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HOT CARRIER POPULATION INVERSION IN p-Ge
S. Komiyama, R. Spies
To cite this version:
S. Komiyama, R. Spies. HOT CARRIER POPULATION INVERSION IN p-Ge. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-387-C7-392. �10.1051/jphyscol:1981747�. �jpa-00221684�
CoZZoque C7, suppZ6rnent au nOIO, Tome 42, octobre 1981 page C7-387
H O T CARRIER POPULATION INVERSION IN p-Ge S. Komiyama and R. Spies
I n s t i t u t fiir Angewandte Physik, UniversitZEt Hamburg, Jungiusstr. 11, 2000 Hamburg 36, FRG
Abstract.- Hall effect measurements a?% pe~formed at 4.2 K on hot carriers in p-Ge (p = 1.5 X 10 /cm ) . The data of the Hall mobility, the Hall angle, and the magnitude of current are compared with calculation to afford definite evidence for streaming motion and population inversion of light and heavy holes. The ratio of population inversion is estimated from an analysis of the results.
1. Introduction.- Streaming motion and population inversion of hot electrons as predicted by Maeda and ~urosawa' have been established in a series of galvanomagnetic experiments on silver halide~.~'~. Re- lating to the population inversion, further interesting phenomena are predicted theoretically. 4-7 Recently we have reported the observation of streaming motion and population inversion of hot carriers in p-Ge on the basis of Hall effect measurements.8 Here we present some addi- tional evidence for the phenomena including the data of mobility and current. From a simple analysis of results, ratio of population inver- sion is estimated and compared with a theoretical expectation.
2. Experimental Methods.- Specimens were cut out of p-Ge crystals with 1 4
acceptor concentration of p = 1.5 X 10 /cm3 into simple rectangular shape (12 X 3 X 0.4 mm 3 ) . Ohmic contacts were prepared by aloying with Au-In (2%) at the ends on the 3 X 12 mm surface so that the dissipa- 2 tive electric field Ex is developed along the 12 mm-long direction in the specimen. The specimen is immersed in liquid helium. Short voltage pulses with a duration in the range 20nsec to 40vsec are applied to the specimen at repetition rate of 30 H z by use of the discharge of either 5052 coaxial cables or LC networks. The principal field E, and the Hall electric field E inside the specimen are detected to an ac-
Y
curacy of 25% by applying a new capacitive probe technique.' Dif fe- rent pulse durations are applied for different voltages so that at each voltage the impact ionization of acceptors is completed within the pulse duration whereas the sample heating effect can be avoided.
The details of the experimental procedure are described in Ref.9.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981747
C7-388 JOURNAL DE PHYSIQUE 3. E x p e r i m e n t a l R e s u l t s . -
The H a l l m o b i l i t y vHr ( l / B ) X(E /E ) measured a t t h e l i m i t
Y X
of low magnetic f i e l d (B<0.03T) Eq. (1)
i s shown a s a f u n c t i o n o f e l e c - t r i c f i e l d Ex i n F i g . 1 . Data
p o i n t s b e g i n t o f a l l s t e e p l y above ~ ~ % 2 0 0 V r / c m c o n v e r g i n g t o sE T=4.2K 5x1014~c\
1
a s o l i d l i n e . The s o l i d l i n e i s I 9 1
drawn a c c o r d i n g t o t h e r a l a t i o n ?
10 I L
h - e 1 h l o1 1 03 - 4 10
U H - --g ( 3 Top) l ( 1 ) l0 (v/c*)
% Fig. l : Plot of t h e Hall mobility p =B-' (Ey/Ex)
H which i s e x p e c t e d when heavy E X '
h o l e s a r e i d e a l l y s t r e a m i n g . Here, m X =0.35 m. i s t h e e f f e c t i v e mass of
fi - 1
heavy h o l e s , e t h e u n i t c h a r g e and Top= (2%hhwop)T (,Ex)-' w i t h t h e op- t i c a l phonon e n e r g y hwOp = 3 7 mev. The good agreement between t h e d a t a p o i n t s and t h e c a l c u l a t e d l i n e s t r o n g l y s u p p o r t s a s i m p l e p i c t u r e of s t r e a m i n g motion of heavy h o l e s i n p-Ge.
E l e c t r i c f i e l d s Ex and E were measured a s a f u n c t i o n o f B a t d i f f e r e n t l e v e l s of a p p l i e d v o l t a g e . Tangent o f t h e H a l l a n g l e , tan9:E Y / E x , i s shown a s a f u n c t i o n o f B i n Pig.2. The v a l u e o f Ex a t B = 0 Y i s n o t e d f o r e a c h c u r v e . The f i e l d Ex was observed t o b e a l m o s t i n d e p e n d e n t of B. A t low Ex l e v e l s below Ex<300 V/cm, t a n 9 i n c r e a s e s smoothly w i t h i n c r e a s i n g B. However, a t h i g h e r f i e l d s above Ex%600 V/cm, where t h e s t r e a m i n g motion dominates t h e h o t c a r r i e r k i n e t i c s , t h e r e a p p e a r s t e p - l i k e s t r u c t u r e s i n t h e c u r v e . The p o s i t i o n of t h e s t e p - l i k e s t r u c t u r e s s h i f t towards h i g h e r magnetic f i e l d w i t h i n c r e a s i n g Ex. Arrows on e a c h c u r v e i n d i c a t e t h e B p o s i t i o n s a t which t h e r e l a t i o n ,
2 2 + i s s a t i s f i e d . H e r e , <l,h' -"loh op Ivy w i t h OP (2hoop/m; ,h)4 , $E ( E ~ + E Y ) / B and t h e e f f e c t i v e mass o f l i g h t h o l e s m~=0.043mo. A s s e e n
from t h e f i g u r e , t a n e b e g i n s t o r i s e s t e e p l y above t h e f i e l d s < h , l = l . We i n t e r p r e t e d 8 t h i s b e h a v i o u r o f t a n 9 above cl=l and ih=1 a s due t o p o p u l a t i o n i n v e r s i o n o f l i g h t and heavy h o l e s , r e s p e c t i v e l y . I n t h i s r e p o r t we w i l l a n a l y z e t h e phenomena i n some more d e t a i l t o o b t a i n c l o s e r u n d e r s t a n d i n g o f thephenomena. I n a d d i t i o n , we p r e s e n t h e r e a n o t h e r e v i d e n c e f o r c a r r i e r a c c u m u l a t i o n observed i n t h e c u r r e n t v s . B r e l a t i o n . The c u r r e n t j a t h i g h E x ' s i s shown a s a f u n c t i o n o f B i n F i g . 3 . I n t h e f i g u r e , t h e v e r t i c a l s c a l e i s g i v e n w i t h t h e u n i t o f d r i f t v e l o c i t y vd a s d e r i v e d from t h e v a l u e o f j by u s i n g v d = j / p e w i t h
Fig.2: Tangent of t h e H a l l a n g l e a s a f u n c t i o n o f B a t s e v e r a l l e v e l s o f E
X -
a s d e r i v e d from t h e magnitude of cur- r e n t and t h e t o t a l c a r r i e r d e n s i t y p , a s a f u n c t i o n o f B.
14 3
p = 1.5 X 10 /cm .With i n c r e a s i n g B a t e a c h E,, vd i n c r e a s e s a t f i r s t and t h e n d e c r e a s e s forming a peak i n the vd v s . B c u r v e . The peak s h i f t s towards h i g h e r B and g e t s b r o a d e r w i t h i n c r e a s i n g Ex. I n a d d i - t i o n , a s l i g h t s w e l l i n g i s n o t i c e d a t t?e lower magnetic f i e l d s i d e of t h e peak i n each c u r v e . The s w e l l i n g s h i f t s a l s o towards h i g h e r B w i t h i n c r e a s i n g Ex. A s seen from t h e arrows i n F i g . 3 , which i n d i c a t e t h e magnetic f i e l d s where t h e r e l a t i o n c l r h = l o r 2 i s s a t i s f i e d , t h e s l i g h t s w e l l i n g and t h e peak a r e r e s p e c t i v e l y l o c a t e d i n t h e B r a n g e s
1 h
of l < < < 2 and l < < <2. These f e a t u r e s i n t h e vd v s . B c u r v e a f f o r d an a d d i t i o n a l evidence f o r p o p u l a t i o n i n v e r s i o n a s d i s c u s s e d i n t h e f o l - lowing.
4 . Disqussion.- L i g h t h o l e s ake a l s o expected t o be s t r e a m i n g a t h i g h e l e c t r i c f i e l d s and B = 0 . The number of t h o s e l i g h t h o l e s i s t h e o r e - t i c a l l y expected t o be a1.5% of t h a t of heavy h o l e s . " Due t o t h e p r e s e n c e of l i g h t h o l e s , t h e H a l l m o b i l i t y i s e s t i m a t e d t o become l a r - g e r t h a n pH h b y - \ . 3 % . T h i s c o r r e c t i o n f a c t o r , however, i s t o o s m a l l t o be d i s c e r n e d i n t h e p r e s e n t experiment.
The a b s o l u t e v a l u e s of t a n 0 i n t h e low B range 5 1 c l can be q u a n t i t a t i - v e l y e x p l a i n e d by t h e scheme of s t r e a m i n g motion of heavy h o l e s . To d e m o n s t r a t e t h i s , a d o t t e d l i n e i s drawn f o r Ex=2.5 kV/cm i n Fig.2
C7-390 JOURNAL DE PHYSIQUE a c c o r d i n g t o t h e r e l a t i o n , 3
-2 -1 1 2 1
- 2 - ~ ) 2 l
t a n e h = 2ch c o s ( l - T < h ) - ( 4 < h ( 3 ) which i s d e r i v e d f o r i d e a l l y s t r e a m i n g heavy h o l e s on t h e assumption o f i s o t r o p i c e n e r g y band. The e x c e l l e n t agreement between t h e d a t a and t h e c a l c u l a t i o n i n t h e r a n g e c l < l l o b t a i n e d a l s o f o r d i f f e r e n t E:s above 300 V/cm,assures t h e s t r e a m i n g motion o f heavy h o l e s . A p o s s i b l e c o r - r e c t i o n f a c t o r t o t a n 9 a r r i s i n g from s t r e a m i n g l i g h t h o l e s i s e s t i - mated a g a i n t o be t o o s m a l l t o be d i s c e r n e d h e r e . The s t e e p r i s e of t a n 9 and t h e r e s u l t i n g d e v i a t i o n from t a n e h above cl%l i s a t t r i b u t e d t o t h e a d v e n t of accumulated l i g h t h o l e s i n a h i g h e n e r g y a r e a K' i n v e l o c i t y s p a c e a s shown i n Fig.4 ( a ) . Though s t r e a m i n g l i g h t h o l e s a l s o e x i s t i n t h e r a n g e l < < < 2 , t h e i r c o n t r i b u t i o n t o t a n 9 1 i s n o t s i g - n i f i c a n t . Then t h e second s t e e p r i s e o f t a n 9 above ch%l i s a t t r i b u t e d t o heavy h o l e a c c u m u l a t i o n i n t o a s i m i l a r a r e a Kh a s shown i n F i g . B ( b ) . C o n t r i b n t i o n o f l i g h t h o l e s i n K 1 t o t a n 9 i s e x p e c t e d r e l a t i v e l y i n - s i g n i f i c a n t i n t h i s r a n g e 5 > l s i n c e t h e y a r e s m a l l i n number a s com- p a r e d t o t h e heavy h o l e s inhKh due t o t h e s m a l l e r s t a t e d e n s i t y i n t h e l i g h t h o l e band. The a d v e n t o f accumulated c a r r i e r s a l s o a f f e c t s t h e d r i f t v e l o c i t y o f t h e c a r r i e r system. The s l i g h t s w e l l i n g i n t h e v
d v s . B c u r v e i n t h e range l < < < 2 i s due t o l i g h t h o l e accumulation i n t o
1 1
K where c a r r i e r s have a l a r g e d r i f t v e l o c i t y . The l a r g e r peak above ch%l i s due t o heavy h o l e accumulation i n t o K ~ . We s h o u l d n o t e t h a t t h e a m p l i t u d e of t h e peak a s measured from t h e v a l u e a t B = 0 i s s a - t u r a t e d above Ex = 770 V/cm. T h i s s u g g e s t s t h e s a t u r a t i o n i n number o f accumulated c a r r i e r s a g a i n s t t h e i n c r e a s e i n Ex above t h i s f i e l d . Andronov e t a l S 7 have r e c e n t l y made a Plonte C a r l o s i m u l a t i o n on t h e d r i f t v e l o c i t y of h o l e s i n p-Ge under t h e c o n d i t i o n of c a r r i e r accumu- l a t i o n . Here o b s e r v e d v a r i a t i o n i n vd i s i n q u a l i t a t i v e agreement w i t h
\
\
\
l I
. L \ ' "p
I
C E '
e e G r /
\'ix
heavy h o l e l i q h t h o l e heavy h o l e l i g h t h o l e
Fig.4: The t r a j e c t o r y o f streaming motion and t h e accumulation a r e a i n v e l o c i - t y space f o r heavy and l i g h t h o l e s . The v e l o c i t y v,, i n d i c a t e s t h e d i r e c t i o n of t h e t o t a l e l e c t r i c f i e l d IE. The p o i n t C denotes t h e c e n t e r o f l c y c l o t r o n o r b i t s , C (%=O, v I- - -V ) . a ) I n t h e range 1<c1<2, t h e accumulation a r e a K appears f o r l i g h t h o l e s whereag o n l y t h e s t r e a m i n g t r a 3 e c t o r y e x i s t s f o r heavy h o l e s . b ) I n t h e range
1<Ch<2, t h e s i m i l a r a r e a Kh appears a l s o f o r heavy h o l e s .
: 5,,=1.54
0.5 -
0 -
/ 0.01 - P
4 1 a05. O l
0.5 1 5 0.5 1 5
lIEl (kV/crn) IEl ( k V / c m )
a) b)
Fig.5: a ) The r a t i o of accumulated l i g h t h o l e s t o t h e streaming heavy h o l e s a t =2.
- 1
Open c i r c l e s denote e x p e r i m e n t a l l y e s t i m a t e d v a l u e s and t h e s o l i d l i n e i n d i c a t e s a t h e o r e t i c a l e x p e c t a t i o n . b) The r a t i o o f accumulated heavy h o l e s t o the streaming heavy h o l e s a t C = 1.54.
h
their result.
Here we estimate the ratio of accumulated carriers from the data of tane. ITost simply, the ratio is estimated from the difference between the observed value of tan0 and the calculated value of taneh in the form 3 ,
where pK is the number of accumulated carriers in K1 or Kh, P: the l,h
number of streaming heavy holes, taneh is defined by Eq. (3) , and ?Fh
the drift velocity of streaming heavy holes along the direction of the total electric field E. Results of estimation for the light and heavy + hole accumulations, respectively, at c1=2 and 5 =1.54, are shown in
K S . h
Fig.5(a) and (b). The ratio pllh/ph 1s saturated against the increase
of [ E \ above 1 kV/cm. This result is consistent with the expectation
from the data of vd. Let us compare the result here with a theoretical expectation. By assuming a simple mechanism for accumulation, we ob- tain' ' 3
where yl'h represents the probability that a heavy hole is scattered into the region KI or Kh after the optical phonon emission and
i:~E
C7-392 JOURNAL DE PHYSIQUE
i s t h e s c a t t e r i n g t i m e o f c a r r i e r s i n K' o r K h . Here, we t e n t a t i v e l y
1 * 3
assume y1 = a X (%)I = 0.01 f o r cl = 2 by c o n s i d e r i n g t h e volume
"'h 1
f a c t o r and t h e r e l a t i v e s t a t e d e n s i t y o f K . For < h = 1 . 5 4 , Kurosawa f 1
h 1
c a l c u l a t e d yh a s a f u n c t i o n o f E t o b e 0 . 0 5 * 0.08. Timp i s e s t i m a t e d t o be 9psec from t h e v a l u e of uH a t low Ex. By u s i n g t h e s e v a l u e s , Eq(5) i s e v a l u a t e d a s a f u n c t i o n o f I E ~ and shown by a s o l i d
l i n e i n F i g . 5 ( a ) and (b). A p p a r a n t l y , t h e e x p e r i m e n t a l l y e s t i m a t e d r a t i o of a c c u m u l a t i o n i-S lower t h a n t h e t h e o r e t i c a l l y e x p e c t e d one a t h i g h e r
K S
E. Though t h e e x p e r i m e n t a l l y e s t i m a t e d v a l u e o f pl,h/ph i s n o t v e r y a c c u r a t e due t o t h e n e g l e c t i o n o f a n i s o t r o p y i n t h e heavy h o l e band, t h e s a t u r a t i o n o f pl,h/PE K a t a h i g h / E l r a n g e can h a r d l y be a t t r i - b u t e d t o t h e i n a c c u r a c y i n t h e e s t i m a t i o n p r o c e d u r e . The s a t u r a t i o n i n pl,h/pg K s u g g e s t s t h a t t h e l i f e t i m e o f c a r r i e r s i n t h e r e g i o n K' o r
i s l i m i t e d n o t o n l y by i m p u r i t y s c a t t e r i n g b u t a l s o , t h r o u g h a s o f a r unknown mechanism, by I E l ( o r B ) .
A s a summary, s t r o n g e v i d e n c e h a s been p r e s e n t e d f o r s t r e a m i n g motion and p o p u l a t i o n i n v e r s i o n of l i g h t and heavy h o l e s i n p-Ge. An a n a l y s i s of t h e r e s u l t s s u g g e s t s t h a t t h e r a t i o o f c a r r i e r accumulation i s s a - t u r a t e d a t h i g h e r E above l kV/cm.
Acknowledgement.- W e a r e g r a t e f u l t o J o r g P. K o t t h a u s f o r h i s encoura- gement o f t h e work.
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