SLATER J. C., Phys. Rev. 81. 385 (195 I ). I7 LATTERR.,Phys.Rev.99,510(1955). I8 LLNDGREN I.,Ar~.Fvs.31,59(1965). 19 LIU L.. Phys. Rev. 126,-l 3 I7 ( 1962). 20 KUNZ A. B.. Phvs. St&us Solidi 29. I I5 ( 1968). 21 OYAMA S. and MIYAKAWA T.. J. phys. Sot. Jnpun 21.868 (I 966). 17 DeClCCO P. D.. Ph.vs. Reu. 153. 93 I (1967). 23 CLARK 7. D. and KLIEWER K. L., Phys. Left. 27A, I67 ( 1968). 24 BOUCKAERT L. P., SMOLUCHOWSKI R. and WIGNER E.. Phys. Rev. 50.58 (1936). 25 BEALL FOWLER W.. Phys. Rev. 132, I591 ( 1963). .I. C. and KOSTER G. F., Phys. Rev. 94. 76. SLATER 149X(3954). D. C., Phys. Rev. 159,717( 1967). 11. LANGRETH 78. LEE T. D.. LAW F. E. and PINES D., Phys. Rev. 90,797 (I 953). K. J. and BALDINI G.. Phys. Rev. 29. TEEGARDEN 155,896 ( 1967). D. M. and WALKER W. C.. 3. opt. 30. ROESSLER Sot. Am. 57,677 ( 1967). G. and BOSACCHI B.. Phys. Rev. 166. 31. BALDINI 863 ( 1968). J. J.. WORLOCK J. M. and PARK K.. 32. HOPFIELD Ph_~s. Reu. Lett. 11, 414 (1963); HOPFIELD J. J. and WORLOCK J. M.. Phgs. Rro. 137A, 1455 (1965). J. W.. BORDERS J. A., BROWN F. C. 33. HODBY and FONER S.. Phys. Rev. Lrrt. 19,952 ( 1967). 34. BRIDGMAN P. W., Proc. Am. Acad. Arts Sri.. 64, 19(1929). 35. FRGHLICH D. and STAGINNUS B.. To be published; PARK J. K. and STAFFORD R. G. Phys. Rev. Leti. 22, 1426 ( 1969). 36. FIJITA H.. GAHWILLER C. and BROWN F. C.. Phyz. Rec. Letf. 22, 1369 ( 1969). 37. FROHLICH D. and STAGINNUS B., Pllys. Rec. Left. 19.496 ( I967 ). 38. MENES M.. Bull. Am. phys. Sm. 14. 429 ( 1969). 39. PAGE L. J., STROZIER J. A. and HYGH E. H.. Phys. Rec. Left. 21. 348 (I 968).