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The mobility of hot electrons in silicon
Volume 36A. number 1
PHYSICS LETTERS
THE
MOBILITY
OF
HOT
ELECTRONS
2 August 1971
IN SILICON
H. P. D. LANYON Eleclrical Engineering Department, Woreesler Polylechnical Instilute, Worcester, Massachusetts 01609, USA
Received 12 June 197l
The mobility of avalanching electrons in silicon pn-junctiuns from 100 to 30O°K is proportional to T-3/2. The measurement shows that the lattice mobility of electrons in silicon is determined by high energy acoustic phonons (zero point scattering) rather than by optical phonons.
Although m o s t f a c e t s of the b e h a v i o u r of hot c a r r i e r s in s e m i c o n d u c t o r s have b e e n studied in g r e a t detail [eog. 1], one a s p e c t which does not a p p e a r to have been c o n s i d e r e d until r e c e n t ly [2] is the i n t e r a c t i o n of hot c a r r i e r s with e l e c t r i c and m a g n e t i c fields at the onset of a v a l a n c h e breakdown. In this note m e a s u r e m e n t s a r e d e s c r i b e d of the t e m p e r a t u r e dependence of the mobility of hot e l e c t r o n s in s i l i c o n by the a v a l a n c h e Hall effect. The m e a s u r e m e n t is of p a r t i c u l a r i n t e r e s t for the r e a s o n that, in cont r a s t to the n o r m a l situation, the c a r r i e r s have a l a r g e kinetic e n e r g y ( a p p r o x i m a t e l y 1o6 eV e s s e n t i a l l y independent of t e m p e r a t u r e ) c o m p a r e d with the l a t t i c e t h e r m a l energy. Since the a v e r a g e v al u e of the e l e c t r o n v e l o c i t y is d e coupled f r o m the l a t t i c e t e m p e r a t u r e , the ' m o b i l i t y ' shows a d if f e r e n t dependence upon the l a t t i c e t e m p e r a t u r e than the n o r m a l l a t t i c e m o bility. Our m e a s u r e m e n t s show that the e l e c t r o n m o b i l i t y for hot c a r r i e r s follows a T -3/2 d e p e n d e n c e within the e x p e r i m e n t a l a c c u r a c y . This is one p o wer of T l o w e r than the n o r m a l l a t t i c e mobility [3] (/ZnO~ T-5/2) implying that the m o m e n t u m r e l a x a t i o n t i m e for mobility is p r o p o r t i o n a l to 1 / v 2 r a t h e r than the n o r m a l 1/v dependence° Such a v e l o c i t y dependence is p r e dicted fo r the s c a t t e r i n g of e l e c t r o n s by a c o u s t i c phonons when the e q u i p a r t i t i o n of energy among phonons b r e a k s down at low t e m p e r a t u r e and is r e p l a c e d by a z e r o - p o i n t s c a t t e r i n g a p p r o x i m a t i o n [4]. It has r e c e n t l y been shown that this s c a t t e r ing m e c h a n i s m is i m p o r t a n t in n - t y p e s i l i c o n f r o m the lack of t e m p e r a t u r e dependence of the s a t u r a t e d drift v e l o c i t y of w a r m c a r r i e r s below 77°K [5]. The p r e s e n t r e s u l t s c o n f i r m the c o n c l u s i o n s of the e a r l i e r work and show that z e r o -
point s c a t t e r i n g d e t e r m i n e s the l a t t i c e mobility of e l e c t r o n s in silicon. A v al an ch e breakdown can o c c u r at field s t r e n g t h s at which an e l e c t r o n gains energy f r o m the e l e c t r i c field f a s t e r than it l o s e s it through phonon c r e a t i o n . At high fields the energy loss m e c h a n i s m is c h a r a c t e r i z e d by a m i n i m u m d i s t a n c e b et w een energy loss events, L ~ . . If Eph is the energy of the phonon c r e a t e d , an e l e c t r o n t r a v e l l i n g down the field d i r e c t i o n will i n c r e a s e its kinetic energy until it can i m p a c t - i o n i z e if the field is g r e a t e r than Eph/qLph. The effect of a t r a n s v e r s e m a g n e t i c field is to deflect a c a r r i e r f r o m its o r i g i n a l path. This i n c r e a s e s the path t r a v e l l e d by a c a r r i e r in c r o s s i n g the s a m p l e analogously to a m a g n e t . r e s i s t i v e effect [e.g. 6]° The e f f e c t i v e magnitude of the e l e c t r i c f i el d E is r e d u c e d by AE given by
& E / E = A l / l =61 02 = 1 ~2H2 This m ean s that the field r e q u i r e d for a given m u l t i p l i c a t i o n c o e f f i c i e n t is i n c r e a s e d by the m a g n e t i c field and c a u s e s an i n c r e a s i n g m a g n e t . r e s i s t a n c e c l o s e to the breakdown v o l t a g e which has been o b s e r v e d by e a r l i e r w o r k e r s in bulk s a m p l e s of n - G e [7] and n-InSb [8]. The magnitudes of mobility e s t i m a t e d in both c a s e s a r e in f a i r a g r e e m e n t with the values quoted for the l a t t i c e mobility. The m e a s u r e m e n t s d e s c r i b e d in this p a p e r w e r e m a d e on a 2N2432 npn s i l i c o n t r a n s i s t o r mounted in a n o n - m a g n e t i c copper f r a m e 14 lead DIP p a c k a g e * . The u s e of such a diode enables a l a r g e e l e c t r i c field to be attained without the * Supplied by Sprague Electric Company, Worcester, Mass. 27
Volume 35A
number 1
IO"
PHYSICS
".~2
-3/Z
~.-Tj :(TA+20)
.
7000 5
&V/V
5
x °TJ 10-2
~o-'soJ * , I00
i iI
T °K
400
Fig. 1. T e m p e r a t u r e dependence of the modulation of
the breakdown voltage of a 2N2432 transistor by an 8kgauss field as a function of temperature. The percentage modulation is shown on the left-hand scale; the value of mobility on the right-hand scale. Both the r e sutls in terms of the ambient temperature and with a correction for the power dissipated to give the junction temperature are shown. j o u l e heating that o c c u r s in bulk s a m p l e s . The t r a n s i s t o r was c o n n e c te d in c o m m o n e m i t t e r c o n f i g u r a t i o n on a t r a n s i s t o r c u r v e t r a c e r and the change in breakdown v o l t a g e m o n i t o r e d , both with and without b a s e c u r r e n t flow. It is well e s t a b l i s h e d that the b r e a k d o w n in t r a n s i s t o r s is d e t e r m i n e d by the m i n o r i t y c a r r i e r in the b a s e r e g i o n , in this c a s e e l e c t r o n s . The change in v o l t a g e r e q u i r e d to m a i n t a i n a constant c u r r e n t was m o n i t o r e d as a function of m a g n e t i c f ie ld and t e m p e r a t u r e . T h e p e r c e n t a g e change was independent of the c u r r e n t l e v e l in the a v a l a n c h e r e g i m e and was p r o p o r t i o n a l to H 2 at all t e m p e r a t u r e s . The t e m p e r a t u r e dependence of the m o d u l a t i o n by an 8 kgauss m a g n e t i c field is shown in fig. 1. The two s e t s of e x p e r i m e n t a l points c o r r e s p o n d to the ' a m b i e n t t e m p e r a t u r e ' of the b r a s s c o n t a i n e r within which the t r a n s i s t o r was
28
LETTERS
2 A u g u s t 1971
c l a m p e d and the junction t e m p e r a t u r e which was a p p r o x i m a t e l y 20°K w a r m e r b e c a u s e of the 150 mW of p o w e r d i s s i p a t e d in the d ev i ce. S i m i l a r r e s u l t s have b e e n obtained with a n u m b e r of d i f f e r e n t types of device. In all c a s e s the m a g nitude of the effect has b e e n c o n s i s t e n t with the e x p e c t e d v al u e of mobility. Th e r e s u l t s show that the hot c a r r i e r m o b i l i ty v a r i e s as T -3/2 r a t h e r than as T - 2 as would be e x p e c t e d f o r s c a t t e r i n g by o p t i cal phonons f o r which the m o m e n t u m s c a t t e r i n g t i m e is p r o p o r tional to 1/v. The T -3/2 law follows i m m e d i a t e ly if one a s s u m e s that it is s c a t t e r i n g by h i g h e r enerj~y a c o u s t i c phonons which c a u s e s the T -5/2 law; a m e c h a n i s m f o r which the s c a t t e r i n g t i m e is p r o p o r t i o n a l to 1Iv 2. Knowing the doping p r o f i l e in the t r a n s i s t o r it is p o s s i b l e to c a l c u l a t e the magnitude of the mobility. This is shown on the r i g h t - h a n d s c a l e of fig. 1 ( a s s u m i n g a l i n e a r l y g r a d e d doping p r o f i l e which is c o n f i r m e d both by c a p a c i t a n c e m e a s u r e m e n t s and the d i f f e r e n c e b e t w e e n the b a s e - e m i t t e r and b a s e - c o l l e c t o r b r eak d o w n v o l t a g e s ) . F o r this c a s e E ~ V 2/3 so that A V / V = 3 / 2 A E / E , w h e r e E is the m a x i m u m f i el d in the junction and V is the breakdown v o l t a g e (approx. 60V).
References [1] E . M . C o n w e l l , in: Solid State P h y s i c s , eds.
F. Seitz, D. Turnbull and H. Ehrenreich, Supplement 9 (Academic Press. New York, 1967). [2] S. C. Mehta and R. Parshad, J. Appl. Phys. 41 (1970) 760.
[3] G.W.LudwigandR.L.Watters, Phys. Rev. 101 (1956) 1699. [4] Reference 1, pp. 116-119. [5] M. Costato and L. Reggiani, Phys. Letters 34A (1971) 84. [6] A.C.Beer, in: Solid State Physics, eds. F.Seitz and D. Turnbull, Supplement 4 (Academic Press, New York, 1963). [7] S. H. Koenig and G. R. Gunther-Mohr, J. Phys. Chem. Solids 2 (1957) 268. [8] E. Muller and D. K. Ferry, J. Phys. Chem. Solids 31 (1970) 2401.
PHYSICS LETTERS
THE
MOBILITY
OF
HOT
ELECTRONS
2 August 1971
IN SILICON
H. P. D. LANYON Eleclrical Engineering Department, Woreesler Polylechnical Instilute, Worcester, Massachusetts 01609, USA
Received 12 June 197l
The mobility of avalanching electrons in silicon pn-junctiuns from 100 to 30O°K is proportional to T-3/2. The measurement shows that the lattice mobility of electrons in silicon is determined by high energy acoustic phonons (zero point scattering) rather than by optical phonons.
Although m o s t f a c e t s of the b e h a v i o u r of hot c a r r i e r s in s e m i c o n d u c t o r s have b e e n studied in g r e a t detail [eog. 1], one a s p e c t which does not a p p e a r to have been c o n s i d e r e d until r e c e n t ly [2] is the i n t e r a c t i o n of hot c a r r i e r s with e l e c t r i c and m a g n e t i c fields at the onset of a v a l a n c h e breakdown. In this note m e a s u r e m e n t s a r e d e s c r i b e d of the t e m p e r a t u r e dependence of the mobility of hot e l e c t r o n s in s i l i c o n by the a v a l a n c h e Hall effect. The m e a s u r e m e n t is of p a r t i c u l a r i n t e r e s t for the r e a s o n that, in cont r a s t to the n o r m a l situation, the c a r r i e r s have a l a r g e kinetic e n e r g y ( a p p r o x i m a t e l y 1o6 eV e s s e n t i a l l y independent of t e m p e r a t u r e ) c o m p a r e d with the l a t t i c e t h e r m a l energy. Since the a v e r a g e v al u e of the e l e c t r o n v e l o c i t y is d e coupled f r o m the l a t t i c e t e m p e r a t u r e , the ' m o b i l i t y ' shows a d if f e r e n t dependence upon the l a t t i c e t e m p e r a t u r e than the n o r m a l l a t t i c e m o bility. Our m e a s u r e m e n t s show that the e l e c t r o n m o b i l i t y for hot c a r r i e r s follows a T -3/2 d e p e n d e n c e within the e x p e r i m e n t a l a c c u r a c y . This is one p o wer of T l o w e r than the n o r m a l l a t t i c e mobility [3] (/ZnO~ T-5/2) implying that the m o m e n t u m r e l a x a t i o n t i m e for mobility is p r o p o r t i o n a l to 1 / v 2 r a t h e r than the n o r m a l 1/v dependence° Such a v e l o c i t y dependence is p r e dicted fo r the s c a t t e r i n g of e l e c t r o n s by a c o u s t i c phonons when the e q u i p a r t i t i o n of energy among phonons b r e a k s down at low t e m p e r a t u r e and is r e p l a c e d by a z e r o - p o i n t s c a t t e r i n g a p p r o x i m a t i o n [4]. It has r e c e n t l y been shown that this s c a t t e r ing m e c h a n i s m is i m p o r t a n t in n - t y p e s i l i c o n f r o m the lack of t e m p e r a t u r e dependence of the s a t u r a t e d drift v e l o c i t y of w a r m c a r r i e r s below 77°K [5]. The p r e s e n t r e s u l t s c o n f i r m the c o n c l u s i o n s of the e a r l i e r work and show that z e r o -
point s c a t t e r i n g d e t e r m i n e s the l a t t i c e mobility of e l e c t r o n s in silicon. A v al an ch e breakdown can o c c u r at field s t r e n g t h s at which an e l e c t r o n gains energy f r o m the e l e c t r i c field f a s t e r than it l o s e s it through phonon c r e a t i o n . At high fields the energy loss m e c h a n i s m is c h a r a c t e r i z e d by a m i n i m u m d i s t a n c e b et w een energy loss events, L ~ . . If Eph is the energy of the phonon c r e a t e d , an e l e c t r o n t r a v e l l i n g down the field d i r e c t i o n will i n c r e a s e its kinetic energy until it can i m p a c t - i o n i z e if the field is g r e a t e r than Eph/qLph. The effect of a t r a n s v e r s e m a g n e t i c field is to deflect a c a r r i e r f r o m its o r i g i n a l path. This i n c r e a s e s the path t r a v e l l e d by a c a r r i e r in c r o s s i n g the s a m p l e analogously to a m a g n e t . r e s i s t i v e effect [e.g. 6]° The e f f e c t i v e magnitude of the e l e c t r i c f i el d E is r e d u c e d by AE given by
& E / E = A l / l =61 02 = 1 ~2H2 This m ean s that the field r e q u i r e d for a given m u l t i p l i c a t i o n c o e f f i c i e n t is i n c r e a s e d by the m a g n e t i c field and c a u s e s an i n c r e a s i n g m a g n e t . r e s i s t a n c e c l o s e to the breakdown v o l t a g e which has been o b s e r v e d by e a r l i e r w o r k e r s in bulk s a m p l e s of n - G e [7] and n-InSb [8]. The magnitudes of mobility e s t i m a t e d in both c a s e s a r e in f a i r a g r e e m e n t with the values quoted for the l a t t i c e mobility. The m e a s u r e m e n t s d e s c r i b e d in this p a p e r w e r e m a d e on a 2N2432 npn s i l i c o n t r a n s i s t o r mounted in a n o n - m a g n e t i c copper f r a m e 14 lead DIP p a c k a g e * . The u s e of such a diode enables a l a r g e e l e c t r i c field to be attained without the * Supplied by Sprague Electric Company, Worcester, Mass. 27
Volume 35A
number 1
IO"
PHYSICS
".~2
-3/Z
~.-Tj :(TA+20)
.
7000 5
&V/V
5
x °TJ 10-2
~o-'soJ * , I00
i iI
T °K
400
Fig. 1. T e m p e r a t u r e dependence of the modulation of
the breakdown voltage of a 2N2432 transistor by an 8kgauss field as a function of temperature. The percentage modulation is shown on the left-hand scale; the value of mobility on the right-hand scale. Both the r e sutls in terms of the ambient temperature and with a correction for the power dissipated to give the junction temperature are shown. j o u l e heating that o c c u r s in bulk s a m p l e s . The t r a n s i s t o r was c o n n e c te d in c o m m o n e m i t t e r c o n f i g u r a t i o n on a t r a n s i s t o r c u r v e t r a c e r and the change in breakdown v o l t a g e m o n i t o r e d , both with and without b a s e c u r r e n t flow. It is well e s t a b l i s h e d that the b r e a k d o w n in t r a n s i s t o r s is d e t e r m i n e d by the m i n o r i t y c a r r i e r in the b a s e r e g i o n , in this c a s e e l e c t r o n s . The change in v o l t a g e r e q u i r e d to m a i n t a i n a constant c u r r e n t was m o n i t o r e d as a function of m a g n e t i c f ie ld and t e m p e r a t u r e . T h e p e r c e n t a g e change was independent of the c u r r e n t l e v e l in the a v a l a n c h e r e g i m e and was p r o p o r t i o n a l to H 2 at all t e m p e r a t u r e s . The t e m p e r a t u r e dependence of the m o d u l a t i o n by an 8 kgauss m a g n e t i c field is shown in fig. 1. The two s e t s of e x p e r i m e n t a l points c o r r e s p o n d to the ' a m b i e n t t e m p e r a t u r e ' of the b r a s s c o n t a i n e r within which the t r a n s i s t o r was
28
LETTERS
2 A u g u s t 1971
c l a m p e d and the junction t e m p e r a t u r e which was a p p r o x i m a t e l y 20°K w a r m e r b e c a u s e of the 150 mW of p o w e r d i s s i p a t e d in the d ev i ce. S i m i l a r r e s u l t s have b e e n obtained with a n u m b e r of d i f f e r e n t types of device. In all c a s e s the m a g nitude of the effect has b e e n c o n s i s t e n t with the e x p e c t e d v al u e of mobility. Th e r e s u l t s show that the hot c a r r i e r m o b i l i ty v a r i e s as T -3/2 r a t h e r than as T - 2 as would be e x p e c t e d f o r s c a t t e r i n g by o p t i cal phonons f o r which the m o m e n t u m s c a t t e r i n g t i m e is p r o p o r tional to 1/v. The T -3/2 law follows i m m e d i a t e ly if one a s s u m e s that it is s c a t t e r i n g by h i g h e r enerj~y a c o u s t i c phonons which c a u s e s the T -5/2 law; a m e c h a n i s m f o r which the s c a t t e r i n g t i m e is p r o p o r t i o n a l to 1Iv 2. Knowing the doping p r o f i l e in the t r a n s i s t o r it is p o s s i b l e to c a l c u l a t e the magnitude of the mobility. This is shown on the r i g h t - h a n d s c a l e of fig. 1 ( a s s u m i n g a l i n e a r l y g r a d e d doping p r o f i l e which is c o n f i r m e d both by c a p a c i t a n c e m e a s u r e m e n t s and the d i f f e r e n c e b e t w e e n the b a s e - e m i t t e r and b a s e - c o l l e c t o r b r eak d o w n v o l t a g e s ) . F o r this c a s e E ~ V 2/3 so that A V / V = 3 / 2 A E / E , w h e r e E is the m a x i m u m f i el d in the junction and V is the breakdown v o l t a g e (approx. 60V).
References [1] E . M . C o n w e l l , in: Solid State P h y s i c s , eds.
F. Seitz, D. Turnbull and H. Ehrenreich, Supplement 9 (Academic Press. New York, 1967). [2] S. C. Mehta and R. Parshad, J. Appl. Phys. 41 (1970) 760.
[3] G.W.LudwigandR.L.Watters, Phys. Rev. 101 (1956) 1699. [4] Reference 1, pp. 116-119. [5] M. Costato and L. Reggiani, Phys. Letters 34A (1971) 84. [6] A.C.Beer, in: Solid State Physics, eds. F.Seitz and D. Turnbull, Supplement 4 (Academic Press, New York, 1963). [7] S. H. Koenig and G. R. Gunther-Mohr, J. Phys. Chem. Solids 2 (1957) 268. [8] E. Muller and D. K. Ferry, J. Phys. Chem. Solids 31 (1970) 2401.