Deformation Potentials for Excitons in Cuprous Halides

Physics and Astronomy Publications

Physics and Astronomy

4-1972

Deformation Potentials for Excitons in Cuprous

Halides

J. B. Anthony

Benedictine College

A. D. Brothers

Benedictine College

David W. Lynch

Iowa State University

, [email protected]

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Deformation Potentials for Excitons in Cuprous Halides

Abstract

The hydrostatic-pressure shifts of the Z1,2 and Z3 exciton peaks were measured in thin films of cubic CuCl,

CuBr, and CuI at 90 K. That of the E1 peak in CuI also was measured. The deformation potentials of all Z

excitons and of the E1exciton in CuI, about -1 eV, are more than twice those of the Z excitons in CuCl and

CuBr. This suggests the two valence bands in CuI may be considerably more mixed than in CuCl and CuBr.

Keywords

hydrostatic-pressure shifts, valence bands, Ames Laboratory

Disciplines

Condensed Matter Physics | Physics

Comments

This article is from

Physical Review B

5 (1972): 3189, doi:

10.1103/PhysRevB.5.3189

. Posted with permission.

LATTICE

DYNAMICS

%'ITH THRE

E

-BODY

FORCES.

II.

KRYPTON

3189

Phys. Letters 34A, 415 (1971).

G. Casanova, R.

J.

Dulia, D. A. Jonah,

J.

S.

Rowlinson, and G. Saville, Mol. Phys. 18,589 (1970).

R.

J.

Dulia,

J.

S.Rowlinson, and W. R. Smith,

Mol. Phys. 21, 299(1971).

J.

A. Barker, D. Henderson, and W. R. Smith, Mol. Phys. 17, 579 (1969).

PHYSICA L REVIEW

8

VOLUME

5,

NUMBER 8 15

APRIL 1972

Deformation

Potentials

for Excitons

in

Cuprous

Halides~

J.

B.

Anthony* and A. D. Brothers

Department of Physics, Benedictine College, Atchison, Kansas 66002 and

D.

%.

Lynch

Department of Physics and Institute

for

Atomic Research, Iowa State University, Ames, Iowa 50010 (Received 6December 1971)

The hydrostatic-pressure shifts ofthe Zl 2and Z3exciton peaks were measured in thin films

of cubic CuCl, CuBr, and CuI at 90K. That ofthe El peak in CuI also was measured. The deformation potentials ofall Z excitons and ofthe Elexcito

.

in CuI, about —1eV, are more

than twice those ofthe Z excitons in CuCl and CuBr. This suggests the two valence bands in

CuI may beconsiderably more mixed than in CuC1and CuBr.

I.

INTRODUCTION

The cuprous halides, CuCl, CuBr, and CuI,

are

insulators whose band structures have been con-sidered tobe related to those ofthe group-IV,

-III-V, and

-II-VI

cubic semiconductors, when account

is

taken of the increased tonicity.

'~

They

are

cu-bic (zinc-blende structure) below about 640 K.s

'

Cardona measured the absorption

spectra

of these

compounds and found two sharp peaks at low energy

and a stronger peak at higher energy. Up to that

time,

it

was believed that the valence band was largely composed ofhalogen-p-like functions. The splitting of the two lowest-energy peaks was

inter-preted as a spin-orbit splitting, since it increased from chloride to bromide to iodide, but because it was smaller than the spin-orbit splitting of the halogen, an appreciable Cu-3d component was

be-lieved to be present in the valence band. The posi-tions of the band edges could not be reconciled with

predictions from the gaps in III-V and II-VI com-pounds.

Song ' calculated energy bands for CuCl. He

found that there were two valence bands, with agap

ofabout 2eV between them

(Fig.

1).

At the center

ofthe zone, the uppermost band

is

derived

primari-ly from Cu'-3d levels

(-30%

of the charge density for the

I'„state)

while the lower

is

predominantly from Cl -3p levels (-SO'/o for the lower

I'„).

He

showed that only the lower band was to be

associ-ated with the valence band in cubic semiconductors,

and

its

position could be extrapolated with reason-able accuracy from band gaps in cubic

semiconduc-tors.

The exciton spectrum of CuCl consists of

a

sharp line called the Z3line, followed at higher energy by

the

_Z»

line, which often

is

split into two by

strain.

According to Bong these originate from the spin-orbit-split

I'„

levels,

I',

and I'8,

respec-tively, both in the upper valence band

(Fig.

1).

The lower valence band gives a

series

of absorp-tion structures, beginning with apeak called E&and extending into the vacuum ultraviolet. ~ _The

_E,

_peak

originates at the lower

I'„

level and corresponds to the lowest-energy

direct

exciton at

I'

in the

group-IV,

-III-V,

and

-II-VI

semiconductors. The

_E,

peak

is

stronger than the Z, 2 and Z3

peaks, and

is

"allowed,

"

while the Z& 3and Z3peaks correspond roughly to dipole-forbidden atomic transitions,

Cu'3d-

Cu'4s, that

are

allowed at k

=0

by the admixture ofCl 3p wave functions. CuBr presumably has a similar set of valence bands,

and so might CuI, but in both of these, the Z, ~

line

is

at lower energy than the Zs line, indicating a reversal of sign ofthe spin-orbit parameter. The gap between the lower and upper valence bands may diminish in CuBr and CuI. (The

Z»

—

E,

separations diminish. ) The two valence bands then

would become more mixed in atomic-wave-function

character.

(In AgC1, the Cl 3p and Ag'4d states

completely mix, except at afew points, into one complicated valence band.

'0)

Extensive mixing

should make the Z, 2and Z3 absorption peaks

stronger with

respect

to the E~peak, but this does not seem to be the

case.

Measurements of the hydrostatic-pressure

simi-3190 ANTHONY,

BROTHERS,

AND

I

YNCH

II-VI compounds

is

not characterized by nearly con-stant values of (BE/BP)r or of (BE/BlnV)r, so

ex-trapolation to the cuprous halides

is

not possible.

II. EXPERIMENTAL

C9

R hl

Lg Lp

1(5

XI

X5 —

Xp

Xp

X5

XI

FIG.

1.

Conduction band and two highest valence bands

as calculated for CuCl by Song (Ref. 7).

lar

binding, the band gaps vary with pressure,

or

volume, with coefficients depending only on the symmetries ofthe states at the edges ofthe gaps.

For

example, in group-IV and

-III-V

semiconduc-tors,

the values of (BE/BP)r are dependent

pri-marily on the symmetry of the states and not on the nature of the atoms.

"

In alkali and silver halides, the deformation potential (BE/B lnV)r seems to be better than (BE/BP)r for characterizing energy gaps, and

it

too seems to depend primarily onthe symmetry of the states at the gap edges.

'

Recent work ' _{indicates that the}

I'„—

_I',

_{transition in}

Thin films of cuprous halides were evaporated

on quartz substrates at room temperature. Film thicknesses were

500-2000

A. The absorption

spectra

ofthese films showed only peaks

charac-teristic

of the cubic phase. Upon cooling to liquid-nitrogen temperature, these peaks became sharper, indicating that differential contraction between film

and substrate was not badly straining the films. A

film of CuBr on a Pyrex substrate gave results in-distinguishable from those obtained onquartz.

Spectra were run on a Cary-14 spectrophotometer

with the samples at 90 K in apreviously

de-scribed"'

high-pressure cell using helium as a pressure fluid. Data were taken in the range

1.

0-3.

5

kbar.

The experimentally observed pressure shifts of the exciton peaks

are

given in Table

I.

Our values for the positions ofthe Z, 3and Z3peaks are not in good agreement with those of Cardona,

but are quite close to those in more recent work on

films and single

crystals.

'

Typical

scatter

of the data about a straight line (Fig. 2)was afew

percent, but sample-to-sample differences in the values of (BE/BP)r ofabout 107ooccurred.

Edwards and Drickamer have measured (BE/ BP)r ofthe absorption edge ofpolycrystalline slabs, using the shift ofthe energy at which the absorp-tion coefficient was 15cm

'

in CuCl. They obtained

(BE/BP)r

=+0.

7&&106eVbar ~

for CuC1, substan-tially in agreement with our value for the

first

exciton peak; approximately zero for CuBr; and

—

_4.

_1&10

6_eVbar

'

_{for CuI. The}

latter disagrees

with our result in sign as well as magnitude. They

also observed apeak at

3.

3eV in CuC1 _{(Z, 2,} Z~,

presumably) and found its pressure shift to be

+0.

74&106

_eVbar,

in agreement with the absorp-tion edge value.

We have converted our values of (BE/BP)r to

de-formation potentials via the equation

TABLEI. Excitons in cubic cuprous halides.

Peak positions at90K (eV) (0bar)

Z1,2

Z3

E) QQ

in10

'

eVbar BP

Z1,2

Z3

Figures inparentheses from Ref. 2.

CuC1

3.288

3.220 (6.45)

+0.

9

+0.

9

CuBr

2.979

3.123 (5. 60, 5.48)

+0.

4

+0.

9

CuI

3.060

3.694

5.10, 4.8

+2.

9

DEFORMATION

POTEN TIAI

S

FOR

EXCI

TONS

IN.

. .

3191

5.223 _5.297

CuCI

80'

K

5.222 3.296

5.22I

O

LLJ

K

LLI

%220

3.295

3.

294

FIG. 2. Position of exciton peaks in CuCl at 90Kvs pres-sure. The film was 1200A

thick. Circles (square) and

crosses (plus) represent data

taken with increasing and

de-creasing pressure,

respective-ly. The lines are least-squares fits and have slopes of

8.

94

and 4.61x10~_eV/atm.

5.2I9

0 1000 2000 3000

PRESSURE (atmospheres)

5.293 4000

BE q(BE

a(lnV) ~~BP, '

where K

is

the compressibility, for which Bridg-man's values2' were used (Table

II).

III. DISCUSSION

One

sees

immediately that the deformation po-tentials (or pressure shifts, since the compressi-bilities vary by

less

than 20/o) for the

first

ezciton peaks in CuCl and CuBr

are

essentially the same,

while that for CuI

is

considerably

larger.

The

higher-energy exciton peak involves ahole in the lower (spin-orbit-split) part ofthe valence band. Since both holes are in states derived from the up-per

I"»

level in

Fig.

1, they are expected to have charge densities with the same spatial symmetry, sothat similar pressure

or

volume shifts are

ex-pected on either an ionic

or

covalent model. Both peaks in CuC1 and Cu.

Br

have essentially the same pressure shifts, while they appear to be different for CuI. (When

errors,

previously estimated as 10%,

are

considered, the two pressure coefficients, in units of 10

eVbar,

are 2.

9+0.

3and

4.

0+0.

5

for the Z, 2 and Zspeaks, respectively, in CuI.) The small shifts for CuCl and CuBr

are

manifesta-tions of the fact that the transitions are quite

lo-calized on one ion. Apeak in CuI at

3.

18eV has been interpreted by Ueta et

al.

as an exciton bound

TABLE II. Deformation potentials (eV) for excitons in cubic cuprous halides.

Zi~2

Z3

Cucl —

_0.

4

—

_0.

₄

CuBr

—

_0.

₂ —

_0.

3

CuI

-1.

0

—

_1.

4 1~1

at an imperfection.

It

has been observed in films

and in single

crystals.

Its

temperature and

pres-sure coefficient

are

similar to those for the

free

exciton Z3 and Z&3peaks.

The E~exciton in CuCl originates from a differ-ent valence band, separated from that for the Z, ~

and Z3excitons by a distinct gap, as

is

probably true for CuBr and CuI.7

Unfortunately the

E,

ex-citon could not be measured in CuCl in the present work, being attoo short awavelength. While the

E,

peak in CuBr (actually adoublet) was in our

wavelength range, the data scattered too badly for

us to estimate

(SE/aP)r.

In CuI the

E,

doublet was not well resolved. The average shift of the pair

with pressure

is

given in Table

I.

The value of (BE/BP)r for the

E,

exciton

is

not in agreement with

that for the

I,

5—

I

~transition in group-IV and

-III-V

semiconductors, + 12&&10 eV

bar,

nor

js

3192

ANHTONY,

BROTHERS,

AND

I,

YNCH

Z3excitons in CuCl and CuBr, but quite close to

that ofthe Z& 2and Z3peaks in CuI. The holes in the valence bands (in CuC1) are partially localized

about different ions: Cu' _{for Z, 3and Zs and} Cl for

E,

.

Such similarities in (BE/BP)r for Z, 2 and

Z3 and Eg excitons in CuI

are

unexpected, and may

be fortuitous, but they could

arise if

the gap

be-tween the uppermost valence bands

is

missing,

or

is

very small, and the ten Cu'-3d electrons and

six

I

-5p electrons form,

or

almost form, one broad valence band complex with

critical

points similar to those in CuCl and CuBr. In this way the exciton spectrum would

persist,

but with altered (BE/BP)r values for all excitons, as observed.

Better

evi-dence

is

needed before this description of the va-lence bands in CuI can be accepted.

Itwould be of value to relate the observed

exci-ton peak shifts to those of the band gap. This

re-quires an estimate ofthe pressure dependence of the exciton binding energy

B.

For

Wannier excitons (BB/BP)r=—2B(8

Inc/BP)r,

where q

is

an effective

dielectric

constant,

For

CuCl, Nikitine et

al.

~v

find

B=0.

189eV and &=

5.

41, between the optical

and static dielectric constants of

3.

8 and

7.

4.

Un-fortunately (8&/BP)r

is

not known, nor

is

8&/8T from which an estimate of (8&/BP)r could be made.

For

alkali halides, (Bing/BP)r

is

about

(5-10)

&&10sbar

',

leading to (BB/BP)r=—₍₂

4)X10-eVbar

',

much larger than our observed shift. This value

is

probably an upper limit for the binding-energy pressure shift, but

it is

so large we cannot equate even roughly the exciton pressure shift with

that of the band gap.

From values of (BE/BT)p, the observed tempera-ture shift, and (BE/BP)r, the measured pressure shift, one can determine what part of the former

arises

from thermal expansion, and what part, (BE/8T)~, from the electron-phonon interaction via the equation

(2)

n

is

the linear-expansion coefficient and K

is

the compressibility. Good values for (BE/BT)~ for the

cuprous halides unfortunately do not

exist.

(BE/

BT)~

is

zero at 0K, and

increases

in magnitude with increasing temperature, reaching a constant value above a temperature

characteristic

of the

phonon spectrum. n also behaves similarly while

Kand (BE/BP)r

are

relatively temperature indepen-dent. Kaifu et af. have shown that (BE/BT)J,

is

constant, +

2.

5&&104eV K

'

between 10and 90 Kfor

the Z~peak in CuC1. Cardona's average (BE/BP)r between 4and 300 K

is

about half this, indicating a possible change in (BE/BT)~ above 90K. (The positive sign

is

not often found in solids. ) One would like touse values of (BE/BT)p and n at a temperature high enough that both are sensibly con-stant. Because (BE/BP)r

is

so small for the Z, peak in CuCl, (BE/BT)v, when evaluated from the above formula using the 3n value given by Mayer, a turns out to oe positive with either Cardona's or Kaifu's value of (BE/BT)~. (BE/BT)„

is

expected to be negative on asimple model, but exceptions

exist.

30

In the

case

where (8E/BT.

_)„

'is

positive,

its

interpretation in terms of the temperature

de-pendence of the electron-lattice coupling

is

quite complicated. 3' Whether such a positive value of (BE/BT)~ really exists for CuC1 (and CuBr) must

await further measurements of (BE/BT)~ on single

crystals above 90K. It

is

interesting to note that for the Z,2peak in CuI, (BE/BT)„

is

about zero

be-tween 4and 80K, leading to a small negative

value of (BE/BT)».

IV. SUMMARY

The exciton deformation potentials we have mea-sured for cuprous halides cannot be interpreted

fully without accurate knowledge ofthe temperature dependence ofthe exciton-peak position. (Data at

4, 78, and 300Kare probably insufficient. ) They

cannot be related to the band-gap deformation po-tentials because the pressure dependence ofthe

ex-citon binding energy may be comparable to,

or

even

larger

than, the small observed exciton deformation potentials. Arough estimate of the pressure shift ofthe exciton binding energy can be made when the pressure (or temperature) dependences ofthe static

and optical dielectric constant

are

available. De-spite this,

it

is

clear

that the exciton in CuI

is

in

one way dissimilar from that in CuCl and CuBr.

ACKNOWLEDGMENT

The part ofthis work performed at Benedictine College was'supported by agrant from the Research Corportion and by the NSF Undergraduate Research

Participation Program, whose support is

grateful-ly acknowledged.

~Work partially performed in the Ames Laboratory of

the U. S. Atomic Energy Commission. Contribution No.

3164.

Present address: Department of Physics, Lehigh

Univers ity, Bethlehem, Pa. 18015.

F.

Herman. ,

J.

Electron. 1, 103(1955).

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J.

Krug and L. Sieg, Z. Naturforsch. 7a, 369 (1952).

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J.

Phys.

Soc. Japan 7, 19 (1952).

M. R. Lorenz and

J.

S. Prener, Acta Cryst. 9, 538

(1956).

K. S. Song,

J.

Phys. Radium 28, 195(1967).

'K. S. Song,

J.

Phys. Chem. Solids 28, 2003 (1967).

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DE FORMATION

POTENTIAL

S

FOR

E

XC

I

TONS

IN.

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1960 (Czech. Natl. Acad. Sci., Prague, 1960), p. 686.

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Bassani, R. S.Knox, and W. B. Fowler, Phys. Rev. 137, A1217 (1965).

P.

M. Scop, Phys. Rev. 139, A934 (1965).

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J.

Appl. Phys. 32, 2082 (1961).

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E.

Slykhaus, and H. G. Drickamer,

J.

Phys. Chem. Solids 11,140(1959).

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1964, edited byM. Hulin (Dunod, Paris, 1964),p. 241;

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M. Ueta and T. Goto,

J.

Phys. Soc. Japan 20, 401

(1965).

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J.

Phys. Soc.

Japan 20, 1022(1965).

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Phys. Soc. Japan 20, 1654 (1965).

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T.

Yashio, and M. Ueta,

J.

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Phys. Soc. Japan 22, 488

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Nakazawa, and M. Koba,

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Phys. Soc.Japan 22, 517

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Gota, T. Takahashi, and M. Ueta,

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Phys. Soc.

Japan 24, 314 (1968).

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and H. G. Drickamer, Phys. Rev. 122,

1149 (1961).

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_{W. Bridgemen, Proc.}

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P HYSICA L REVIEW

8

VOLUME

_5,

NUMBER 8 15

APRIL

1972

Elastic Constants

of

RbF

from

300

to

4.

2

K

C. R.

Cleavelin, D. O. Pederson, and

B.

J.

Marshall

Department of Physics, Texas Tech University, Lubbock, Texas 79409 (Received 19November 1971)

The adiabatic elastic constants have been measured in single crystals ofrubidium fluoride over the temperature range of 300 to 4.2Kto investigate a discrepancy with neutron scattering data. The values ofthe elastic constants at 0 K, extrapolated from 4.2K, are C&&

=6.

527, C44 =0.952, and C&2

=1.

255 in units of10&&dye/cm

.

The Debye characteristic temperature at OK

(8&)as calculated from the elastic constants is SO=221.0K. A summary ofDebye temperature

values calculated from elastic constant data at0Kispresented forall previously measured

face-centered-cubic alkali halides.

INTRODUCTION

Recently Raunio and Rolandson' have obtained

the phonon dispersion relations for RbF by using

inelastic scattering ofthermal neutrons. The

elas-tic

constants of RbF were calculated at 80K by

utilizing the dispersion curves for small wave

vec-tors.

A comparison ofthese

elastic

constants with

those obtained from an extrapolation of Haussahl's room-temperature thermoelastic constants showed

acceptable agreement for C44and C&~ but not for

C».

Because of the large discrepancy in the two

cal-culated values of

C»,

we have measured the tem-perature dependence of the elastic constants ofRbF from 300to

4.

2K.

EXPERIMENTAL

The single-crystal specimens used in these

measurements were approximately cylindrical with

a diameter of

1.

6 cm and alength of

2.

5cm. The crystals were obtained from Semi-Elements Inc.

with the end

faces

of one crystal oriented perpendic-ular to the

[100]

axis and the end

faces

of the other oriented perpendicular tothe

[110] axis.

The

ori-entations were checked in this laboratory by means of

x-ray

Laue backreflections. Since RbF

is

ex-tremely hygroscopic, preparation ofthe crystal was accomplished in a helium atmosphere. The crystals were aligned towithin

1'

ofthe desired directions. The end

faces

were polished parallel to within +

0.

001 cm.

Because ofthe symmetry ofa cubic crystal, there

are

only three independent elastic constants:

_C»,