Energy relaxation of warm electrons in (100)-Si-MOSFETs under substrate bias

Solid State Conununications, Printed in Great Britain.

Vol.bl,No.9,

pp.679-683,

1984

0038-I098/84 $3.00 + .00 Pergamon Press Ltd.

E N E R G Y R E L A X A T I O N OF WARM E L E C T R O N S IN ( 1 0 0 ) - S i - M O S F E T s UNDER S U B S T R A T E BIAS W.

H 0 n l e i n a nd G.

Landwehr

M a x - P l a n c k - I n s t i t u t f(ir F e s t k b r p e r f o r s c h u n g , Hochfeld-Magnetlabor, F - 3 8 0 4 2 G r e n o b l e , F r a n c e , a nd Physikalisehes Institut der Universit~it Wfirzburg, D-8700 W0rzburg, FRG

(Received ii October 1983, in revied form 6 June 1984 by B. bCflhlschlegel) We h a v e s t u d i e d the e n e r g y l o s s of w a r m e l e c t r o n s i n a t w o d i m e n s i o n a l e l e c t r o n g a s a t the S i - S i O 2 i n t e r f a c e of ( 1 0 0 ) - S i M O S F E T s . T h e a p p l i c a t i o n of a n e g a t i v e s u b s t r a t e b i a s d e c r e a s e s the i n e l a s t i c e l e c t r o n - p h o n o n i n t e r a c t i o n c o n s i d e r a b l y a t T 1 = 1 . 8 K. T h e t e m p e r a t u r e d e p e n d e n c e of t he e n e r g y l o s s s u g g e s t s t h a t a t z e r o s u b s t r a t e b i a s and l ow t e m p e r a t u r e s d e f o r m a t i o n p o t e n t i a l s c a t t e r i n g by S i - b u l k p h o n o n s c a n n o t e n t i r e l y a c c o u n t f o r the e x p e r i m e n t a l f i n d i n g s .

I.

Introduction

mit.

It w a s s h o w n 5 t h a t r e f l e c t e d -

and R a y l e i g h -

t ype p h o n o n s c a n be n e g l e c t e d a nd t h a t the o n l y m o T h e e l e c t r o n - p h o n o n i n t e r a c t i o n in n - and p - c h a n nel inversion layers

d i f i c a t i o n of t he t h r e e - d i m e n s i o n a l s c a t t e r i n g a r i s e s r f r o m the t w o - d i m e n s i o n a l n a t u r e of t h e e l e c t r o n

of M O S F E T s h a s b e e n s u b j e c t

of m a n y e x p e r i m e n t a l

and t h e o r e t i c a l i n v e s t i g a t i o n s 1.

In the h i g h t e m p e r a t u r e

gas.

limit scattering may be

c a u s e d by i n t r a v a l l e y a c o u s t i c a l and o p t i c a l p h o n o n s and b y i n t e r v a l l e y o p t i c a l p h o n o n s . perature

The experimentally

d e t e r m i n e d v a l u e s of

~ph

by K a w a g u c h i a nd K a w a j i 8 c a n n o t be e x p l a i n e d b y t he s u r f o n t h e o r y u n l e s s u n r e a s o n a b l e

In the l o w t e m -

l i m i t o n l y i n t r a v a i l e y a c o u s t i c phonon

T h e d e d u c t i o n of tl~e e - p h s c a t t e r i n g

scattering is relevant.

If the c a r r i e r

too h i g h ,

the e l e c t r o n s

a r e c o n d e n s e d in the l o w e s t

electrical

s u b b a n d and no i n t e r s u b b a n d s c a t t e r i n g

assumptions

a r e m a d e f o r t he d e f o r m a t i o n p o t e n t i a l c o n s t a n t s .

energy is not

temperature

time from the

d e p e n d e n c e of the c o n d u c t i v i t y i s c e r -

tainly dubious because other temperature mechanisms

dependent

a r e u s u a l l y p r e s e n t 9.

has to be taken into account. Acoustic phonon s c a t 2 t e r i n g has f i r s t been t r e a t e d by Ezawa et al. for

Under

a s t r e s s - f r e e ideal s u r f a c e of an i s o t r o p i c e l a s t i c

concept is applicable I0, the energy-loss

continuum in the high t e m p e r a t u r e limit. These

carriers,

a u t h o r s calculated the phonon c o r r e l a t i o n function in the deformation potential approximation and i n t r o -

can give valuable information about inelastic scatii tering processes . Hess et al. 4 determined expe-

the assumption

that the electron-temperature

heated up by an external electric field,

duced the s u r f on modes a s a consequence of the

rimentally the energy-loss

boundary condition at the s u r f a c e . The calculated

type inversion layers.

s c a t t e r i n g times a g r e e only qualitatively with the

investigated the energy-loss

e x p e r i m e n t a l l y observed

nel inversion layers at low temperatures

T and they d i s a g r e e still

by a f a c t o r of 2 if additional s c a t t e r i n g m e c h a n i s m s 3 a r e taken into account . M o r e o v e r , the contributions,

riety of samples

a r i s i n g f r o m the s u r f a c e - i n d u c e d

of the energy-loss

phonon

amination

scattering,

of their data shows

varies considerably

t i m e ~c ~ w h i c h i s d o m i n a t e d by t h e b u l k - p h o n o n pn 2 interaction .

Recently,

4

and r e c e n t l y S h i n b a e t al.

5,6,7

carriers

of w a r m surface li-

of electrons in n-chanfor a vaAn

ex-

that the dependence

on the lattice temperature for different samples

r1

and gives 2 to 3.

Shinba et aL 6 calculated the energy-loss electrons model.

energy-loss

679

in p-

and L a n d w e h r 12

values of the exponent roughly ranging f r o m

ex-

t e n d e d the s u r f o n t h e o r y to the low t e m p e r a t u r e

of w a r m

Neugebauer

with different mobilities.

do not s i g n i f i c a n t l y i n f l u e n c e the t o t a l s c a t t e r i n g

H e s s et al.

of w a r m

at low temperatures

u s i n g t he

T h e y found a T14 d e p e n d e n c e o f t he

at small

AT -- T c - T 1 (T c = e l e c t r o n

W A R M ELECTRONS

680 temperature)

IN (100)-Si-MOSFETs

U N D E R SUBSTRATE

BIAS

Vol.

51, No. 9

1/ 2

in contrast to the experimental data. 2 e o s Si~d(NA -ND)

Chain and W h e e l e r 13 first pointed out that the ener-

Ndepl

gy loss decreases

N A,

if the electrons are forced closer

=

(3)

2 ¶e

ND = a c c e p t o r ,

donor concentrations,

to the interface by application of a negative sub-

solute dielectric constant,

strafe bias.

c o n s t a n t of Si.

This entirely unexpected result w a s

subsequently investigated in detail b y the authors 14. We

at T 1 = I. 6 K,

the energy loss drops b y if the substrate bias

voltage is increased to approx.

- 8 V.

and s t r o n g i n -

by

"0"I.G Cd

substrate bias hardly influences the energy loss whereas

eSi = relative dielectric

low temperatures

v e r s i o n ~d c a n be a p p r o x i m a t e d

found that at T 1 = 4.2 K the application of a

nearly one order of magnitude

For

eo = a b -

It w a s point-

=

EG

+

e

(4)

Usb

= energy gap of Si. Using the variational funct-

ion of F a n g and H o w a r d 16 for the z-dependent part of the electron w a v e function in the lowest subband

ed out that deformation potential scattering by bulk

b a 1/2

silicon phonons cannot entirely account for the ex-

~o(Z) = (-~-)

i

z exp(-g bz)

(5)

perimental results, but that at least one additional relaxation channel m u s t

exist. W e

proposed

the average extension of the carriers in z-direction

that

the energy relaxation might be caused by two-levelsystems

in the a m o r p h o u s

which is perpendicular to the interface is given b y

SiO 2 via surface- and

3

z°

oxide charges located at the interface.

= ~ ;

(6)

It is the

purpose of this paper to present n e w experimental

with

data supporting this m o d e l and to discuss the re6 sults of Shinba et al. including the possible influ-

b =

12 m z e2 (N de pl + 1 1 / 3 2 N s ) i / 3 (7) ~ o e S i "l~2

ence of the Si-SiO 2 interface. N

2.

= s u r f a c e c a r r i e r concentration, m = z-corns z ponent of the e f f e c t i v e m a s s • T h u s the a p p l i c a t i o n

Theory

of a n e g a t i v e s u b s t r a t e b i a s i n c r e a s e s b and f o r c e s

If t h e e l e c t r o n - e l e c t r o n

interaction is much strong-

er than electron-phonon

scattering

the e l e c t r o n s c l o s e r to the i n t e r f a c e . It should be

energy gain from randomized racterized

( t e e << Tph),., t h e

an external electric

f i e l d ESD i s

in the e l e c t r o n g a s w h i c h c a n be c h a 10 b y a n e l e c t r o n t e m p e r a t u r e Tc > T1

noted that for substrate experiments,

z

o

w a s c h a n g e d b y a b o u t 0. 1 n m o r

5%.

T h e m e a n e n e r g y g a i n p e r e l e c t r o n c a n e a s i l y be

To d e t e r m i n e

evaluated by

t he o s c i l l a t o r y t r a n s v e r s e

f i e l d = e p ( E s D ) For

small

deviations

2 ESD

- P;

/u

AT = Tc - T 1 and s m a l l

w e m a y u s e /u ° i n s t e a d of /U(EsD). state,

= mobility

(t)

T 1,

In the s t a t i o n a r y

the e l e c t r o n t e m p e r a t u r e

nikov-de Haas-effect,

SdH) a s a t h e r m o m e t e r .

of t he a m p l i t u d e of a c e r t a i n

< dE "~- > coll

(2)

i c f i e l d s 15,

bias can

formula for low magnet-

and no B - d e p e n d e n c e

was observed,

a p p l i e d the n e g a t i v e m a g n e t o - r e s i s t a n c e where

This

SdH o s c i l l a t i o n w i t h

electric field and substrate

be f i t t e d w e l l w i t h A n d o ' s dE < d--~-> field =

T we u s e c" m a g n e t o - r e s i s t a n c e (S hub-

p r o c e d u r e has b e e n d e s c r i b e d e l s e w h e r e in d e t a i l 12, 14 • A l t h o u g h t he e x p e r i m e n t a l l y d e d u c e d c h a n g e

temperature,

we h a v e

b i a s e s u s e d i n t he p r e s e n t

we

i n t he w e a k -

< d E / d t >coll describes the energy loss due l y l o c a l i z e d r e g i m e a s a n a l t e r n a t i v e m e t h o d of T c 19 e v a l u a t i o n . T h e r e s u l t s a r e in good a g r e e m e n t

to inelastic e-ph-interactions. T h e a p p l i c a t i o n of a n e g a t i v e s u b s t r a t e t h e b a c k of th e s i l i c o n s u b s t r a t e pletion layer

thickness.

bias Usb at

e n h a n c e s the d e -

The d e p l e t i o n l a y e r c h a r g e

per unit area Ndepl is given by

w i t h the S d H - d e d u c e d e l e c t r o n t e m p e r a t u r e ,

thus in-

dicating that a quantizing magnetic field does not h a v e a s i g n i f i c a n t i n f l u e n c e on the e n e r g y r e l a x a t 20 ion

WARM ELECTRONS

Vol. 51, No. 9

IN (100)-Si-MOSFETs bias

3. Results and Discussion

is observed.

creases

However,

considerably

The experiments have been performed on Si-(100)-

closer

MOSFETs with a channel length ranging from 400

tron-phonon

to 1000 /urn and a channel width of 4 0 - 5 0 /urn. In

scattering

order to avoid contact problems, the voltage drop

drops

along the channel at constant current was detected

681

UNDER SUBSTRATE BIAS

to the interface. interaction rate

as

electrons

Thus,

K,

T inc pushed

are

the inelastic

decreases

increases.

The

while

elec-

the total

of T c vs. ESD i s l o w e r e d 14. In F i g . 2,

the temperature

we plotted

at T 1 = 1.8

if the

the normalized

slope

energy

loss

as

a function

via potential probes. The total power input was kept well below 10 -6 W to ensure that the lattice

PS1004/5

was at bath temperature. Changes of the substrate

/~

u~: ov

¢~- :

:

__

u~:-~v

bias were c a r r i e d out at temperatures T 1 > 100 K

//Y//

in order to achieve stable potential conditions at the substrate. The change in Ndepl was checked by

.

3.6K

monitoring the threshold voltage Uth as a function

3.OK

of substrate bias with Uth deduced from SdH o s c i l lations at constant magnetic fields B < 5 Tesla. The gate voltage U was adjusted to keep the ing version l a y e r electron density N s Ug -Uth constant. The mobility p at N = 2 - 4 . 1 0 1 2 cm -2 was ap! & proximately 7'000 cmZ/Vs for all samples under investigation. Fig.

1 shows

the electron

t i o n of t h e e l e c t r i c peratures

field

temperature ESD for

and two different

At T 1 = 3.6

K,

!

hardly

as

a func-

two different

substrate

an influence

tem-

bias

voltages.

of t h e

substrate

'" I

2

Fig. 2

PS 3~/11b ° U~b= OV • U~b=- OV

1.5

1.0

/

/

°

4

5

•2

3

4

TcIK]~

Normalized energy loss as a function of the c a r r i e r temperature T

at different c l a t t i c e temperatures for zero substrate

T, t

3

1.8K

/

//

bias Usb(left) and -4 V (right) of the electron temperature for two different subs t r a t e biases. As indicated by the arrows, we extrapolated the curves for T c - T l - ~ 0 12. Fig. 3 shows extrapolated values of the energy l o s s as a function of the negative substrate bias for different lattice temperatures. As it is already suggested by

0.5

Fig.

1, the substrate bias hardly influences the

energy l o s s at T1 = 4.2 K, whereas at T 1 = 1.8 K P / ( T c - T I) drops considerably. At Usb = - 9 V, a 01

Fig.

1

02

0.3

Temperature A T as between

a function

Usb

3.6

of w a r m

probes

at lattice K and

minimum can be seen with a slight i n c r e a s e for 5 higher substrate bias voltages. Shinba et al. cal-

=twrm""cm ]~

of the electric

the potential

and

03

increase

(100)MOSFET of 1 . 8

0.4

E

electrons

PP temperatures

2 substrate

numerC

i c a l l y in the framework of the surfon theory. F o r

field for

culated the energy l o s s as a function of T

a

small

T1

lytical expression (see equation A5 in ref.

biases

AT and low temperatures, they gave an ana6) from

which we calculated the energy l o s s indicated by the dashed lines in Fig. 3. Although this expression is

682

WARM ELECTRONS IN (100)-Si-MOSFETs UNDER SUBSTRATE BIAS ,

i

i

,

,

,

,

,

,

,

however,

PS 304111a

that

x drops

~-----~~-o--38K

a

tried

further

perature

energy

Tc --# T 1 as bias as

- Usb

extrapolated

a function with

The

values

for

of the substrate

the lattice

a parameter.

calculated

loss

temperature

dotted

after

lines

ref.

indicate

Moreover,

stence

of irregularities phonon

to a c h a n g e

good

agreement

strate

biases

and

for

with

o f z o,

it gives

the experimental

at the minimum

T 1 ~ 4.2

surprisingly data

for

sub-

for low temperatures

K in the whole

substrate

bias

range. In Fig.

4,

loss

a function

as

versus

we plotted

good

of the lattice

the substrate

corresponding

bias

surfon

x

with theory.

1 4

of the energy

temperature

voltage.

to t h e m i n i m u m ,

agreement

by the

the exponent

T1

At the voltage

we find x = 4 in

the T 1 dependence

predicted

At low substrate

bias

voltage,

~

4

Exponent versus

should ions

extension

an

bias

from

existence rise

x

-Usb

in a region average

of l a t t i c e

wave

of charges

phonon

of the interface,

the Si-crystal

and

the amor-

function

energy

we

excitat-

to t h e a v e r a g e

at or near

to additional

the with two

the amorphous

( t is comparable

of t h e e l e c t r o n

inversion

layer

e.g.

z ). o

the interface

relaxation

~ ' "~ ' 12

electrical

energy as

be

channels

a function

for

27.

interaction

to screening

electrons

induced

3,

interaction

pile

biases

would

and

the

relaxation belonging

Si-crystal remain.

at

thus

by

therefore

in z . Pushing o would result

energy

the bare

This

up

the

typical

generation

subjected

interface

substrate

which to the

to phonons

to changes

Fig.

Due

passing of the

might

sitive

surface loss

phonon,

electrons

layer

to the

of ions

SiO 2 26.

ac-field,

couple

A

the distance

system

version

er

electrons.

modulates

in the

however,

x of the normalized

of substrate

sides

to

emit-

to c r o s s interact

by the

a mixture

SiO 2 layer

be able inversion

phonons

Thus,

on both

originating

4 nm

temperature

.

to h a v e

mirror-charge

~' ~.' ~,'

for 14

be characterized t

hand,

en-

scatter-

impinging

might

be able

characteristic

expect

phous

ively

lattice

may

wavelength

for

-~b[V]-Fig.

systems

at low temperatures

interface,

P

should

strongly

scatter

other

velocithe exi-

to resonant

into the SiO 2 and

state which

sound that

Phonons

and

high,

~

2'

o'

enter

level

gives x

On the

ted by hot electrons interface,

The

due

the

be very and

as

to t h e

low tem-

at the interface

s t a t e s 2 2 ' 2 3 , 24.

electrons.

for

clear

the SiO 2 onto the interface

layer

such

theory 21,

become

into the Si-crystal

of the

modes

According

should

transmission

ing by surface

intrude insensitive

the possibility

densities

it has

hances

from

6.

to the si-

mismatch

to n o t t o o d i f f e r e n t

ties.

it should

phonon

phonons

we pro-

At first

of the Si-SiO 2 interface

acoustical

data,

adjacent

at the interface.

of the acoustic

transparency

Normalized

reduces

of s u r f a c e - i n d u c e d waves

if the

of t h e p h y s i c a l

licon

results

3

the experimental

that an SiO 2 layer

Rayleigh

Fig.

for

the influence

of the Si-SiO 2 interface.

crystal

indicates

substantially

be mentioned

due

This

No. 9

is decreased.

o

to account

existence

-U~blV]~

z

to estimate

perties

< 3.

is modified

distance

In o r d e r

-°-

to v a l u e s

e-ph-scattering

average

Vol. 5 1 ,

would,

mobile be

of

effect-

very

electrons

insenclos-

in screening

out

channels.

Thus,

to the

minimum

deformation-potential

in

Vol.

683

W A R M E L E C T R O N S IN (100)-Si-MOSFETs U N D E R SUBSTRATE BIAS

51, No. 9

Inelastic surface

scattering as described

in the present experiments m / m ° as determined

above

should be influenced s t r o n g l y by phonons o r i g i n a t -

f r o m SdH-data s h o w s an increase with substrate

i n g f r o m the a m o r p h o u s SiO 2. It i s w e l l known

bias in the low bias range.

that,

due to the e x i s t e n c e of t w o - l e v e l - s y s t e m s ,

p h o n o n d e n s i t y of s t a t e s in a m o r p h o u s nificantly different from crystalline peratures

T < 1 K 28, 25.

e n c e in c r y s t a l s ,

solids is sig-

FET

e.g.,

to the T 3 - d e p e n d -

T h e s p e c i f i c h e a t of v i t r e o u s SiO_ a t T = 2 z29 1 K shows approximately a T -dependence , thus

good a g r e e m e n t

data (Fig.

electrons

substrate

in a n S i - ( 1 0 0 ) - n - c h a n n e l M O S -

closer

bias.

P u s h i n g t he

to t he i n t e r f a c e r e s u l t s of t he c a r r i e r

temperature

b i a s i s s e e n a t T 1 = 3 . 6 K.

4).

for zero substrate

temperatures.

m a d e to e x p l a i n the e x p e r i m e n t a l l y

and i n t e r a c t i o n w i t h t w o - l e v e l - s y s t e m s

of

/~T f o r t e m p e r a t u r e s

mass

observed in-

T 1 = 3 . 6 K.

calculated.

Data for lower temperatures There

is,

i n t he a m o r -

p h o u s SiO 2 c a n q u a l i t a t i v e l y a c c o u n t f o r t he e x p e r i mental findings.

surfons

Acknowledgment

scattering can ac-

c o u n t f o r the c h a n g e in e l e c t r o n t e m p e r a t u r e T 1 = 3. 6 K.

bias at low

It i s s u g g e s t e d t h a t s u r f a c e c h a r g e s

s u b s t r a t e b i a s in c o n -

j u n c t i o n w i t h s c a t t e r i n g by 2 D - p h o n o n s , and the s u r f a c e r o u g h n e s s

V a s s 30

of the e l e c t r o n e f f e c t i v e

m/m ° with increasing

The temper-

b i a s can be e x p l a i n e d by an a d d i t i o n a l e n e r g y r e laxation mechanism

p o i n t e d out t h a t a d e c r e a s e

at

w h e r e a s h a r d l y a n i n f l u e n c e of the

I t s h o u l d be n o t e d t h a t a n o t h e r a t t e m p t h a s b e e n

crease

in a c o n -

a t u r e d e p e n d e n c e of the e n e r g y l o s s u n d e r s u b s t r a t e

w h i c h i s in

with our experimental

electrons

under negative substrate

T 1 = 1 . 8 K,

states.

1,

we have i n v e s t i g a t e d the e n e r g y l o s s

siderable increase

i n d i c a t i n g a c o n s t a n t d e n s i t y of

r e d u c i n g the e x p o n e n t b y a p p r o x .

In c o n c l u s i o n , of w a r m

solids at tem-

The specific heat,

i s n e a r l y l i n e a r in T in c o n t r a s t

the

however,

at

T h e a u t h o r s a r e g r a t e f u l to D r .

w e r e not

the d i f f i c u l t y t h a t

Forschungslaboratorien,

G.

M~nchen,

Dorda,

Siemens

for providing the

samples. References

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unpublished and ref.1, p.539

1983, p.114