treatment of the

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Solid State Cormnunlcatzons, Vol 54,No.7, pp.567-571, 1985 0038-1098/85 $3.00 + 00

Printed in Great Britain Pergamon Press Ltd.

F A R - I N F R A R E D T W O - P H O T O N I N T R A B A N D T R A N S I T I O N S IN n - l n S b P D i r n h o f e r , W. B o h m , * W. P r e t t l , a n d U. R o s s l e r

N a t u r w z s s e n s c h a f t l z c h e F a k u l t a t II - P h y s z k ,

U n z v e r s z t a t R e g e n s b u r g , D - 8 4 O O R e g e n s b u r g , F e d e r a l R e p u b l i k o f G e r m a n y

( R e c e i v e d F e b r u a r y 19, 1 9 8 5 b y M. C a r d o n a )

T w o - p h o t o n c y c l o t r o n r e s o n a n c e a n d t w o - p h o t o n s h a l l o w d o - n o r t r a n s i t i o n s h a v e b e e n e x c i t e d b y a h i g h p o w e r p u l s e d D 2 0 l a s e r e m i t t i n g A = 119 ~ m a n d 66 ~ m l a s e r l i n e s a t t h e s a m e t l m e . T r a n s i t i o n s i n v o l v i n g t h e a b s o r p t i o n o f t w o 119 ~ m p h o t o n s o r s i m u l t a n e o u s l y o n e 119 ~ m a n d o n e 66 ~ m p h o t o n w e r e o b s e r v e d . T w o - p h o t o n s e l e c t i o n r u l e s a r e d l s - c u s s e d b y a r i g o r o u s t r e a t m e n t o f t h e s y m m e t r y o f t h e f r e e e l e c t r o n i a m l l t o n l a n .

I. Introduction

The development of hlgh power far-infrared (FIR) molecular lasers initiated a growing number of nonlinear optlcal investigations in the long wavelength infrared spectral range Includ- ing the saturation of cyclotron resonance and shallow impurlty transltlons zn semlconductors [I-5] and second harmonic generation [6]. FIR two-photon transltlons between Is~2s shallow do- nor levels and two-photon cyclotron resonance were first observed in n-GaAs wlth the highly sensitive method of magneto-photoconductlvlty [7] In thls paper we report first experimen- tal results concerning two-photon intraband ab- sorption in n-lnSb and their interpretation by means of two-photon selection rules. Evidence for two-photon transitions stems from the good agreement of observed peaks zn magneto-photoconductlvlty wlth calculated energy separations and from the nonlinear power dependence of the slg- nals.

II. Experlmental Technlque

The measurements were carried out at liquid hellum temperature In a superconductlng magnet.

The sample was a 0.5 m m thick high purity n-lnS6 crystal of N D - N A = 9.1013 cm -3 effective donor concentration. It was mounted in an integratlng cavity with the [I00] crystallographic direction parallel to the magnetic field. The b e a m of a TEA-CO 2 laser pumped D20 laser emitting radiation of 66 ~m and I19 ~ m wavelengths at the same time was transmitted through a metallic wave guide which was terminated by a cone to concentrate the radiation on the sample. A small fraction of the power decoupled by a beam splitter was detected by a n-GaAs photoconductor of 2 ns response time to monitor the peak power. The GaAs detector was calibrated by a pyroeleetrlc energy meter which was assumed to be linear in the power range of the measurement. Photoconduetlvlty was measured by irradiating the power of both laser lines on the samples or by using the 1|9 ~ m llne only. In the latter case radlatzon of 66 ~m was eliminated by a cold KCI single crystal filter. The peak power of the total emission was about ]O k W and that of the I19 ~ m llne after filtering was ap- proximately 2 kW The power was varied by an ap- erture stop of variable diameter reducing the power of both laser lines in the same way.

* p r e s e n t address Siemens AG, Central Research and Development, D-8OOO Munchen 83, Federal Republic of Germany

567

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568 F A R - I N F R A R E D T W O - P H O T O N I N T R A B A N D T R A N S I & I O N S IN n - l n S b Vol 54, No 7 III. R e s u l t s

In Fig. I the m e a s u r e d p h o t o c o n d u c t i v e signal e x c i t e d b y b o t h w a v e l e n g t h s zs p l o t t e d as a f u n c t i o n of the m a g n e t l e f i e l d s t r e n g t h B for v a r i o u s z r r a d z a t l o n p e a k powers. In o r d e r to i d e n t i f y the o b s e r v e d p e a k s the m a g n e t i c f i e l d d e p e n d e n c e of the two l o w e s t L a n d a u l e v e l s for b o t h spin o r i e n t a t i o n s c a l c u l a t e d w i t h the T r e b l n - R o s s l e r - R a n v a u d m o d e l [8] and s h a l l o w d o - n o r t r a n s i t i o n e n e r g i e s as far as k n o w n f r o m i z C e r a t u r e [9] are I n c l u d e d zn Fig. I. P o s s i b l e one- a n d t w o - p h o t o n t r a n s i t i o n s c o i n c i d e w i t h o b s e r v e d s t r u c t u r e s zn p h o t o c o n d u c t z v l t y and are

cl 11 Z o

5

u') 0

S

7 "

13-

q

40

2O

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LIJ Z ILl

20

F i g

66 ÷119 IJ

1191J 2"1191J

3 4

ICIC2

66#

1 2 I

ICI L'2

'

/ A II I" o'

T:4 2, m .

O5

000%110-

2 3 4

MAGNETIC FIELD (Tesla)

(a) M a g n e t o - p h o t o c o n d u c t z v z t y of n - l n S b e x c i t e d b y the I = 119 ~ m and 66 p m lines of a D 2 0 laser. N u m b e r s o n the right side d e n o t e laser p o w e r zn r e l a - tive units, I c o r r e s p o n d s to a b o u t 10 k W (b) L a n d a u l e v e l s a f t e r [8]

(c) S h a l l o w d o n o r t r a n s i t i o n e n e r g i e s a f t e r [9] S i n g l e and d o u b l e a r r o w s zn (b) and (c) i n d i c a t e p o s s i b l e o n e - and t w o - p h o t o n t r a n s i t i o n s , r e s p e c t i v e l y

z n d l c a t e d b y arrows. The c y c l o t r o n r e s o n a n c e s + ] +

O ~ and O- ~ I- - c o r r e s p o n d i n g to (2-2-00) (3+3+11) and (]-I-11) ~ (2+2+20), r e s p e c t i v e l y , zn the full n o t a t i o n of ref. [8] - and the zm- p u r i t y t r a n s i t i o n s (OOO) + ~ (]|O) + glve rise to s t r o n g o n e - p h o t o n s i g n a l s a r o u n d B = ! 4 T and 2.6 T for % = I]9 ~ m and 66 ~m, r e s p e c t i v e l y The i n d i v i d u a l lines are not r e s o l v e d due to the rapid s a t u r a t i o n of o n e - p h o t o n s i g n a l s p a r t i c u - larly at low m a g n e t i c field s t r e n g t h for I = ]19 ~ m and b e c a u s e of the large t h i c k n e s s of the sample, w h i c h w a s c h o s e n to f a c i l i t a t e the d e t e c - tlon of w e a k s t r u c t u r e s zn p h o t o c o n d u c t z v z t y A b o v e a b o u t B = 3 T no o n e - p h o t o n t r a n s i t i o n is e x p e c t e d to o c c u r In this m a g n e t i c field range the signal d e p e n d s s u p e r l l n e a r l y on the e x c i t i n g p o w e r P e a k s | and 2 (Fig. la) agree well w l t h the t w o - p h o t o n c y c l o t r o n r e s o n a n c e t r a n s i t i o n s w i t h o u t

+ + - _

spln fllp O ~ I and O ~ I , r e s p e c t i v e l y , in- v o l v i n g two I]9 ~ m p h o t o n s A n i m p u r i t y t r a n s i t i o n

(119 p m and 66 ~m) m a y also c o n t r i b u t e to p e a k 2 The w e a k s t r u c t u r e d e n o t e d by 3 in F i g la just c o r r e s p o n d s to the m a g n e t i c field s t r e n g t h of

0 +

t w o - p h o t o n c y c l o t r o n r e s o n a n c e ~ I w l t h spln fllp g e n e r a t e d by s i m u l t a n e o u s l y a b s o r b i n g one 119 ~ m and one 66 ~m q u a n t u m

In the range of B b e t w e e n 3 3 T and 4 T (shaded zu Fig Ic) the p h o t o c o n d u c t i v e s i g n a l m u s t be g e n e r a t e d by two d i f f e r e n t p h o t o n s

(119 ~ m and 66 ~m) b e l n g due to t r a n s i t i o n s f r o m the d o n o r e n e r g y level series (OOB) + and (oT6) + b e l o w the lowest L a n d a u level to final s t a t e s in the series (lIB')- and (I16') + The i n v o l v e d do- n o r b o u n d state q u a n t u m n u m b e r B,~' c a n n o t be r e c o g n i z e d b e c a u s e of the low r e s o l u t i o n of the m e a s u r e m e n t r e s u l t i n g f r o m the large t h i c k n e s s of the sample

In Fig. 2 the p h o t o s z g n a l of the I = 119 ~ m line alone is s h o w n for v a r i o u s p e a k p o w e r s between B = 2.5 T and 3.5 T In this range of m a g - n e t i c fleld s t r e n g t h no r e s o n a n t o n e - p h o t o n t r a n s z t l o n s are p o s s i b l e w i t h r a d i a t i o n of thls w a v e l e n g t h The lines I and 2 are a g a i n I d e n t l -

+ +

fzed as t w o - p h o t o n c y c l o t r o n r e s o n a n c e s 0 ~ ] a n d O ~ I , r e s p e c t i v e l y . The peaks 5 and 6 m a y be a t t r i b u t e d to t w o - p h o t o n a b s o r p t i o n of s h a l l o w d o n o r s T h e s e t r a n s i t i o n s are h l d d e n in F i g 1 b y the s t r o n g 66 ~ m o n e - p h o t o n e x c i t a t i o n

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Vol 54, No. 7

A

_J

<

Z

oo O I- o_

FAR-INFRARED TWO-PHOTON IhTRABAND TRANSITIONS IN n-lnSb 569

0 T = L 2 K 5 119p

6 I 1 2

I I I

Fig 2.

I I I

25 30 35

MAGNETIC FIELD (T)

M a g n e t o - p h o t o c o n d u c t z v l t y measured for X = l]9 Bm Numbers on the right side denote irradiation power in relative unlts, ] corresponds to about 2 kW

IV Two-Photon Selection Rules

Two-photon transztlons require intermediate states belng connected by electric dlpole matrlx elements wlth the znltlal and final states of the transltlon Thus selection rules for two-photon transztlons between Landau levels (free states) or between impurity levels attached to Landau levels (bound states) can be obtalned from the corresponding one-photon selection rules. Dipole selection rules for transztlons between free states have been formulated rigorously on group- theoretical grounds [8]. The dlpole operator can be represented as the derivative of the Hamilton- tan wlth respect to the Landau operators a + (cyclotron resonance active polarization e+), a (e_) or the m o m e n t u m component E parallel to the magnetic field (e3). Therefore, as the H a m i l t o n z a n contains parts with axial, cublc or tetrahedral symmetry, dipole transitions become allowed w i t h the weight MI, i = O,.. ,5 of these terms. M ° (e±) type transitions follow from axial symmetry and correspond to a change in the L a n d a u quantum number by ~| M 2 can be classlfled as w a r p i n g in-

duced and M 4 as inversion asymmetry Induced transitions Whlle Mo, M 2 and M 4 denote transltzons at E = O, those of type M], M 3 and M5, respectively, refer to E # O transztlons. Fig. 3 shows possible two-photon transztlons for cyclotron and combined resonances It turns out that comblned two-photon resonance zs posslble only wlth two quanta of e± polarlzatzon, while two-photon cyclotron resonance is possible only, If one photon is e 3 polarlzed. The p r o b a b l h t y of type M 2 .M 5 transltzons owing to their cublc or tetrahedral symmetry should depend on the orientation of the magnetic field wlth respect to the crystallographic axes [IO] This could be shown in a properly designed experiment

Donor states in small gap semlconductors with magnetlc field have been descrlbed by Zawadzkl [|I] in the frame of Kane's band model The impurlty states, attached to a Landau level are classzfled by quantum numbers for the Landau level n, the angular m o m e n t u m component M, the bound state B, and spln s. One-photon transitions between impurity states are possible wlthout or wlth change of the internal quantum number B (see Fig 4) In the former case ( A B = O ) the transztlon connects different Landau levels, while in the latter case ( A B # O ) the transition takes place between different impurity levels attached to

the same Landau level. These transitions can be

1"

0-

0*

B II [100]

Ml(e-] Ml{e.)

M4(e , 'lA(e.)

~0(e. ~0(e.)

M5(e_', '.,IL(e_)

~(ej

M1(e-i

%%1 MsL ~

I

i

, L

M 4(~, M l(e_: I

IM M 1 (e÷) 4(e-)

MOI ~)

N~K~O p 2 ~ 2 ~ 2 0

3* 3* 1 1

1- 1- 1

2 - 2 - 0

Fig. 3 Two-photon cyclotron and combined resonance. Virtual transltlons are char- acterlzed by the welghts M I [8].

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570

classified as M ° and MI, respectively Transl- tlons of types M 2. .M 5 do not occur in this model, which does not consider terms of cubic or tetrahedral symmetry in the Hamiltonlan. Possible two- photon transitions between impurity states are shown in Fig 4

V Conclusions

In summary two-photon intraband transitions in the far infrared spectral range have been oh-

FAR-INFRAP~D TWO-PhOTON INTRABAND TRANSITIONS IN n-lnSb Vol 54, No 7 served in InSb. Two photon cyclotron resonance and combined resonance could clearly be zdentl- fled in agreement with two-photon selectlon rules In an improved experlmenta] arrangement e.g. using thinner samples and applying well defined po- larlzatlon configurations excited shallow impurity states not accessible to conventional one- photon spectroscopy might be resolved

W e thank H R. Trebln for valuable discussions

co I

-- 111- I" ~'-/'////~/ M ~

111"/"1

Mole.) Mo(e.)

o-

o. i

M,Ie3)T L....-~OOO.

40(e.) 110-

001- 000-

Fig 4

(b) As A M An A ~

e 3

0 0 0 _+I

M~

-I +I +I 0 M o

+I -I -I

0 M o

e+

e_

0 +I +I 0 M o

*I 0 0 _+I M I

0 -I -I 0 M o

-I 0 0 _+I M~

Two-photon shallow donor transitions de- duced from electric dipole selection rules after Zawadzkl [II]

References

[I] T. Murotanl and Y Nlslda, J Phys Soc Japan 32, 986 (1972)

[2] E Gornlk, T.Y. Chang, T J Bridges, V T Nguyen, I D, McGee, and W Muller, Phys Rev, Letters 40, I151 (1978)

[3] K Muro, N Yutanl, and Sh Narlta, J Phys Soc. Japan 49, 593 (1980)

[4] C R Ptdgeon, A Vass, G R. Allan, W Prettl, and L Eaves, Phys Rev Letters 50, ;309

(1983)

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Vol 54, No. 7

[5] W Prettl, A. Vass, G R. Allan, and C.R. Pldgeon, Int. J. Infrared and Mllll- meter Waves 4, 561 (1983).

[6] F Kellmann, 7th Int. Conf. on Infrared and M1111meter Waves, Marsellle ]983.

[7] W. Bohm, E. Ettllnger, and W. Prettl, Phys.

Rev. Letters 47, 1198 (1981)

[8] H.R. Trebln, U. Rossler, and R. Ranvaud, Phys. Rev. B 20, 686 (1979).

FAR-INFRARED TWO-PHOTON INTRABAND TRANSITIONS IN n-lnSb

[9] R. Kaplan in Lecture Notes in Physlcs,

[10]

[11]

571

vol. 133, p. ]38, ed. W. Zawadzkl (Sprlnger, Berlln - Heldelberg - N e w York, 1980).

M. Braun, U. Rossler, J. Phys. C, in prlnt.

W Zawadzkl in Theoretlcal Aspects and N e w Developments in Magneto-Optlcs, Nato Ad- vanced Study Instltutes Serles, P l e n u m Press

1980, p 347.