NON MAXWELLIAN ELECTRON DISTRIBUTION FUNCTION IN GaAs

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NON MAXWELLIAN ELECTRON DISTRIBUTION FUNCTION IN GaAs

G. Lindemann, E. Gornik

To cite this version:

G. Lindemann, E. Gornik. NON MAXWELLIAN ELECTRON DISTRIBUTION FUNCTION IN GaAs. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-399-C7-405. �10.1051/jphyscol:1981749�.

�jpa-00221686�

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-399

NON MAXWELLIAN ELECTRON D I S T R I B U T I O N FUNCTION IN GaAs

G. Lindemann and E. Gornik

Institut fttr Expexn-mentalphysik, Universitat Innsbruek, Austria

Résumé : La distribution des électrons chauds est étudiée dans 1' AsGa de type n d'une qualité très pure. Les électrons sont stimulés par des champs électriques continus dans des champs magnétiques.

L'émission cyclotron, qui en résulte, est analysée par des détecteurs photoconducteurs d'une résolution de 0.25 cm

. En configuration E//B les températures des électrons, distribués entre des niveaux de Landau, s'élèvent extrêmement jusqu'à 180 K. La distribution des électrons devrait dévier considérablement d'une loi maxwellienne. La bande d'électrons diffère d'une parabole plus que prédit théori- quement.

Abstract: Hot electron distribution functions in a magnetic field are investigated in high purity n-type GaAs. The electrons are excited by d.c. electric fields and the resulting Cyclotron emissi- on is analysed with narrowband photoconductive detectors with a resolution of 0.2 5cm~1. Extremely high interoand electron tempera- tures up to 180K are observed in the E//B configeration indicating strong deviations from a Maxwellian type distribution function. The nonparabolicity of the conduction band is found to be considerably

stronger than theoretically predicted.

1. Introduction:

The electron system in a semiconductor at very low temperatures is no longer in thermal equilibrium with the lattice when a moderate elec- tric field (some V/cm) is applied. In this case the electron distri- bution is described in terms of a temperature T

higher than the lattice temperature /1,2/. This hot electron concept has been treated in various ways with special emphasis on n-type InSb. One approach involving electric field (E) modulated cyclotron resonance, was persued by the group of Otsuka /3,4,5/. The electric field dependent far infrared transmission was used to determine hot electron distri- bution functions in a quantizing magnetic field (B). The intensity and lineshape of the cyclotron resonance lines was analysed. Two

types of electron temperatures (inter- and intrasubband) were found necessary to describe the observed data. Deviations from a Maxwellian type distribution were found in the manner pedicted by Yamada and Kurosawa (YK) /6/ and Kotera 111 with stronger deviations for the E//B, than for the EiB case. However the observed T values

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981749

C7-400 JOURNAL DE PHYSIQUE

(Te(max) * 50X) are considerably lower than the calculated (Te (calcula- ted)

100-200K) ones.

A second approach to the hot electron problem uses the analysis of cyclotron emission spectra /8-10/. This technique,which has previously also been applied to InSb yields information on the distribution of electrons in higher Landau levels. Xiiller et a1 /9/ have performed detailed experimental studies for the case E I B and found the electron heating to depend on the free carrier concentration of the sample.

Part1 et a1 /10/ could show that this effect is due to the role of electron-electron scattering in combination with optical phonon emission. They included electron-electron scatterinq and Landau level broadeninq into the theory of YK /7/ and improved significantly the agreement between experimental and theoretical Te values. The deviati- on of the distribution

function from a Maxwellian type is found to be small

K

due to the randomizing o n effect of the electron-

(U

electron scattering in E

n-type InSb.

4.

'

The present work is an ex- tension of the emission

technique to n-type GaAs . ^cl

GaAs in available in con- siderably higher purity

than InSb. In the absence Magnetic F l e l d ET1 of significant electron-

Fig.1: Emission spectra as a function of electron scattering strong

maqnetic field for electric fields of 40 deviation from Maxwellian and 300 V/cm. The detector response is

set at 60,4cm-1 and at 83,5cmW1. The type are ex- correspondinq transitions are labeled.

pected. In this paper we

will show for the first time that electron temperatures between 100K and 200K can be achieved in high purity GaAs in agreement with the predictions of YK / 7 / .

11. Experimental results

Investigations of electrically excited recombination radiation between

Landau levels were carried out in the magnetic field range between 3.0

and 8.0 T at.liquid helium temperatures. The experimental systemused

has been described in detail previously / g / . It consists of two super-

conducting magnets-positioned in a distance of 26cm. One holds the

emitter sample the other the photoconductive detector. A high purity

r a d i a t i o n . The narrowband r e s p o n s e ( l i n e w i d t h 0.25cm-1) o f t h e l$-2p ( m = + l ) t r a n s i t i o n i s used f o r s p e c t r a l a n a l y s i s between 40 and 1 0 0 c m - ~ ; t h e broadband r e s p o n s e due t o t h e Is-continuum t r a n s i t i o n s a t B=O f o r measurements of t h e i n t e g r a l i n t e n s i t y .

Two d i f f e r e n t GaAs samples a r e i n v e s t i g a t e d : sample R137 w i t h a f r e e c a r r i e r c o n c e n t r a t i o n o f n0=2. 10' 3cm-3 and a t o t a l i o n i z e d i m p u r i t y c o n c e n t r a t i o n N I = 8 . 1 o1 3cm-3 and sample L83 w i t h no=l ,2.101 4cm-3 and

~ ~ = 2 . 1 0 ~ ~ c m - ~ .

Emisssion s p e c t r a a s a f u n c t i o n o f t h e e m i t t e r magnetic f i e l d a r e shown i n Fig.1 f o r two e l e c t r i c f i e l d s i n t h e E / / B c o n f i g u r a t i o n . The d e t e c t o r i s s e t w i t h a c o n s t a n t magnetic f i e l d t o a c e r t a i n f r e q u e n c y . Resonant peaks o c c u r i n t h e s p e c t r a when,the e m i s s i o n l i n e c o i n c i d e s w i t h t h e d e t e c t o r l i n e . A s t h e e m i t t e r i s t u n e d , h i g h e r e n e r g i e s a p p e a r on t h e low magnetic f i e l d s i d e , lower a t t h e h i g h f i e l d s i d e . The s p e c t r a f o r t h e lower emissi-on f r e q u e n c y c o n s i s t s of up t o 4 l i n e s , f o r t h e h i g h e r o f two l i n e s .

From t h e i n c r e a s e i n i n t e n s i t y o f t h e a d d i t i o n a l l i n e s w i t h i n c r e a s i n g e l e c t r i c f i e l d s we c o n c l u d e , t h a t t h e s e l i n e s a r e due t o t r a n s i t i o n s b e t w e e n h i g h e r Landau l e v e l s . I n t h e s p e c t r a f o r t h e lower magnetic f i e l d t h e i n t e n s i t y due t o e m i s s i o n from t h e N=2 Landau l e v e l i s domi- n a n t . The s e p a r a t i o n o f t h e e m i s s i o n spectrum i n s e v e r a l i n d i v i d u a l l i n e s r e v e a l s a s t r o n g e f f e c t of n o n p a r a b o l i c i t y . To d e m o n s t r a t e t h e e f f e c t of n o n p a r a b o l i c i t y we have p l o t t e d i n F i g . 2 t h e energy of h i g h e r Landau l e v e l t r a n s i t i o n s

w i t h r e s p e c t t o t h e l o w e s t t r a n s i t i o n (N=l t o N = O ) ,

which i s simply t h e appea-

r e n t s p l i t t i n g from t h e f i r s t l i n e i n cm-l . ^The

0 )

d e v i a t i o n from a p a r a b o l i c +

U

2

b e h a v i o u r amounts up t o

cl 5% f o r t h e N=3 t o N=2

X t r a n s i t i o n . The f u l l m 1

L

c u r v e s i n F i g . 2 show c a l -

c c u l a t e d energy d i f f e r e n c e s W

between Landau l e v e l s u s i n g t h e f o r m u l a r g i v e n

0 2 4

6 8 10

M a g n e t i c F i e l d C T I

by et 2'' F i g . 2: Energy d i f f e r e n z (cm-l) o f Landau

The c a l c u l a t e d s p l i t t i n s s t r a n s i t i o n s 2-1 and 3-2 i n r e s p e c t t o 1-0.

a r e s i g n i f i c a n t l y s m a l l e r , The f u l l c u r v e s a r e t h e o r e t i c a l p l o t s a c c o r d i n s t o / l 2 / .

however t h e tendency i s if f e r e n i symbols r e p r e s e n t d a t a from

d i f f e r e n t samples.

C7-402 JOURNAL DE PHYSIQUE

well described. At present we do not have an explanation for the ob- served nonparabolicity.

111. Analysis and discussions

The emitted radiation is due to transitions between different Landau levels. Since the spin splitting in GaAs is very small /12/ the ob- served lines correspond to transitions from electrons with increasing quantum number: line 1 from N=l, line 2 from M = 2 and so on. The ob- served emission line intensities reflect directly the population in the initial Landau levels. The appearent strong nonparabolicity of the conductionband of GaAs provides the basis for our data analysis: Since each emission line appears at a different energy (or magnetic field in the spectra) we can determine the originating Landau level of the elec- trons and the distribution function in a wide energy range. However the density of states peaks at the Landau level edges so that we track predominantly the population at the bottom of each level.

Electron temperatures can be defined in two ways: one is the inter- subband temperature defined by the relative population of Landau sub- bands. This temperature is determined from the ratio of the individual intensity peaks. The other is the intrasubband temperature defined through the distribution within each subband. Our present analysis is restricted to the calculation of intersubband temperatures. Informa- tion on intraband temperatures could be obtained from a linewidth ana- lysis of the individual lines / S / . This is not possible from our ex- periments since the separation between the lines is comparable to their width.

The emission intensity from Landau level N is given by IN=C.Y.n~ where C is a constant (if B, E are constant) and nN is the population of the level N / 3 / . The interband electron temperature TN,~-l is defined through the relation: nN=nN-1 exp(-hc+/kBT~,~-l) where wc is the cyc- lotron frequency and n N m l the population in the (N-l) level. Combining the two relations we obtain for

Interband temperatures T2,1 for sample R137 as derived from emission spectra at several magnetic fields are shown in Fig.3 as a function of the applied electric field for the E//B configuration.

There are two ranges of emission frequencies which show the same be-

haviour: For frequencies between 40 and 65cm-' (corresponding to mag-

netic fields between 3.5 and 5 T) the electron heating is independent

of magnetic field. Extremely high temperatures between 75K and 180K

i s r e d u c e d , b u t a g a i n n e a r l y independent of magnetic f i e l d .

*

I n t h e p i c t u r e of t h e YK 2

t h e o r y / 6 / f o r t h e E / / B

a,

c o n f i g u r a t i o n t h e e l e c t r o n

2 150 h e a t i n g i s governed by an m i n t e r p l a y o f energy g a i n

i n t h e e l e c t r i c f i e l d and a c o o l i n g due t o o p t i c a l t- phonon e m i s s i o n . The mag-

0

n e t i c f i e l d o n l y i n f l u e n c e s : ⁵⁰

t h e s c a t t e r i n g r a t e f o r

E//B which c o u l d r e d u c e t h e C

e n e r g y gain. However t h i s 0 -

e f f e c t seems n o t t o be s i g -

E l e c t r i c F i e l d C V / c m l n i f f c a n t i n GaAs, s i n c e

Fig.3: I n t e r b a n d t e m p e r a t u r e T2,1 a s a the heating is ^in- _-ion of e l e c t r i c f i e l d f o r sample R137 dependenL of B. The en- f o r s e v e r a l e m i t t e r magnetic f i e l d s (T) : hanced e l e c t r o n c o o l i n g D ( 3 1 6 8 ) r V ( 4 t 1 3 ) + ( 4 , 7 4 )

A ( 5 , 1 8 )

* ( 5 r 6 6 )

0 ( 6 , 1 4 ) .

e f f e c t . I t i s i n t e r e s t i n g 3 ¹⁵⁰

t o n o t e , t h a t t h i s enhan- 1x1 above a c e r t a i n magnetic

f i e l d s u g g e s t s t h e e x i s t e n - 2 200 c e o f a r e s o n a n t c o o l i n g

a,

t e m p e r a t u r e s a r e lower E l e c t r i c F i e l d CV/cm]

t h a n f o r t h e E / / B c a s e Fig.4: T 2 , , a s a f u n c t i o n of e l e c t r i c f o r a l l f r e q u e n c i e s . No f i e l d f o r sample R1 37 (open symbo1s)and

sample L83 ( f u l l symbols). The magnetic d i f f e r e n c e i n t h e hea- f i e l d s of t h e symbols a r e t h e same a s i n t i n g i s o b s e r v e d when t h e F i g . 3 .

E L B

c e d c o o l i n g c o i n c i d e s w i t h

t h e N=4 Landau l e v e l pas- n ₁₀₀ s i n g t h r o u g h t h e o p t i c a l t-

phonon energy.

0

I n t e r b a n d t e m p e r a t u r e s T2

1 :

⁵⁰

f o r t h e E I B c o n f i g u r a t i o n a,

a r e shown i n F i g . 4 f o r W

sample c o n c e n t r a t i o n i s

i n c r e a s e d from 2,O. 1 01 t o 1,4.1 o1 4cm-3 i n d i c a t i n g no s i g n i f i c a n t

- T

0 -

19 8

+ ^!+++

A&4+. • 6 l

A

..

-

I L

i n f l u e n c e o f t h e e l e c t r o n - e l e c t r o n - s c a t t e r i n g i n c o n t r a s t t o t h e r e -

sample R137 and L83. The

': l B

s u l t s i n InSb / 4 / . A s y s t e m a t i c d e c r e a s e i n h e a t i n g w i t h i n c r e a s i n g

magnetic f i e l d i s o b s e r v e d . The o b s e r v e d b e h a v i o u r might be due t o an

C7-404 JOURNAL DE PHYSIQUE

enhancement o f a a c o u s t i c phonon e m i s s i o n w i t h i n c r e a s i n g magnetic f i e l d . To o b t a i n a rough p i c t u r e o f t h e d i s t r i b u t i o n f u n c t i o n , t h e knowledge o f T I , O i s n e c e s s a r y . T I , O can b e d e r i v e d from t h e e l e c t r i c and magnetic f i e l d depen-

dence of t h e i n t e g r a l i n - t e n s i t y a c c o r d i n g t o t h e approach o f M u l l e r e t a 1

/ g / . W e have u s e d t h i s - ₈

t e c h n i q u e t o d e t e r m i n e

+

T I , ~ i n t h e f r e q u e n c y _W - ^\

r a n g e between 50cm-I and

70cm-I u s i n g a GaAs de- +

t e c t o r a t z e r o magnetic f i e l d . The i n t e r b a n d tem- p e r a t u r e s T l , O o b t a i n e d a r e

c o n s i d e r a b l y lower t h a n t h e t e m p e r a t u r e s between h i g h e r l e v e l s .

Fig.5: Normalized e l e c t r o n d i s t r i b u t i o n A 'lot

distribution f u n c t i o n o v e r e l e c t r o n energy f o r BG4.74T f u n c t i o n f o r a e l e c t r i c and B=5,66Tas o b t a i n e d from t h e a n a l y s i s . field of 150 V/cm and two The p o s i t i o n of t h e Landau l e v e l s and t h e

o p t i c a l phonon e n e r g y a r e marked.

d i f f e r e n t magnetic f i e l d s

i s shown i n F i g . 5 f o r t h e E//B c o n f i g u r a t i o n . Our d a t a y i e l d o n l y i n - f o r m a t i o n a b o u t i n t e r l e v e l t e r n p e r a t u r e s , ~ ~ t h a t we can o n l y draw s t r a i g h t l i n e s between t h e l e v e l s . I n t h e f i g u r e T3,2 v a l u e s o b t a i n e d a r e a l s o i n c l u d e d . The r e s u l t i n g d i s t r i b u t i o n f u n c t i o n s a r e e x t r e m e l y non Maxwellian and r e v e a l c l e a r l y t h e r u n away e f f e c t i n t h e l o n g i t u - d i n a l c o n f i g e r a t i o n . The d r o p on t h e h i g h energy s i d e i s due t o o p t i c a l phonon e m i s s i o n . I n t h e E I B c a s e t h e form of t h e d i s t r i b u t i o n func- t i o n s i s s i m i l a r however t h e d e v i a t i o n s from a Maxwellian t y p e a r e s m a l l e r .

F u r t h e r e x p e r i m e n t s t o d e t e r m i n e i n t h e i n t r a Landau l e v e l t e m p e r a t u r e s from a l i n e w i d t h a n a l y s i s a r e i n p r o g r e s s .

Acknowledgment: T h i s work was sponsored by t h e Fonds z u r Forderung d e r w i s s e n s c h a f t l i c h e n Forschung, AUSTRIA ( P r o j e k t S22/05).

W e t h a n k A.Foglar, H.Edlinger and C - P a r i g g e r f o r t h e i r h e l p i n analy-

s i n g t h e d a t a .

/l/ E.M.Conwel1, Solid State Phys. (Ed. F-Seitz, D-Turnbull and H.Ehrenreich), Supplement 9, Academic Press (1967)

/2/ G.Bauer, Springer Tractsin llodern Phys. 74, (Ed. G-Hohler) ,

Springer Verlag (1 97 4)

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40, 819 (1979) -

/6/ E-Yamada and T.Kurosawa, J.Phys.Soc.Jap. 34, 603 (1973) /7/ N.Kotera, E.Yamada and K.F.Komatsubara, J.Phys.Chem.Solids 33,

1311 (1972)

/8/ E.Gornik, W.Muller, F-Kohl, Proc. 12 intern.Conf.Phys.of Semicon- ductors, Stuttgart (1974), (Ed.M.M.Pilkuhn) p.839, B.G.Teubner /9/ W.Muller, F.Koh1, S-Part1 and E.Gornik, Solid State Electronics

21, 235 (1978) -

/IO/H.Partl, W.Muller, F.Kohl and E-Gornik, J.Phys.C: Solid State Phys. 11, 1091 (1978)

/11/This detector was5 provided by Dr.E.Bauser (Max Planck Institut Stuttgart)

/12/G.E.Stillmann1 Semiconductors and Semimetals 2, (Willardson and Beerj , Academic Press (1 977)

/13/E.GornikI Optics and Laser Technology z, 121 (1975) and J. of Magnetism and Magnetic Materials 11, 39 (1979)

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Electronics 2, 1455 (1978)