Physica 134B (1985) 179-183 North-Holland,Amsterdam 179

HOT HOLE MAGNETOPHONON RESONANCE IN P-InSb IN HIGH MAGNETIC FIELDS

Koji YAMADA and Noboru MIURA*

Saitama University, Shimo-ohkubo, Urawa, Saitama 338, Japan *Institute for Solid State Physics, University of Tokyo, Roppongi, minatoku, Tokyo 106, Japan

Hot hole magnetophonon resonance was investigated in p-lnSb at low temperatures 4.2-30 K in pulsea high magnetic fields up to 32 T. Two series of resonances for heavy holes were simultaneously observed corresponding to Landau level-acceptor state transition and inter-Landau level transition.

I. INTRODUCTION

The magnetophonon resonance is a useful means

for i nves t i ga t i ng e lec t ron band structures,

phonon states and electron-phonon interaction

mechanisms. Not much has been done on the

magnetophonon resonance in p-type materials in

comparison to n-type materials, because of the

complex valence band st ructures and the low

hole mobi l i t ies. The f i r s t observation of the

magnetophonon resonance in p-lnSb was done by

Amirkhanov et al. l More extensive work was made

by Eaves et al. 2 and Tsidi lkovski i and Yakunin 3.

Recenly, Shimomae et al. observed a magneto-

phonon o s c i l l a t i o n series associated wi th the

t ransi t ion of holes at l i gh t hole Landau levels

to acceptor states by the emission of LO phonons

in hot hole condition at 40 K. 4 In this paper,

we inves t iga te the magnetophonon resonance of

hot heavy holes in p-lnSb at low temperatures

supplying a large bias current in high magnetic

f ie lds up to 32 T. By vir tue of the hot carr ier

effect and excellent resolution in high f ields,

we can investigate the magnetophonon effect and

the band parameters at low la t t i ce temperatures.

2. EXPERIMENTAL PROCEDURES

The samples were cut out from a p-type InSb

crystal doped with Ge. The mobi l i ty p and the

carr ier concentration NA-N D of the samples were

= 3400 cm2/V.s, NA-N D = 5.68 x lO 14 cm -3 at T

= 90 K and u= 57000 cm2/V.s, NA-N D = 1.6 x 1014

cm -3 at T = 25 K where the m o b i l i t y takes the

maximum value. The l o n g i t u d i n a l magneto- res is -

tance was measured f o r magne t i c f i e l d d i r e c -

t i o n s p a r a l l e l to the <I00> and the < I I I > axes

in the t e m p e r a t u r e range between 4.2 and 30 K.

The pulsed magnetic f i e l d s were generated by a

copper w i re wound pulse magnet w i t h a r i se t ime

( T/4 ) o f about 5 ms. A b ias c u r r e n t was

s u p p l i e d to the sample by a p p l y i n g a pu lsed

e l e c t r i c f i e l d of 10-30 V/cm.

In order to observe a smal l o s c i l l a t o r y par t

superposed on a monotonous m a g n e t o r e s i s t a n c e ,

we employed a measur ing system as shown in

F ig. l . Using RC c i r c u i t s we d i f f e r e n t i a t e d the

s i g n a l f r o m t h e s a m p l e t w i c e . S i g n a l s

corresponding to the magnetoresistance R(t) ( the

s i g n a l V 0 in Fig. l ) , the f i r s t and the second

d e r i v a t i v e s of R(t) (V 1 and V2) and the magnetic

f i e l d change d B / d t were reco rded in a d i g i t a l

memory w i t h an acc racy of 12 b i t s . T y p i c a l

examples o f the reco rded da ta are shown in

F i g . 2 . In such a manner , we o b s e r v e d

o s c i l l a t o r y changes of R(t) c l e a r l y in the dR/dt

and d2R/d t 2 s i g n a l s . The p o s i t i o n o f each

ext remum in - d 2 R / d t 2 a p p r o x i m a t e l y c o i n c i d e s

w i t h t h a t in R(t) . However, such a doub le

d i f f e r e n t i a t i o n t e c h n i q u e u s u a l l y has a phase

s h i f t p rob lem due to i n c o m p l e t e d i f f e r e n t i a -

t ion . Therefore, we processed the recorded data

w i t h a computer to o b t a i n the r e a l s i g n a l o f

0378-4363/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

180 K. Yamacla and N. Miura / Hot hole magnetophonon resonance in p-InSb

v 0

CRYOSTAT

o

dB/dt

sAMpLE ~ "0

CPU ~ I

Figure 1 Block d iag ram of the measur ing c i r c u i t o f the magnetophonon resonance, VO, VI, V 2 and d B / d t are recorded in CPU.

4J

e <

~ l=dR/dt

10

T i m e t ( m s e c )

Figure 2 An example of the reco rded t r a c e s f o r dB /d t , Vo=R(t), V l : d R / d t , V2:dZR/d t z. The s t e p - w i s e change in R(t) near the s t a r t i n g po in t is caused by the bias vo l tage pulse app l ied to the sample.

d2R/dB 2 in t he f o l l o w i n g way. F i r s t , we

c a l c u l a t e d R(t) n u m e r i c a l l y from the V 2 s ignal

by s o l v i n g d i f f e r e n t i a l e q u a t i o n s f o r the RC

c i r c u i t s , The obtained r e s u l t is equ iva len t to

the s i g n a l V O, bu t c o n t a i n s more i n f o r m a t i o n

about the o s c i l l a t o r y par t because of the f i n i t e

r e s o l u t i o n of the A-D c o n v e r t e r . Then we

ca lcu la ted f ( t ) = dR/dB = ( d R ( t ) / d t ) / (dB/d t )

us ing the data f o r reco rded data of dB /d t .

F i n a l l y , d2R/dB 2 was o b t a i n e d by c a l c u l a t i n g

( d f / d t ) / ( d B / d t ) , The c a l c u l a t e d f i n a l r e s u l t

fo r d2R/dB 2 is considered to be f a i r l y accurate,

because the two t r a c e s o b t a i n e d f o r the up-

r i s i n g and d o w n - f a l l i n g magnetic f i e l d s almost

co inc ide showing no phase sh i f t .

RESULTS AND DISCUSSION

At temperatures below 40 K, no magnetophonon

o s c i l l a t i o n s were observed when the bias cur rent

was smal l . Increasing the bias cu r ren t to ra ise

the hole temperature, w e l l - d e f i n e d o s c i l l a t i o n s

were observed in the d2R/d t 2 s i g n a l . To avo id

Jou le h e a t i n g o f the sample, the pu lsed b ias

v o l t a g e was a p p l i e d j u s t be fo re the magne t i c

f i e l d pu lse was s t a r t e d . At low t e m p e r a t u r e s

be low T = 6 K, the ho les f r e e z e out i n t o the

acceptor s tates, and the sample r e s i s t i v i t y was

as h igh as 2 × 103 ~.cm a t B = O. By a p p l y i n g a

large bias vo l tage pulse, the sample r e s i s t i v i t y

dropped by a f a c t o r of about 400 due to impac t

i on i za t i on . The measurement was ca r r i ed out fo r

thus exc i ted holes. At temperatures above lO K,

the sample r e s i s t i v i t y change by the b ias

vo l tage was smal l .

The o b t a i n e d r e s u l t s f o r -d2R/dB 2 are

p l o t t ed in Fig.3 as a func t ion of magnetic f i e l d

p a r a l l e l t o t he <lO0> a x i s f o r v a r i o u s

t e m p e r a t u r e s . The maxima in -d2R/dB 2 are

c o n s i d e r e d to co r respond to the magnetophonon

resonance peaks. There is a s l i g h t t e m p e r a t u r e

dependence of the peak pos i t ions . They s h i f t to

h igher f i e l d by about l , l Z fo r the temperature

change f rom 4.2 K to 28 K, As the t e m p e r a t u r e

K. Yamada and N. Miura / Hot hole magnetophonon resonance in p-lnSb 181

ffJ

('.4 4 ~

c'4

I

I I 1 1 I l I I I I I 1

7

i00>

28

i I I [ I l I I t I I t i I

0 5 10 15 20 25 30 35

B (T)

Figure 3 -d2R/dB 2 v,s, magne t i c f i e l d curves oba ined by the data processing w i t h a computer fo r Bff <I00> at var ious temperatures,

(~ 4~ "o

"0 I

I i I I i i i i i i 1 i i 1 1 I

i00 >

I I I ! I I I I I t I I I

10 20 30

100 B(T)

Figure 4 -d2R/dB 2 v ,s . I /B fo r B//<IO0>

,]::[ C~

I

1 I I I

T=4.2 K

B// < I l l >

,I I I I 5 10 15 20 25

i00 B-TTT

Table 1 F i e l d p o s i t i o n s o f the r e s i s t a n c e maxima, In the r i g h t columns the assignment of the peaks in the a- and b -se r ies is shown w i t h the harmonic numbers,

B / / < I O 0 > B / / < I I I >

B(T) IO0/B(T) B(T) IO0/B(T)

27.7 3,6 20,0 5.0 a2 25.0 4.0 b3 18.5 5.4 b5 23.5 4.25 15.2 6.6 b6 22,0 4.55 14,9 6,7 a3 19.6 5.1 b4 13,2 7,6 b7 16,4 6,1 a2 11,9 8.4 a4 15.4 6,5 b5 11.4 8,8 b8 12.8 7.8 b6 9.9 I0. I b9 11.8 8.5 a3 9.5 10,5 a5 I I . 0 9,1 b7 9.1 I I . 0 blO 9,3 10,8 a4 8,0 12.5 a6 7,6 13,1 a5 6,5 15.4 a6 5.6 17.9 a7 4,9 20.4 a8 4,4 23,0 a9

Figure 5 -d2R/dB 2 v .s . I / 8 f o r B / / < l l l >

182 K. Yarnada and N. Miura / Hot hole magnetophonon resonance in p-lnSb

25

20

15

lO

I I F' I I I I I I

p-InSb / l=h.2 K / "

. , I i

B#<loo> -

/-a~B//<lll>

+'z~'f I I I I I I I I 1 2 3 4 5 6 7 8 9 l@

N

Figure 6 The inverse of the resonant f i e l d pos i t i ons I /B v .s . the hamonic number N.

was lowered, much f i n e s t r u c t u r e was observed in

a d d i t i o n to the main o s c i l l a t i o n s e r i e s . In

o r d e r t o u n d e r s t a n d t he p e r i o d i c o s c i l l a t o r y

behav iour more c l e a r l y -d2R/dB 2 is p l o t t e d as

a f u n c t i o n o f I / B in F i g . 4 and F i g . 5 f o r

B / /< IO0> and B / / < I I I > axes, r e s p e c t i v e l y . As

can be seen in t he f i g u r e s , the o s c i l l a t i o n is

a l m o s t p e r i o d i c w i t h r e s p e c t t o I / B f o r bo th

f i e l d d i r e c t i o n s . The p o s i t i o n s o f t he r e s i s -

t i v i t y maxima are l i s t e d in Table I.

I t was found t ha t the peaks can be c l a s s i -

f i e d i n t o two s e r i e s f o r each f i e l d d i r e c t i o n .

The v a l u e s o f I / B a re p l o t t e d as a ha rmon i c

number N i n F ig .6 f o r t he two s e r i e s f o r bo th

B / / < I O 0 > and B / / < I I I > . The o s c i l l a t i o n

a m p l i t u d e f o r the p r i n c i p a l s e r i e s a i s much

l a rge r than t h a t f o r the se r ies b. The I /B vs N

l i n e s a re a l m o s t l i n e a r f o r the b - s e r i e s , bu t

t h e y show a s l i g h t bend ing upward f o r t he a-

s e r i e s . When the l i n e s f o r the b - s e r i e s a re

e x r a p o l a t e d to N = O, t h e y c ross the o r i g i n o f

t he graph. The l i n e s f o r the a - s e r i e s , on the

o t h e r hand, c ross t he a b s c i s s a a t abou t N = -

I / 2 . The g r a d i e n t o f t he l i n e f o r the a - s e r i e s

i s abou t 1.7 t i m e s l a r g e r than t h a t f o r the b-

se r ies f o r each f i e l d d i r e c t i o n , These r e s u l t s

sugges t t h a t t he a - o s c i l l a t i o n s e r i e s a r i s e s

f rom ho le t r a n s i t i o n s f rom Landau l e v e l s t o

acceptor s ta tes obeying the f o l l o w i n g cond i t i on ,

-~FwLO = ( N + I / 2 ) I ~ c + Eb(B), ( I )

whereas the b - o s c i l l a t i o n se r ies o r i g i n a t e s from

i n t e r - L a n d a u l e v e l t r a n s i t i o n s w i t h t h e

cond i t i on

-FE~LO = N1~wc, (2)

where Eb(B) i s t he m a g n e t i c f i e l d dependen t

i o n i z a t i o n e n e r g y o f t h e a c c e p t o r s t a t e s .

Usual ly , in the magnetophonon resonance of hot

c a r r i e r s , the in te r -Landau l eve l t r a n s i t i o n s are

no t obse rved , bu t i t is made o b s e r v a b l e by the

h igh m a g n e t i c f i e l d s emp loyed in the p r e s e n t

e x p e r i m e n t . Using t he va lues~wLO = 24.3 meV

and Eb(O ) = 9.8 meV, 5 we can o b t a i n t h e

e f f e c t i v e masses o f ho les m ~ = 0.34 m 0 f o r

B / /< IO0> and m* = 0.43 m 0 f o r B / / < I I I > a t 4.2 K.

The v a l u e s o b t a i n e d f o r t he two s e r i e s a re in

good ag reemen t w i t h each o t h e r , i f we n e g l e c t

the s m a l l bend ing a t h igh f i e l d s . These v a l u e s

are f a i r l y c lose to but s l i g h t l y d i f f e r e n t from

the prev ious data I -3 ,

The Landau l e v e l s in the v a l e n c e band is

r a t he r compl ica ted due to the k H e f f e c t and the

non-un i fo rm spacing f o r low quantum numbers. 6'7

Because of the la rge app l i ed e l e c t r i c f i e l d , the

ho l e t e m p e r a t u r e i s c o n s i d e r e d t o be f a i r l y

high, In fac t , from m o b i l i t y measurements i t is

e s t i m a t e d t o be abou t 30 K. A c c o r d i n g l y , t he

heavy hole Landau l eve l s are regarded as a lmost

u n i f o r m l y spaced f o r the observed in te r -Landau

l e v e l ( b - s e r i e s ) magnetophonon peaks. The non-

p a r a b o l i c i t y o f the heavy hole Landau l eve l s is

K. Yamada and N. Miura / H o t hole magnetophonon resonance in p-InSb 183

sma l l in InSb. For the a - s e r i e s peaks, the ho le

t r a n s i t i o n takes p lace near k H = 0 because o f

the predominance o f t h i s component in the

a c c e p t o r wave f u n c t i o n , so t h a t the k H e f f e c t

should be s m a l l . These f a c t s make the magneto-

phonon resonance s t r u c t u r e r a t h e r s i m p l e in

sp i te of the compl ica ted valence band s t ruc tu re .

Moreover , the change in the a c c e p t o r b i n d i n g

energy by magnetic f i e l d is qu i te smal l , due to

the l a r g e Eb(O ) and l a r g e e f f e c t i v e masses. I t

is es t imated to be 1 meV at 30 T, and is in good

agreement w i t h the exper imenta l observat ion in

the bending o f the I / B vs N l i n e .

In c o n c l u s i o n , the p r e s e n t e x p e r i m e n t

a l l o w e d the s i m u l t a n e o u s o b s e r v a t i o n o f both

in ter -Landau leve l t r a n s i t i o n s and Landau leve l

to acceptor t r a n s i t i o n fo r the f i r s t t ime owing

to the e x c e l l e n t r e s o l u t i o n in h igh magne t i c

f i e l d s .

REFERENCES

I. Kh.I.Amirkhanov, R.I.Bashirov and Z.A.

Ismailov, Sov. Phys. Smicond.2 (1968) 356.

2. L. Eaves, R.A. Hoult, R.A. Stradling, S.

Askenazy, R.Barbaste, G.Carrere, J.Leotin,

J.C.Portal and-J.P.Ulmet, J.Phys. C IO (1977)

2831.

3. I .M.Tsidi l 'kovski i and M.V.Yakunin, Sov.Phys.

Semicond. 13 (1979) 178.

4. K.Shimomae, K.Senda, K.Kasai and C.Hamaguchi,

J. Phys.Soc.Jpn. 4__99 (1980) IO6O.

5. R. Sharan and E.L.Heasell, J.Phys.Chem.Solids

3__!I(1970) 541

6. C.R.Pidgeon and R.N.Brown, Phys.Rev. 146

(1960) 575.

7. J.C.Hensel and K. Suzuki , Phys.Rev. B9 (1974)

4219.

APPENDIX: Additional figure.

.Q

c~

(-q

~o

'I 3

.i b b a b b a b a a a '[. 3 ~ 4 2 s 6 3 7 4 54, 6¢

2~#5 b a , b 5 b \ /a R//<Inn

lJ 6W3 b h # 9 ¢ Iu '~, o

r i I r I I J I I h, I

4 5 6 7 8 9 10 11 12 13 14 15 16

100 B(T)

FIGURE 7 Expansions of Fig. 4 and Fig. 5 at the small part of IO0/B. very small to identify as the exact locations of them.

Note that some of the peaks are