VOLUME 4, NUMBER PHYSICAL REVI EW LETTERS FEBRUARY 15, 1960

T. H. Geballe and F. J. Morin, Phys. Rev. 95,1085 (1954).

5S. H. Koenig, J. Phys. Chem. Solids 8, 227 (1959).SP. J. Price (unpublished).~P. Fisher and H. Y. Fan, Bull. Am. Phys. Soc.

3, 128 (1958).S. H. Koenig, Phys. Rev. 110, 988 (1958).See W. S. Hoyle and K. F. Rodgers, Jr. , J. Opt.

Soc. Am. 49, 66 (1959).

M. A. Lampert, Signal Corps Report (unpublished).E. Burstein, J. S. Picus, and ¹ Sclar, Proceedings

of the Conference on Photoconductivity, Atlantic City,November 4-6, 1954, edited by R. G. Breckenridgeet al. {John Wiley and Sons, lno. , New York, 1966},p. 353.

S. H. Koenig, Phys. Rev. 110, 986 (1958).A. Karagounis, Suppl. Bull. inst. intern. froid,

Annexe 1956-2, p. 195.

INTERNAL IMPURITY LEVELS IN SEMICONDUCTORS: EXPERIMENTS IN P-TYPE SILICON

Solomon Zwerdling, Kenneth J. Button, Benjamin Lax, and Laura M. RothLincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts

(Received December 29, 1959)

The concept of the existence of multiple sets oflocalized impurity levels, each associated withthe various band extrema pertinent to its donoror acceptor character, has been verified by theobservation of newly-found transitions to ac-ceptor excited states in silicon lying near thesplit-off p~, valence band maximum, within thecontinuum of the p» bands as shown in Fig. 1.The detection of such discrete internal impuritylevels should provide information about the loca-tion of higher bands in energy-momentum spaceas well as about their characteristics and band

parameters.In the case of the impurity levels of the valence

bands in silicon, transitions are possible be-tween the ground state, which is primarily asso-ciated with the p» band (band a), and the excitedstates of the p~, band (band b), as shown inFig. 1. If we treat the coupling between the p»and pz, bands as a perturbation, the transitionprobability from level 1 of band a to level 2 ofband b has the form

a(1)b(1) -„a(1)b(2) 8 b(1)b(2)

(b)

FIG. 1. Valence energy bands of silicon showingthe acceptor ground state (1) and excited states (2) forboth the degenerate p&& bands {a}and the split-off p~2band (b) . The internal '

impurity levels are thosenearest band b. The unlabeled arrows are alsoallowed transitions.

a(2)b(2)-g

) ( )a(l)a(2)

Here D is a momentum operator between groundand excited states, ' X'z~ is a matrix element ofthe effective-mass Hamiltonian connecting thetwo bands, and h~b is the energy separation ofthe levels. Transitions are also allowed betweenthe ground state of band 5 to an excited state ofband a permitting, in principle, population in-version between state 2 and state 1 of band 5 or

between state 1 of band 5 and state 2 of band a.Optically-induced transitions of holes from the

ground state in the forbidden gap to excitedstates associated with the split-off p„, valenceband were observed at 4'K in the wavelengthrange 10-15 microns using both boron- andaluminum-doped silicon. The spectra for boron-doped silicon in Fig. 2 show clearly-definedabsorption maxima for transitions to the firstfew quantum levels, and the energies appearedto fit a Rydberg series for both boron and alumi-

173

UM&&R 4~or.~M~ 4 & ~~gypRSR PRIE~H YS&~&1- yg, 19~0P+gRU+R+

O.9

C

~ p.8~a

hIO

p.6—0CANXCOX

p54J

I-

o,4

p8.9 ItG

SI (boyoA)

e g (11'I)E J„B

4 go K

I

p.p89 p.o9PI

O.P84I

OP8~

entsectra ur eminfrared sPthe jpnlga

F ompre ' "'tates in sllaccep 0

e an~ ~

excited sb d maxim™

pn

f holes to th I s'he transitwn

tn". ted bY extrapo

h (boron)

n energy oplat j.ng

hoton energiesg a].uminum)p ppp3) ev and

t s iitting of th(() O441 ap

in orbit sp ~Then t"ebands ln sl 1valence b

p ppp4) ev.(p O44& ~g —g

gIce ~Where I

l„de a correc ~ "ary tp in u

alenceit is necess

pf the p~2 vaever 1

1' character 0the nonparat' nshlp beco

bpllc cpmesat the rela 10bs,nd, sotg)

(2g = g ~I 2(3nn

(3)Iha, veFlg.A for plotting .

Owave cyclo-In evaluating

d from mlcrowaused m = . 's andthe v

n~ dedoce

pf Q Pb-us

ance results,ted Hydberg

tron resonans, plot o

.3) in th™

f the uncorran-

taine& fr mshown in»gashed ].ine s

ination o

formu» (dThis deter m

lidribed above

(3) and is v»ve the same v

correctipn terentire Rydberg oth the correct

since the enso that bot

me seriesishes a,s

lpts yield the s' the

or r ected P~

n fpr hl y

and uncorhe expression

d ls

't &~ Frpm t

valenc~ ban

limit I 'ss for the Pe2

y—y3. 6

fective mWhere ry

hple e~

(p 25~0.pl)ma~„'=l3o have

fo„nd to beature value' oev an d a lp~-temPe

2 a'i&~= "'as

g The

ded tp ' clcu]ation

e p "turbation theo y

as taken usingthen a p

f the latter' essen-

expectatlp. The coefflcie "4

n val~e oA. ls esfunctions'

f 'ent pf the Pensionless coe band param-

led' ate scale

ence anb ds- In the P oh aluminumfprt e

f the boron3, the or ln

tp that oa,cceptothe greater dep

prdi-acc p o

t The in«rcep o"O882~O Opp].

grpund st t '

@ *(boron) =(gives di(O l/30+ O. pev and inurn)

I

p.p88I

p.p81

p.5—

OP86p.p85

e and atIe~tyof) yolt)

ld {sol&d l~

p.p8&

pTpN « "G"' "'ns a«ero '

internalobtained for

und ~tate in t g' ion through a

ntal spectru oacceptor gro

the trans~

Experimen am the boron ac

the ratio of t e

FyG. 2.line}

inate scale ik logauss {d

nd. The ordif a pure san'p

88. &

of the p&2 ban ~

15yoo) to that o

ited states o=6x10

exc&{t=3doped sa~P e

174

VOLUME 4~ NUMBER PHYSICAL REVIEW LETTERS FEBRUALRY l 5 y~ 960

0.089

0.088

$- Si (boron)0.0882

~-Si (A)0.1130 0 113

o~ 0 087Ohe

0.086

CO

0.085Uf

+o 0.084

0.083

ectedberges0)

—0.112

—0.111

—0.110

—0.109

—0.108

00092

Oo090

O

0+088

0.0820 0.1 0.2

—0.107

FIG. 3. Plot of Rydberg series (dashed line) andEq. (2) (solid line) using experimental energy valuesfor optical transitions from acceptor ground state inthe forbidden gap to internal acceptor excited statesfor boron-doped Si (left scale) and Al-doped Si (rightscale). For Si(boron), N+=6 &&10@/cm~ and i=3 mm,and for Si(AI), Ng = 2 x 10~5jcma and t = 6 mm.

Oe086I-O

00084

5p

been used. This mass value is in good agree-ment with the cyclotron resonance result. ' Ifthe P' term were not included, this hole effectivemass would be 0.205nz„ indicating a correctionof about 20 /c due to the influence of the P» bands.

The Zeeman effect of these internal excitedstates is shown in Fig. 4 and, in addition, transi-tions apparently associated with bound Landaulevels within the P~, band are evident above 81 .The spectral structure associated with the transi-tion to the 2p' level is probably due to spin split-ting superimposed on the Zeeman splitting. Simi-lar spin splitting may be present in the boundLandau level spectra. Since the gap ground statepresumably has a Zeeman effect also, furtherinvestigation will be needed for confirmation aswell as for an evaluation of the pz, band effectivemass from the magnetospectroscopic data. Ten-tatively, the interpretation as spin splittingyields g =3. It may also be possible to determinethe sign of the valence band parameter B fromthis g value.

Oe082—2p

I I I I

t0 20 50 40MAGNETIG FIELD, 8, {kilo gauss)

An analogous experiment should be possiblefor the valence bands in p-type germanium.Here the values of 8 ~ would be larger by afactor of about four, and the first term of Eq. (1)

175

FIG. 4. Zeeman-effect energy level diagram show-ing the splitting of the internal impurity levels in amagnetic field. With increasing magnetic field thehighest states become linearly dependent on field abovethe ionization limit, similar to Landau levels. Thetransition to the 2P' level shows structure probablydue to spin splitting.

VOLUME 4, NUMBER P HYSICAL REVI EW LETTERS FEBRUARY 157 1960

is dominant. The transition probability shouldbe of the same order of magnitude as in Si. Un-fortunately, the absorption corresponding to thistransition overlaps by -0.01 ev the steep absorp-tion edge of the intervalence band transition at-0.28 ev. Nevertheless, by using an optimumimpurity concentration and high spectral resolu-tion, it may be possible to observe this transi-tion in Ge as well as in InAs, GaAs, InP, GaP,AlP, AlAs, and A1Sb where intervalence bandtransitions have been studied. ~

In principle, it should be possible to observetransitions between impurity levels of bandshaving extrema at different values of k. Suchan indirect process probably involves the emis-sion of phonons at low temperature. Thus inheavily-doped n-type Ge, transitions from thelowest (111) impurity level to the ( 000) and(100) impurity states may be observable. Theformer should occur at -0.15 ev, and the latterat -0.22 ev. ' Similarly in GaSb, a transitionfrom the (000) to the (111) impurity level shouldoccur at -0.08 ev." However, since the transi-tion probability is low, impurity concentrationsof the order of 10"/cm' may be required.

We have profited greatly from theoretical dis-

cussions with Professor G. F. Koster.

This work was performed with the joint support ofthe U. S. Army, Navy, and Air Force.

G. F. Koster has shown that the operator II is givenby H =mo&pX, where Z is either the 4 && 4 or the 2 x 2

effective-mass Hamiltonian corresponding to ~+ or&b. The operator lI' is similarly obtained from X',which is associated with the overlapping elements ofthe 6 && 6 matrix.

E. O. Kane, J. Phys. Chem. Solids 1, 82 (1956).R. N. Dexter, H. J. Zeiger, and B. Lax, Phys.

Rev. 104, 637 (1956).4H. J. Hrostowski and R. H. Kaiser, J. Phys.

Chem. Solids 4, 148 (1958); S. Zwerdling, K. J. Button,and B. Lax (to be published).

A rough estimate of & —0.05 ev has been madefrom measurements of intervalence band transitionsby L. Huldt and T. Staflin, Phys. Rev. Letters 1,313 (1958).

Based on the data of M. Cardona, W. Paul, andH. Brooks, J. Phys. Chem. Solids 8, 204 (1959).

R. Braunstein, J. Phys. Chem. Solids 8, 280(1959).

S. Zwerdling, B. Lax, L. M. Both, and K. J.Button, Phys. Rev. 114, 80 (1959).

SR. Braunstein, A. R. Moore, and F. Herman,Phys. Rev. 109, 695 (1958).

A. Sagar, Westinghouse Research Report 6-40602-3P5, June, 1959 (unpublished) .

UPPER LIMIT FOR THE ANISOTROPY OF INERTIA FROM THE MOSSBAUER EFFECT

G. Cocconi~ and E. E. SalpeterCornell University, Ithaca, New York

(Received January 21, 1960)

The question of anisotropy of inertia has beendiscussed in some detail in a previous paper';here we recall only that one of the consequencesof Mach's principle could be that the value of theinertial mass M of a body depends on whetherits acceleration is in the direction towards thecenter of our galaxy or in a direction perpendi-cular to it. The variation ~ of mass withdirection, if it exists, was found to satisfy~/M &10 '.

The recent discovery of the Mossbauer effect'has opened the possibility of comparing frequen-cies of y rays, emitted by long-lived states ofnuclei, with extremely high precision. Therecoil-free resonant absorption of the 14-kevnuclear y ray of Fe' has already been investi-gated'&4 with sufficient resolution to observe thehyperfine structure splitting in the excited state.The Mossbauer effect has been proposed as an

excellent tool for detecting the gravitational redshift' and for many other purposes. Here wepropose its use in a search for anisotropy ofinertia. Precision studies of the line shapes ofthe Fe'7 transition, for instance, will serve thispurpose if the atomic spins can be aligned, bothin the emitter and in the absorber, and theorientation of the spins relative to the directiontowards the Galactic center is varied.

Consider an Fe" nucleus in a nuclear statewith total angular momentum quantum number Jin a ferromagnetic sample with atomic spinsaligned parallel to an external magnetic field.The effect of some anisotropy of inertia is thenvery similar to that discussed in reference 1 forthe atomic Zeeman effect, except that we arenow dealing with the motion of the nucleon in theFe" responsible for the y-ray transition ( atleast on a shell-model picture) instead of the

176