The scattering of electrons by surface oxide charges and by lattice vibrations at the silicon-silicon dioxide interface

SURFACE SCIENCE 32 (1972) 561-575 © North-Holland Publishing Co.

THE SCATTERING

OF ELECTRONS

BY S U R F A C E

O X I D E C H A R G E S A N D BY L A T T I C E V I B R A T I O N S AT THE SILICON-SILICON

DIOXIDE INTERFACE*

C. T. SAH, T. H. NING** and L. L. TSCHOPP Department of Electrical Engineering, and of Physics and Materials Research Laboratory, University of lllinois, Urbana, Illinois 61801, U.S.A.

Received 17 February 1972; revised manuscript received 24 April 1972 The electron conductivity mobility on weakly inverted Si surfaces covered with thermally grown oxide is reported at temperatures from 30 to 300°K as a function of the surface oxide charge density. The scattering of electrons by surface oxide charges and by ionized impurities is reformulated by treating the random spatial fluctuations of charge density as the perturbation. The theory is applied to a simple two-dimensional model with a random Poisson distribution of oxide charges, giving /q pc T/N~, in agreement with experiment at low temperatures. Surface phonon scattering mobility is extracted from the experimental data and the two-dimensional ion scattering theory using 1 / i l L = l / / t e x p - -- 1/,ui. The theory of two-dimensional surface acoustical pbonon scattering is extended to include surface optical or intervalley phonons. Comparison with experimental values of gL indicates that for T > 150°K the effects of optical or intervalley phonons must be included. 1. I n t r o d u c t i o n

The m o b i l i t y o f electrons in n-type inversion channels on silicon surfaces has been studied extensively in recent years1-10). The observed m o b i l i t y a n i s o t r o p y is interpreted fairly satisfactorily in terms o f the effective-mass a n i s o t r o p y 9-11). H o w e v e r , i n t e r p r e t a t i o n o f the d e p e n d e n c e o f the m o b i l i t y on t e m p e r a t u r e a n d on surface electric field requires an u n d e r s t a n d i n g o f the scattering m e c h a n i s m s involved 12,13). Early analyses were generally in terms o f the classical t h e o r y o f diffuse or p a r t i a l l y diffuse surface scattering14-16), but the deficiencies o f this t h e o r y have a l r e a d y been p o i n t e d putt6). The o t h e r m e c h a n i s m s that have been t r e a t e d at some length, either classically or q u a n t u m - m e c h a n i c a l l y , are the scattering by fixed charges 13,17-21) a n d the scattering by lattice vibrations Z2,23). However, the scattering by lattice vibrations is i m p o r t a n t only at relatively high temperatures. * Supported in part by the Air Force Office of Scientific Research and the Advanced Research Projects Agency. ** IBM Postdoctoral Fellow. 561

562

C . T . SAH, T . H . NING AND L . L . T S C H O P P

Experiments showed a strong correlation between the mobility and the built-in surface charge density 4-6, 8); and Kamins and MacDonald 6) have demonstrated a quantitative correlation between the localized surface charges and the change in the diffusivity of the surface at room temperature. However, just as in the study of bulk mobility, detailed temperaturedependence data are needed to give information on the relative importance of the various scattering mechanisms involved. In this paper, we study the electron mobility on weakly inverted silicon surfaces covered with thermally grown oxide at temperatures from 30 to 300°K as a function of the surface oxide charge density. The results are interpreted in terms of simple two-dimensional theories of scattering by the spatial density fluctuations of the surface oxide charges and of scattering by surface acoustical and optical phonons. It should be pointed out that the weakly inverted surfaces of this study are nondegenerate, i.e. the conduction-band edge at the surface lies more than 2 k B T above the Fermi level, so that the mobility is given by # = (e/mu) ( E ~ c ) / ( E ) ,

where r is the relaxation time, E is the energy, m u is the mobility effective mass, and ( ) means the averaged value using Boltzmann statistics. Thus, through the dependence of r on E, # is generally a strong function of the temperature T. On the other hand, for a strongly inverted surface where degenerate statistics apply,

Ft =

(e/mu) v

( Ev),

where ~ is evaluated at the Fermi energy E v. Thus, except for the explicit dependence ofv on T, # for a strongly inverted surface is generally a relatively weak function of T. The experiments of Fang and Fowler ~) confirmed this for strongly inverted silicon surfaces. Hence we expect a weakly inverted surface to provide more information on the scattering mechanisms than a strongly inverted one. It should be further emphasized that even though the surface is weakly inverted, the electrons are concentrated in a narrow surface channel of less than 100 A wide so that the two-dimensional model is a valid approximation.

2. Experiments and results The devices used are shown in fig. 1. They are linear MOS transistors for conductivity mobility measurements and MOS Hall transistors for Hall mobility measurements. The devices were fabricated on p-type Czochralski silicon crystal with a boron concentration of approximately 2 x 1014 cm-3.

563

SCATTERING OF ELECTRONS

#DRA,N

SOUROE

GATE

GUARD GATE

HALL

(GATE

(SOURCE ,

# ,,

,

TRANSISTOR

\

fDRAIN !

...............................

i.........

)

:::::::::::::::::::::::::::::::::

! . . . . . . . . . . .

LINEAR TRANSISTOR ~ALUMINUM GATE I

•

I

ISOURCE(INSULATORDRAIN~ (SIC2) [ P TYPE SILICON ..

SUBSTRATE EXPANDED CROSSECTION

Fig. 1. Top view and cross-section of experimental devices.

Crystal surfaces of (111), (110), and (100) orientations were used, with the current flow in the [110] direction. The linear transistor geometry is reentrant so that the n-type drain is completely enclosed by the gate electrode to prevent unwanted surface leakage current. The channel length is kept small to minimize frequency effects. For the Hall transistor, a guard gate is used to pinch off the surface channel immediately surrounding the control gate to prevent fringing fields from interfering with the electrical measurements. An oxide thickness of about 5000 A was used. Phosphorus gettering was used to stabilize the devices. The surface oxide charge was generated by heat treatment in oxygen at 600-1000 °C while the control samples were annealed in argon ambient at 1000 °C 24).

564

c.T. SAH, T. H. NING AND L. L. TSCHOPP

The surface oxide charge density was obtained by comparing an experimental capacitance-voltage curve with the appropriate theoretical one at flat-band condition (Us= UF). A typical family of C-V curves for a 10 mil circular capacitor is shown in fig. 2. The total surface oxide charge density is calculated as (Qo + QOT + Qss) = Co (Vcxp - Vth¢o -

4~

(2.1)

q~ms),

/

~--

~

-

"

US = UF

^LDEVJCE RUN 0 4 7 1 - H A L L < z (110) SURFACE (D ~ T = 2 5 ° C ! f = I MHz NA = 8 x 1014cm- 5 I h - - i - - - a

.

:

.

: _ _ 1

!

_ _ L

i

a

~

__

- 2 4 - 2 2 - 2 0 -18 - t 6 -14 -12 - t 0 - 8 - 6 -4 GATE VOLTAGE (volts)

~.

-2

t

~

0

2

4

Fig. 2. H i g h f r e q u e n c y c a p a c i t a n c e - v o l t a g e c u r v e s for M O S c a p a c i t o r s w i t h different s u r f a c e o x i d e c h a r g e densities. O H T = o x y g e n h e a t t r e a t e d ; A H T -- a r g o n h e a t t r e a t e d .

where Qo is the areal density of fixed oxide charges, QoT that of slow oxide traps, Qss that of fast surface states24), Co is the oxide capacitance, q~ms is metal-oxide-semiconductor work function difference, and (Vexp--Vtheo) is the voltage shift at flat-band condition. For this study, we have 0.4 x 1011 < (Qo + QOT + Qss)/q < 9 x 1011 c m - 2. The amount of surface oxide charge that exists as oxide traps, QOT, and fast surface states, Qss, is estimated to be less than 10~o so that the fixed oxide charges, Qo, are the dominant scattering centers. Small signal ac conductance measurements were made on the linear transistors as a function of the gate voltage and temperature. The surface conductivity mobility/t c is related to the drain conductance gd bY 1)

Pc = (L/Z) gd/qN,

(2.2)

where (L/Z) is the channel length-to-width ratio, qN= C o ( V c - V6-r) is the induced electron charge in the channel per unit area, Co is the oxide capacitance per unit area, VC is the applied dc gate voltage, and VGT is the threshold gate voltage. Fig. 3 shows a typical family of gd versus Vc curves. For 1 < ( V G - VGT) <3 V, gd is a linear function of ( V 6 - VGT), so that the slope dgd/dV6 can be used to give #c. The corresponding induced electron density was about 1 . 0 × 1 0 H c m -2, and the surface electric field was about 2.5 × 104 V/cm. Under these weak inversion conditions, the surface channel

565

SCATTERINGOF ELECTRONS I000

. . . . .

,,,/,

,

~ ~

90C NA=2.2xI014cm-5 / ' /

80C T=25°C

,

,

/

7

,/

1

,/

,<

/ / "

oO # E 50C

2oc

,

¢o~/

,47

, /2/

Pz

4oc

#/

-12 -II -I0 -9

#/

~

#/3

3/

?/?/

-8 -7 -6 -5 -4 -3 -2 -I GATE VOLTAGE (volls)

0

I

2

5

Fig. 3. Conductance-voltage curves for various surface oxide charge densities.

remained nondegenerate, which can be proved using either a three-dimensional or two-dimensional model. Room-temperature Hall voltage measurements were made on the Hall transistors as a function of the gate voltage at B = 104 gauss. The Hall mobility was determined from #n = ( L / Z ) V , / B V s D , (2.3) where VH is the Hall voltage, and VSD is the source-to-drain voltage. We found that r = I~H/pc = 1.0 4-0.05 at room temperature. Fig. 4 shows the surface conductivity mobility at 300°K as a function of the surface oxide charge density. It shows that the mobility is independent of surface orientations, but is dependent on the surface oxide charge density. Fig. 5 is a similar plot at 80°K. It shows that the surface conductivity mobility is inversely proportional to the surface oxide charge density, indicating that scattering by surface oxide charges is the dominant relaxation mechanism at low temperatures. Fig. 6 shows the temperature dependence of pc as a function of surface oxide charge density. At low temperatures and high surface oxide charge density, where lattice scattering may be neglected, we found that/2¢ oc T.

3. Scattering by surface oxide charges There have been several theoretical treatments on ion scattering in surface inversion layers, taking into account screening by the induced carriers 17-21, 25). However, an analytic expression for the electron mobility is usually either impossible or in such a complex form that comparison with experiment is difficult. Recently Greene et al. 26) obtained expressions for

566

C . T . SAH, T. H . N I N G A N D L . L . T S C H O P P 1200

~

,

I

I

~

I

SURFACE

fill} (110) (100)

A E} 0

I]O0 - ~

I

'

I

I

CHANNEL

(1~0) (110) (110)

032670 O11970 031170-

NA= 212 × 1014cm-3 T : 25°C ;OOO

-

Oo

v°--{ 9°° 800i 7OO

6OO

5OO 0

2

4 QO + QOT + Qss

6

8

I0

IQ II

([-6m ~

Fig. 4. Surface conductivity mobility of electrons as a function of total oxide charge density at 300°K. the electrostatic potential fluctuations near a semiconductor surface due to localized surface charges, but no application to surface mobility calculation was made. In this section we shall use a somewhat different approach to derive a simple expression for the surface mobility due to scattering by surface oxide charges which is applicable to the present experimental conditions. Since intraplanar electron mobility anisotropy for a weakly inverted silicon surface with less than 5 x 101° surface states per cm 2 was observed only at T < 3 0 ° K I ° ) , it is reasonable to assume that for a high-surface-state surface, anisotropy is unimportant or removed because of the additional scattering, except may be at about liquid helium temperatures. This is consistent with the experimental observations given in figs. 4 and 5 which show no surface orientation dependence. Therefore, we shall assume that the conduction band may be described by a spherical effective-mass m*. The effective-mass Hamiltonian is then is) H =-

( h Z / 2 m *) V 2 - e 4 ( r ) ,

(3.1)

where q~(r) is the local electrostatic potential given by Poisson's equation V2(#

=

--

4xp ( r ) / e .

(3.2)

SCATTERING OF ELECTRONS

567

104 '

I

'

'

'

'1

'

I 0

Io

'

'

I

'

~ ' '-

SURFACE (100)

CHANNEL (llO)

(111)

(110)

iO s

0.5

0.5

1.0

2.0

0 0 +QoT + QSS

5.0

I0.0

I0 II

Fig. 5. Surface conductivity mobility of electrons as a function of total oxide charge density at 80°K.

Here e is the dielectric constant and p (r) is the local charge density which is the sum of the induced electron density, the depletion layer ion density, and the surface oxide charge density. It should be emphasized that if p were a function only of z (the S i - S i O 2 interface is assumed to lie in the z = 0 plane), as usually assumed in energylevel calculations, then q~ would also be a function only of z, and H would contain no perturbation in the x - y plane to cause the scattering of the electrons. This would correspond to a superlattice, in x - y dimensions, of the surface oxide charges and depletion layer ions; and it is well-known that a superlattice gives no drift velocity randomization and hence divergent mobility. Therefore, we shall reformulate the scattering problem by treating the spatial random fluctuations of the charge density as the perturbation. Let ( )av denote the average over the x - y dimensions, and let pO (z) be the charge density thus averaged, i.e. po (z) -

.v.

(3.3)

568

C.T. SAH, T.H. NING AND L.L.TSCHOPP

i

I

i

'

i

I

,

ill

N A : 2.2 X 1014cm- 3

'

I

1

't

E s = 2 . 5 x l O 4 volts/cm

qo + QOT+ QSS (crn-2)

q.

t~

4.0 x I0 I0

•

1.0 X I0 II

Z~

II e

•

•

A

Z~.A

•

A

o~

2 . 0 x I0 II

AA • X XX X

x

x

x

X ~ 0

D

× ×AA • OOOo

zL

,o~ IO

Fig. 6.

,

/

~

~

I

I

,

2O 30

9.0 x I0 II

I ~ ~ ,,I

5O

tOO T(°K)

,

I

I

2O0 300

,

5OO

Experimental surface conductivity mobility of electrons as a function of temperature.

Then the charge density fluctuation is 6p (r) = p (r) - pO ( z ) .

(3.4)

If tk°(z) and @~(r) are the electrostatic potentials due to p ° ( z ) and 6 p ( r ) respectively, then (3.1) may be rewritten as H = H ° + H',

(3.5a)

H ° = - (h2/2m *) V 2 - eq~° ( z ) ,

(3.5b)

H ' = -- e a4, ( r ) .

(3.5c)

where and H ° determines the average properties of the electronic states in the channel, but it is H' that scatters electrons in the x - y directions. There are two contributions to @ (r), and hence to H'. First, there is the spatial density fluctuations of the depletion layer ions and the surface

569

SCATTERING OF ELECTRONS

oxide charges; and second, there is the induced electron charge density due to these fluctuations (screening effects). For a weakly inverted and nondegenerate surface, we shall neglect screening. Furthermore, since scattering by surface oxide charges is the dominant mechanism, we shall neglect scattering by the depletion layer ions. With these assumptions, then (.) = e [., (.) - N, (z)],

(3.6)

where n~(r) is the local surface oxide and surface state charge density and N~ (z) is the density averaged over the x-y plane. Let AV/be an elementary volume located at r=R~= (Pl, z~), then the perturbation corresponding to (3.6) may be written as 27)

H' (r)

=

-

(e/O Y, zxv,

( R,)II

- R,I

i

(3.7) = -- (e/~,) ~ A Q i / l r -- R , I , i

where g=½(esi+eox ), AQi=AVI6p(RI), and the sum is over all of the elementary volumes. For an inverted surface in the classical approximation, the induced electrons are concentrated near the interface12,28). If we assume the local electron density to be proportional to e x p ( - e E s z / k B T ), where E s is the surface electric field and eEsz is the electron potential energy, then for the present experimental conditions, where Es~-2.5 × 104 V/cm, most of the electrons are localized within a distance of XcH from the interface, where XcH-~ 10 A at 30°K and XcH_~ 100 A at 300°K. Now the average thermal energy of an electron in the surface channel is kBT, so that its thermal wave length is

~',h = 2~/k = 2re (h2/2m*kBT) ~ . Using m* = 0.323 mo, we obtained 2th (30 ° K ) = 424 A and 2th (300 °K) = 134A. That is, 2th~>XcH, SO that the motion of the electrons perpendicular to the surface may be neglected. Therefore, we shall assume that the induced electrons form a two-dimensional gas at the interface, so that the matrix element between two plane-wave states [k> and Ik'> is

= - (e/g) ( 2 ~ / A q ) ~ A Q i e x p [ -

qlzi[ + i p i ' q ] ,

(3.8)

i

where q = ( k ' - k ) , and A is the area of the surface. Now, for a two-dimensional r a n d o m Poisson or Gaussian distribution of the surface oxide and surface state charges, we have
and

<(AQi)Z)av = e2Nl(zl) AVi,

570

c . T . S A H , T. H . N I N G A N D L. L. T S C H O P P

so that <]
= (e2/g) 2 (2n/Aq) 2 ~, N, (zl) AV/exp [ - 2q IzJ] i

(e~l~) ~ (2n/q) 2 (l/A) f N~ (z) exp [ - 2q Izl] dz.

(3.9)

The transition rate is given by the golden rule as

Ck, k, = (2n/h) (l(t,'l H ' Ik)12),v a (Ek -- E~,),

(3.10)

and the momentum relaxation time z is given by

1/~ = y, rk~, (1 - cos 0),

(3.1 l)

k'

where 0 is the angle between k and k', and the sum is over the two-dimensional wave-vector space. Substituting (3.9) and (3.10) into (3.11), we obtain

l/~ (e) = (2~/h) (e2/~) 2 (1/e) ( NI (z) ~ (e, z ) d z ,

(3.12)

where E = Ek= hZ k2 /2m *, and

G(E, z) - t- e x p [ - 4kiz[ sin qS] dq~.

(3.13)

I/

0

The electron mobility is given by

tq = (e/m,,) ( E r ) / ( E ) ,

(3.14)

where m s is the mobility effective-mass and (A)-

t'Aexp(-E/k"T)

dE

(3.15)

0

is the average value of A for a two-dimensional nondegenerate gas. We have evaluated/~ for three assumed distributions of oxide charges:

(i)

N,(z) = N I '0(Z)

;

(ii) N l ( z ) = ( N J ~ ) exp (--z/s); (iii) Nl(z ) = (Niz/fl 2) exp ( - z/fl) ; where z is measured from the interface into the oxide. The results are plotted in fig. 7. Case (i) is of particular interest. It corresponds to the case where all the surface oxide charges lie in the oxide at the oxide-semiconductor interface. The mobility in this case takes a simple form of I~ = (e/mu) (~/e2) 2 (2hk.T/n2NO • (3.16)

SCATTERINGOFELECTRONS 103

I

I

~

....

I

~

~ ~ ~ I

571

~

I

(2) (I)

.....

I./

.,'

/"

J ~

/'--'~74

.,,./4

/

/

_%/ Nz = 9.0 x IOllcm-2 :

2

~Oz'


m*: 0.323 % ....

/'

m/z =U.tz)~m o

/

/q

/'/

•v," ,,q,.. /

:8.o

,,

/

//

~"./,,L<-,; / ~/.',; ~ / ~

.

.. / ¢ ,

/-.:

,C;U< /

,.3

E i "> o 102 v

..

/ # / /y

1

I

/ -I |

/

i

i

I.I //.7 iI

5

/./

I0

II

A 20

I /2 / I //./// ,j,.~///

I 30

X

/

I

50

~ k

~1 I00 T(°K)

z/

"~,u.:wT

~

I 200

I 500

500

Fig. 7. Theoretical two-dimensional electron mobility due to scattering by surface oxide charges for three different charge distributions. (1) Ni(z)~ Nil(z); (2) Ni(z)= - (NUa) exp ( -- z/a); (3) N1 (z) -- (Niz/• 2) exp ( -- z/fl).

The same result could also be obtained from the two-dimensional scattering cross-section in the Born approximation given by Stern and Howard:8). The dependence p ~ o z T / N i agrees with the experimental observations (figs. 5 and 6). However, using the bulk values of mu=0.259 mo and ~=8.0, we found that p~ given by (3.16) is about 20 times smaller than the observed values at low temperatures. The discrepancy could be due to one or all of the following factors. (i) There is some screening effect. (ii) The surface oxide charge has a finite distribution into the oxide layer (see fig. 7). (iii) Surface layers of Si and SiO2 may be more polarizable than bulk Si and bulk SiO2, so that the effective value of ~ is larger than its bulk value. (iv) The distribution of the surface oxide charges may be correlated instead of random. Correlation has the effect of reducing scattering and hence increasing the mobility which will be discussed in detail in a future paper. It appears that (ii) and (iv) are the important factors. We note that eq. (3.16) differs from the two-dimensional Rutherford formula 8) of fq = (3 x/2/4\/~) (g/e) ~/(kBT/m*)/N,,

572

C . T . SAH, T . H . N I N G

AND L.L. TSCHOPP

which is inversely proportional to the coupling constant, (e2/g), and has a x/T temperature dependence, while (3.16) gives p~-loc(eZ/~)z and has a T dependence.

4. Scattering by surface phonons At high temperatures and/or for low surface oxide charge densities, scattering by lattice vibrations cannot be neglected. This is evidenced by the measured mobility deviating from its linear dependence on T (fig. 6). A n estimate of lattice scattering mobility, /~L, may be extracted from experimental data by using the approximation 29). 1/pL = l/,Uexp- l//ui.

(4.1)

Here Pi is assumed to be proportional to TIN I and fitted to the low-temperature data. The resulting #L is shown in fig. 8. The theoretical curve, which we shall derive below, is based on a two-dimensional model. For T < 100°K,

105~_

I

'

I i i bi l

i

I

I

/

I ( Q o + Q +Qss )

•

I x I011 crn-2

×

2 x IOIIcm~2

D

4 x l O I I c m~2

q_

9 x I011 crn-2

5 ~

I

\

4 x IolOcrn-2

~.

-

~

Q

5

+ °~ J,

2

°

103 I

20

50

,

I

50

i i i i I

I00 T(°K)

i

I

200

iX

500

Fig. 8. Surface phonon scattering mobility of electrons as a function of temperature. The theoretical curve is given by eq. (4.8) of text, with Pae = 7.4 × lO~/Tcm2/V-sec, he)a/k~ = 650°K, and ZR/ZA = 2.3.

SCATTERINGOF ELECTRONS

573

where only surface acoustical phonons are important, the data can be fitted to pL=7.4xlOS/TcmZ/V-sec. For T>150°K, it can be fitted to PL= 108/T2 cm2/V-sec. The additional high-temperature mobility drop can be interpreted in terms of surface optical or intervalley phonon scattering. In the two-dimensional deformation potential theory of surface phonon scattering, the electron-phonon coupling matrix elements have the same form as that of the three-dimensional bulk case. Thus, for scattering by surface acoustical phonons we have 3°) I(k'i Hac Ik _ q ) [ 2 = (ZEhe)q/ZpAu 2) (Nq q- ½ ~ ½) (~k',k±q,

(4.2)

where he)q is the acoustical phonon energy with_ wave vector q, Nq is the phonon occupation number, p is the areal mass density of the semiconductor sheet, ut is the longitudinal sound velocity, A is the area of the surface, and Z A is the acoustical surface deformation potential in eV. Eq. (4.2) is identical to the form used by Kawaji 2a). The relaxation time and the mobility derived from (4.2) are 2a) l/'rac = (m*Z2kBT/h3pu2), (4.3) and ,uac = (e/m,) (Ezac)/(E) = (eh3pu~/m*muZ2kBT), (4.4) where ( ) has the same meaning as in (3.15). Thus, at low temperatures the mobility due to surface acoustical phonon scattering is proportional to T-1. It is noted that (4.4) also holds, within a factor of order unity, for scattering in a three-dimensional thin film 22). At higher temperatures, optical or intervalley phonons cannot be neglected. The corresponding matrix elements may be written as 31) I(k'l Hop rk _+ q)l 2 = (Z~he)R/2pAu~) (NR + ½ -T- ½) 6k, k±q,

(4.5)

where he)R is the phonon energy which is assumed to be independent to q, NR is the phonon occupation number, and Z R is the electron-optical or intervalley phonon coupling constant in eV. The relaxation time is

l/zop = (m*Z2he)R/2h3pu 2) INR + (NR + 1) u (E - he)R)],

(4.6)

where u(x) is the unit step function, u ( x < 0 ) = 0 and u ( x > 0 ) = 1. For combined acoustical and optical phonon scattering, the relaxation time is given by a°) 1/'CL = 1/Za¢ + l / t o p = (1/rat) {I + (Za/ZA) 2 (he)R/2kaT) [Na + (NR + l) U (E -- he)R)]}. (4.7) The mobility for this two-dimensional case can be obtained explicitly since

574

C . T . SAH, T . H. N1NG AND L. L. TSCHOP P

the density of state is independent of energy. This is given by

PL = (e/m,) / = /~a¢

1 -- [1 + X0R]exp(-- X0R) [1 + XR °] exp(-- xR°) -1 +--I~[Z,/ZA] - - - - ~ 5 . ~ 5x,N, ~ 7 + 1 + ½[ZR/ZA] 2 XOR[2NR + i] f '

(4.8)

where x°=he)R/kB T. Using (4.8), p.~=7.4x 105/T cm2/V-sec, he)R/ka= =650+50°K32), and ZR/ZA =-2.3, a good fit to the experimental data was obtained as shown in fig. 8. The physical significance of this good fit may be explored by comparing the theoretical P,c of (4.4) with the low temperature data of pat=7.4 X 105/TcmZ/V-sec. Using the bulk values of m*=0.323 too, m=0.259 mo, u t = 9 . 0 x 105 cm/sec, and ZA=7.65 eVa3), we obtain p = (30 A) x (2.33 g/ /cm 3 )= (30 A) x Pb,~k, where /)bulk=2.33 g/cm 3 is the mass density of bulk silicon. This gives an effective thickness of the surface layer of 30 A, which is similar to the thickness of the electron layer of Xcn=kBT/eEs= 10-100 A. This agreement further substantiates the two-dimensional model of electronacoustical phonon interaction involving surface sound waves. 5. Discussion and conclusions

We have demonstrated that on a weakly inverted silicon surface covered with thermally grown oxide, electron mobility is determined mainly by surface oxide charge scattering and by surface phonon scattering. These scattering mechanisms can be described by the classical two-dimensional models. The effects of the surface oxide charge should be less important for a heavily inverted and degenerate surface because of screening, but then the surface mobility becomes relatively independent of temperature, giving us less information on the scattering mechanisms. Our analysis of ion scattering emphasizes the spatial density fluctuation of the scattering centers. The two-dimensional theory accounts well for the temperature and density dependences of the surface conductivity mobility. Such spatial density fluctuations would also cause fluctuations in the surface potential and the quantum energy levels in the surface channel which will be presented in a future publication. We have extended the two-dimensional theory of surface acoustical phonon scattering to include optical or intervalley phonons. Comparison with experiment indicates that for T> 150°K, the effects of the optical or intervalley surface phonons are important.

SCATTERING OFELECTRONS

575

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