Frequency dependence of the photo-EMF of strongly inverted Ge and Si MIS structures—I. Theory

FREQUENCY DEPENDENCE OF THE PHOTO-EMF OF STRONGLY INVERTED Ge AND Si MIS STRUCTURES-I. THEORY R. S. NAKHMANSON Instituteof Semiconductor Physics. Novosibirsk.

U.S.S.R.

(Receioed RJuly 1974) Abstract-The dependence of the real and imaginary photo-EMF components on frequency of the incident light modulati,m has been calculated for Ge and Si MIS structure in the inversion regime at the interface. It is shown that physical parameters of a semiconductor, such as doping levels in the bulk and near the interface, the rate of surface recombination proceeding at the backside of the sample. diffusion coefficient of minority carriers and light absorption coefficient can be derived from the experimental results.

I

NOTATION E dielectric E,, n A A,

8.86~

constant IO-l4 Fcm-’

r

I,,,

C,Kl(C 9 1)/C, n,In =p/ni doping parameter semiconductor effective A value near the interface

O<.u
of the semiconductor

bulk lifetime

of

.I M

in the region

N n(n)

of the semiconductor

p(p.) q R,

(fl cm)

(set)

R

oW

A

(EEQ)’

Maxwell

(set-‘) relaxation

S s frequency

(sect’)

q2n,/kT (F cm-‘)

/.I x4wN) C,, capacitance ity carriers

I&

the

CJG, (set) effective bulk lifetime near the interface in the region 0 < x < d at zero band bending (set)

‘ID/X2 (set-‘) w angular frequency

I&and

7’

f of the quasi-neutral

r X .Y

region for minor-

(F cm- ‘) C,, C,,(W= 0) (F cm -‘) COn C,(w 9 7-l i 0) (F cm-‘) C, capacitance of the space-charge region (Fcm-‘) C, capacitance of the insulator layer (F cm-‘)

Y,,

absorption

depth

of light

in the semiconductor

(cm) effective values of I estimated by different methods (see section 5) (cm) metal v’(l + iw) number of electron-hole pairs generated by light per unit time (cm-? set-‘) bulk (surface) electron concentration (cm-‘) bulk (surface) hole concentration (cm-‘) elementary charge (= I.6 x IO I9 C) series resistance of the bulk of semiconductor and contacts (it may also include the insulator losses) (0 cm’) semiconductor surface recombination velocity on the backside of the semiconductor wafer (cm set ‘) absolute temperature (“K) time (set) \/CD/T) (cm set ‘) semiconductor wafer thickness (cm) coordinate (cm): semiconductor wafer thickness (cm) admittance of the quasi-neutral region for minority carriers (R ’ cm ‘)

D diffusion d E Go G,, G, G, I I

coefficient of minoritv carriers (cm* V secC’) width of the space-charge region (cm) photo-EMF (V) conductance of the quasi-neutral region for minority carriers (0-l cm-*) G,,(w = 0) (a-’ cm-‘) G,(w %r-‘+fl) (a-’ cm-‘) recombination conductance of the space-charge region (R-’ cm- ‘) insulator photocurrent (A cm-‘)

1. INTRODUCTION

Measurements of the photo-EMF provide a possibility to derive some information about physical processes occurring in the MIS structures. The proposed method requires more complex apparatus as compared to the widely adopted admittance (capacitance and conductance) measurements. But when one measures photo-EMF, the requirements for specific insulator capacitance, series resistance in the bulk and contacts to a semiconductor are immediately reduced. It is possible, in particular, to make measurements using air gap and on heavily doped semiconductors as well. Moreover, there appears an additional possibility to measure with various spectral compositions of the incident light.

i xi-l)

k Y L L,

Boltzmann’s constant (= I ,38 x 10mZ2 J”K ‘) light effective depth of diffusion of carriers from surface (cm) intrinsic Debye length (cm)

the

617

61X

R. S. NAKH~I~NVI~

Maximum

information

may be extracted

in the same

corresponds

to strong inversion.

i.e. when the condition

way as when measuring admittance if one investigates in a

p$ > II is fulfilled for the n-type. while the condition

wide

is fulfilled for the p-type semiconductor

frequency

range

covering

the

main

transient

n, > p

(p\ and II, are the

processes at the surface and in the bulk of semiconductor

surface concentrations).

and also measures both the real and imaginary

is small compared to the thickness of the semiconductor

signal

wafer

components. This paper presents strongly ments

inverted

a theory

MIS

of the photo-EMF

structures.

Methods

and some of the experimental

will

of

carriers

be

considered for Ge and Si MIS structures in a subsequent

T i\ the bulk lifetime).

\ (D/w).

The condition

OJ under discussion

seems to be much more simple for analysis because it

condition

(0

eliminates the effect of the surface states. It would forstal

relaxation

frequency.

inversion,

band bending

study.

Moreover,

value

is maximum,

the photo-EMF

The

excitation) MOUEI.

is

I. The model

measured

semiconductor.

is one-dimensional.

from

the

interface

case

semi-transparent

w reaches semiconductor

metal

electrode

is sufficiently

semiconductor equilibrium.

so that

across a

non-degenerate

wafer

of

deviation

from

Ge or Si of distinct

10

I is to be

this

restriction

(bulk

fabricated

may be represented

by a

circuit as shown in Fig. 21 I. 21. Here (

is the insulator capacitance. C, and G, are the capacitance

the quasi-neutral include

conductance region

of

the

in the bulk.

the insulator

photocurrent

3pace charge

II- or

ha5 both

consideration) frequency

and contacts (it

losses as well):

I

iq the

pairs generated by the light per unit time.

The current

;I

dirccc component

and alternating

(omitted

component

with

from

angular

(I). The voltage on the free terminals of the MIS

E will be the photo-EMF

structure

The band bending at the interface

of

R, is the eerie\

source: I = qN. where N is the number of

electron-hole of

p or p > II; n and p are the bulk electron and

hole concentrations).

to

the Maxwell

the MIS structure

semiconductor

and recombination

may

the state of the is composed

i\

resistance of the bulk of semiconductor

pairs. The light

is, on the average, close to thermodynamic The semiconductor

homogeneous p-type (n >

low,

corresponds

’

CL),, = (tt,,/~)

region. C,, and G,, are the capacitance and conductance

,bI and transparent

insulator layer I generating electron-hole intensity

the

The light .Y modulated over intensity with

the angular frequency

tend\

will be discussed in section 5.

simple equivalent

The

into

of minority

depth of the light absorption

on an rl-type

A physical model of the MIS structure is schematically in Fig.

coeficient

At ~7 ‘a I(f)

and approximately

Under these restrictions.

shown

11.1: 1. :

regarded much lower than X and II,1 (surface excitation).

which

sitive cells.

distance

depth of

surface

tl c \ (D/COI restricts the range of

CO,,.where

The effective

at strong

seems interesting when using MIS structures as photosen-

2. PHYSICAL

the

carriers.

paper. The case of strong inversion regime at the interface

an arbitrary

from

\ [ 07/( I + iw~)] (n is the diffusion

of measure-

results

X and to the modulus of the effective

diffusion

of

The space charge region width d

sought. In

21 general

case E is delayed against I over the phase, i.e. in the case one considers exp f-

I and E to be the coefficient5

and ImE c 0.

i;‘l

A simple photo-EMF w

equivalent

d

1

ILI

X 0

Fig. I. Metal-insul;ltor-semiconductor \tructure under illumirxtion. 2’. light modulated by intensity with angular frequency w: ./1. semi-transparent metal electrode: I. transparent insulator layer: S. semiconductor: cL space charge region width: 1. effective depth of light absorption: L. effective depth of diffusion of carrier\ generated near the \enliconduct0r-ln~ulator interface: A’. semiconductor wafer thickne\\. *Each of the following contributes to it: the contact to a more heavily doped semiconductor of the same type of conductivity or MIS contact with a zero or accumulated band bending; high safe1 thickness; high measurement frequency; high recombination velocity at the contact; large area of the right-hand contact (compared to the area of semi-transparent metal electrode). For proofs. see ref. [7]. iBelow we \hnll u\e the umt \&ace area in the calculations throughout.

circuit

of

Fig.

1 reflects

the

resulting from the separation of electrons and

holes at the inversion 0 i)

of the

(id ). and I to be real and positive. for (I) 2 0. ReE :z 0

barrier of the MIS structure

in the

left-hand

side of the sample of Fig. I. In a general case.

the EMF

may also appear on the right-hand

side of the

sample but it will be regarded as negligibly

small. Thi$

condition

may be readily ensured in practice.*

because of the difference

between

Moreover.

the diffusion

coeffi-

cients. electrons and holes ma) separate in the bulk of the sample. thus creating the so-called ‘Dember EMF’. small excitation intrin4c

semiconductors

only.

and may be taken

:tc
circuit

onI) to

of the source of the photo-

w they can be neglected when calculating author\

for minori(\

for Y,,

in into

1I].

(‘, and R, gibe the contribution impedance

The rxprej\ion

But at

levels the Dember EMF ij important

E.

G,,+ hC,, has been derived by

from the solution of ;I diffui;ion

carriers IWK. foi- c\amplr.[!])::

equation

Photo-EMF

,=G,+i~C”=AAuM~~“,i~~l’+“s’sf:1~~:

619

of MIS structures-I

(2) 5 6 1. ‘Thin’ wafer. Conditions of recombination at the right-hand boundary are essential; & ti I. By putting sh[ = 5 + t3/6 and ch[ = 1 + 5’/2 in equations (2) and (3),

(1)

Here A = M = d/(1 + km), s is the surface recombination velocity at the right-hand boundary of the sample, X is the semiconductor wafer thickness.

5’+s$ G,=Ah$

++s

-=AA-

(8)

At WTe 1

(9)

D 7 v’(sh[ch[ t 5) t 2vssh2[ + s’(sh[ch[ (vch[ + ssh<)2

- 5)

AAJ(T)?

Here and below by Coo and C, low-frequency limits of Go and C,. At WT 9 1

we

.

Here Go and C, approach G, and C, only at w 2 fi (w 2 2tm2). The effect of the bulk recombination in this case is IdatiVely unimportant. Thus 7 is not contained in the expression for C, at all. Equation (5) gives the correct value of C, in the whole frequency range 0 s w e wM; in particular, as w -+O it reduces to equation (9). Equation (4)

(3)

denote

the

b’(sh26-sin2b)+bs$(ch2btcos2b)ti G,,= AAd(p)

(4) b’(ch26 t cos 26) t bsg(sh2b

t sin 2b) t i

b2(sh2b+sin2b)+bs$(ch2b-cos2b)ti C, = AA J(F)

(5) b2(ch2b+cos2b)tbs~(sh2btsin2b)t~

At

~0781

and wPfl

Y0 = AA d/(iDw),

gives the correct value of Go in the whole frequency range as well, if one adds AAX/T to the right-hand side of this equation. (2a) s % D/X. High surface recombination velocity.

(oJTB~~~-~)

Go = GO, = WC”= WC& =AA

J(

s.

(6)

>

Here and below by Cm and C,= we denote the high-frequency limits of Go and Co. Let us consider some of the particular cases allowing simplification of equation (1): (1) 5 % 1. ‘Thick’ wafer. Conditions of recombination at the right-hand boundary are not essential; a27 < 1.

Yo=AA J(f)

d/(1 + iam),

Go = AA J(g) G,=AA

g[q(l

J(

$,

>

(2b) s < D/X. Medium and low surface recombination velocities.

t

C02T2) t

Cm=AAF.

11,

do

=

AA@)\/[\/(l

t

02T2)-

11;

(7)

620

R. S. NAKHMAMJN

I”

A

L-_ w

w”

,‘-

/

/’

I/?

,’

,,’

20

I

I--

c 3 ^ IO0 VI?

,,

,l

I,

/’

,

,

,

,,;+

/’

lC2

Fig. -1. Equivalent circurt of MIS \trucfure under strong inversion C,. insulator capacitance; C, and G,. capacitance and recombination conductance of the space charge region; C,, and G,,. capacitance and conductance of the quasi-neutral region: R,. series resistance of rhe semiconductor bulk and contacts (losses in the insulator included here, if necessary); L. photocurrent source: E, photo-EMF.

/’

,’

,,’

,’

’

,’

,,

,

,’

,

_*’

,’

/’

,:’

w

2

,s ,’

,’

’ ,“W

,,’

,/,

L ~~_~ ~~~

n

Fig. 3. G,, (full line\) and WC,, (dashed line+) as function\ of CO:. Relative unit?. (I) (B I; (21) [=@I: .SBD/X: 1%) f -0.1. s 4 X/T.

and similar capture cross-sections for electrons and hole\ are responsible for the recombination region. Reference

in the space charge

[2] gives more general expressions

for

G, and T,, which in turn. may be extended to the cases of G,, = AA

sh?b-sin2h

+“+

ch2hicos2b

7

s

several recombination

’I ’

complications

levels or continuum.

are not of principal

The simplest assumption fabricating

is A, = A and T, = T. Rut in

the MIS structure, the region of a xemiconduc-

tor near the surface is frequently wc,,

=

AA

But all theje

importance.

enriched in defects. and

7, + 7 [h, 71. A long-time high-temperature result in a redistribution

of impurities

treatment

ma)

and change in the

doping level near the surface. Depending on the type of impurity

A, > A or A,
Using equations (h), (7). (IO) and relation

at g I)

_

c,

we can write the

C,,, =cI

At room temperature while

( I I)

between G,, and G, and between C,, and c’, as:

for

7, = 7 and d/(07) = IO formulas

’ for usual Ge samples.

A = IO-‘-IO

usual Si samples A = IO “-10 ‘. For

‘cm it follows

that for germanium

from

Go 5 G,, while C,,,, and C,

may be values of the same order of magnitude. decreases with frequency. cies.

For

quantities

silicon

Since C,,

C,)@ C’, at very high frequen-

C,,< C,, while

of the same order

increases with frequency,

A, -c A.

the above

G,, and G,

of magnitude.

may

be

Since G,,

Go+ G, at very high frequen-

cies. If T\ < T, then for germanium G, may he comparable to Go at low frequencies,

while G, % G,, for silicon up to very

high frequencies. In the p-type semiconductor

A ’ in the formulas

A would be replaced bq

of this paper, while

in Fig. 2 the

621

Photo-EMF of MIS structures--l and photo-EMF the reverse.

directions would be changed to

current

3.FREQUENCY DEPENDENCES OF PHOTO-EMF

According to the circuit of Fig. 2 the real and imaginary parts of the photo-EMF are:

Go+ G,

Re’ ='(G,, +G,)'+

(12)

w*(C,,+ C,)” CJ

-w(co+

ImE=r(G,,+G,)2+~*(C,i+C,)2'

(13)

At very low frequencies:

Re E = I&

00

Intersection of plot log (Re E. (Gal+ G,)/(C,HIf At very high c,, = c,,, e c,,

I

x coo, Im E = I

the low-frequency -Im

E)/log

C,). frequencies

-w(G+cI) (Gta+ G,)*

x wI ’ (14)

Fig. 4. Re E (full lines) and -1m E (dashed lines) as functions of ~7 at 119 I, qt 9 I and various 5 and s. The MIS structure on Ge. relative units. For explanations, see Table I.

asymptotes on the when w = w2 =

w occurs

when

Ge = Goi+ GI and

Intersection of the high-frequency asymptotes on the plot log (Re E, -Im E)/log w occurs at w = wj defined by: G,Jw,) = WC,, from which on account of equations (6) and (IO) one has:

From equations (6) and (7) one can deduce the following relation for wl: C&w,) = c,,

w37 = 2$,

Tj = Q

%$I

Fig. 5. Re E (full lines) and -1m E (dashed lines) as functions of wr at 7 e 1 and different 5 and s. MIS structure on Ge, relative units. For explanations, see Table I. This figure presents the frequency dependences of the photo-EMF for the MIS structure

on Si as well.

(17)

CW* C, (n = -\/S x 10’ % 1, to condition C,< C, (7 = 0.1 < 1). In the former case inequality 75 * 1 is required for condition C, ti C, to be fulfilled when 5 4 I. The curves are listed in Table 1; the exponents w and intervals OT corresponding to the straight linear parts of the log-log plot are given in the same Table. The last column of Table I gives w+values. For usual Si samples with A = 10-4-10-6 C,< C,, and equations (12) and (13) can be simplified. Since this corresponds to condition n
The behaviour of the photo-EMF in the intermediate frequency region is complex. Below we shall consider some particular cases of frequency dependences of the photo-EMF for Ge and Si samples. As mentioned at the end of Section 2 for usual Ge samples with A = 10~‘-10-2, Go % G, when 7, = T, and equations (12) and (13) can be simplified. Re E and Im E depend on the values & s (through Go and C,) and also on the relation between Cm and Cl. Figures 4 and 5 show particular shapes of the frequency dependences of Re E and Im E for Cases (1) (5 9 I), (2a) (5 = O.l+ I, s 9 D/X) and (2b) (5 = 0.1 < 1, s 4 X/T) of section 2. Figure 4

to condition

75 = q/s x 10 % I), while Fig. 5 corresponds

622

R. S.

NAKHMANSON

Table I. Intervals of Re E and Im E power dependences on o. Germanium and silicon MIS structures, “surface” excitation Re E 5 and s

rl

Interval ~7

Im E Exp

Interval ~7

Exp

Case (I) 59 I

Case (?a) (
-1

Case (2b) (Gl S
0 -I/? -3/2

Case (I) tBl

0

Case (2a) f
-2 -312 0 -2

Case (2b) f
0

~312

In particular, this leads to an increase of the frequency w?. In a limiting case G, > Go,,. which is typical of room temperature, w2 = GJC, = 7,-‘. The absolute wz-values for Si samples are naturally much smaller than for Ge samples

because 4.

of too small

A.

DETERMINATION OF PHYSICAL PARAMETERS OF THE MIS STRUCTURES

In this section we will try to answer the question, how one can determine some characteristic physical parameters of the MIS structure semiconductor from the photo-EMF dependence on frequency modulation. If Re E and Im E are known from the experiment, Go+ G, and w(CU+ C,) can be calculated for each frequency o from the formulas which are the reverse form of equations (12) and (13):

“+

IReE G’ = (Re E)‘+ (Im E)"

w(c0+ C,) =

-1ImE (Re E)* + (Im E)*'

(18)

*Calibration of I may be performed, for example, using simultaneous photo-EMF and capacitance measurements when the latter offers no difficulties. For a Ge sample by putting A = A, the absolute I-value can be determined from the photo-EMF measurements (see subsequent paper).

Since Go = G,,, x w”’ and C,, = C,, J We“’ at high frequencies, and G, and C, are frequency-independent, one can determine G,, Go(w), C, and C,,(O), separately, from the results of measurements of Re E and Im E over a rather wide frequency range. Equations (18) contain the absolute photocurrent value I (A/cm’). If this value is known,* one can find A. A,, r, and T, from equations (6) (10) and (11). However, u priori determination of the absolute I-values may turn out to be complicated, perhaps, because of the lack of data concerning transmittance and reflectance of light by metal and insulator layers. In this case one can determine, firstly, G,,(w). C,,(w), G, and C, values in relative units, while the absolute 7, value can be so far determined from equation (11). If A is a priori known or A = A, with sufficient probability, then on finding wX from the experimental results and condition C&W~) = CI, one can determine A, from equation (16) and 7, from equations (II). The parameters 7 and s are relevant to low frequency Go and Co-values; they can be determined from relative values Goa, CL,, Go= and C,,= only, even if I and A are not known. In view of equation (6) equations (2) and (3) can be rewritten as: GM _ vshcf + schc$ vch.$ ssh<' Go?(0)

’

f

623

Photo-EMFof MIS structures-l

coo 1 u*(sh@h[ + 5) + 2vssh*.$ + s’(sh[cht -=_ (uch[ + ssh[)* Co=(~) 5

- 5)

,
=2%<3.

(20) By solving equation (19) with respect to s (GdGo4R))chS - .$sht ’ = ’ [ch[ - (G,/G&))sh<’

(21)

and substituting equation (21) in (20), one transcendental equation for the definition of 5:

gets

a

Inequality (27) is related to the interval of possible s-values 0 s s < x. On determining S, from equation (24) one finds 5’ and 7. (2a) s + D/X. High surface recombination velocity. Equations (24) and (25) reduce to:

AL=, G,&)

’ >=O.

2 CW ?i??Qij=

fin7 J( T. >

J( >

(23)

From equations (23) and (6) one has:

-=- p+sg

(24)

( 1 ( >

1+sgtj

D~

2

.

(25)

1ts;

X

-I+

&=2.

(29)

In this case r and s cannot be defined individually. One can find from equations (29) the effective lifetime ? only:

T

X

Go=(Q 2

G C,x,’

The last of the equations (29) could be also used for a more accurate definition of D or X. If the Con-value cannot be found experimentally on account of condition C,+ C,, one may try to use the Go value for intermediate frequencies besides Gu, for the definition of r and s. Thus at .$ < 1(Case (2)) equations (4) and (6) for b = 1 imply: 2

Go@) -= Go=(R)

2.718+3.346~$+2,268 2’ 3.346+4536s$+2.089

This formula may be used instead of equation (25).

5.BULK EXCITATION

x* so

From the last equation one finds s:

s=-

__&=~‘+s~,

1,s:’ Xl

c,m

(28)

The values r and s fall out of the equations and cannot be However, relation (28) may be used for determination of R and then for a more accurate definition of X or D, if necessary. (2b) s
?

Each of these equations can be used for determination of 7. The knowledge of the exact &value is not necessary because equations (23) are independent of a. The value s in this case plays no role, and cannot be defined. We note that condition G,/G,% = C,,/C, is not sufficient for the Case (1) and relations (23) to be fulfilled. It only implies 5 > 1.2. (2) 5 < 1. ‘Thin’ wafer. Conditions of recombination at the right-hand boundary are important. Using equations (8) and (9) one obtains:

c,4

3’

Coo R _!,l+S=__=L!

Goo 2 c,,=;.

GO0 Go=@)

2

(22) defined.

On solving the last equation we find & 7 = 2/fl[’ and s from equation (21). Let us consider the particular cases, which admit simplifications: (1) 5 9 1. ‘Thick’ wafer. In this case equations (19) and (20) reduce to: GW m=

-=-Gil ’ c”,(n)

J[$-u)]

(26)

We have so far assumed that the light is absorbed in a thin layer near the interface, i.e. the effective absorption depth I is much smaller than X or L. If this condition is not fulfilled, then at the same I-value the photo-EMF would be smaller in amplitude and different in phase. These differences are due to the fact that only some of the generated carriers would reach the space charge region, especially at high light modulation frequencies. and the moment when carriers appear in the space charge region will be delayed in time against the moment of their

R. S.

624

NAKHMANSON

generation. Solution of this problem in a general form was given in ref. [7]. In practice. one often has I e X. In this case the equivalent circuit of Fig. 2 still remains valid when instead of I one uses the “effective” current[7]: I Icn = 1t(//L)’

D I-’

I I

1(l) dl I([) Idl

Re E(o,J = 0.

(35)

At I

% I and usually fulfilled relation 7, 9 7. from equations (35) and (33) one has:

If the numerical value of the sum in parentheses of equation (36) is greater than unity, cos and arc cos must be replaced by ch and arc ch. respectively. The o,,-value can be greater or smaller than 7 ~‘. At

and at (31) I v’(h)

I<-

Equation (30) has been derived for a monochromatic light with characteristic depth of absorption 1. If the incident light is non-monochromatic, then to derive the expression IeRit is necessary to substitute the spectral density I(I) in the right-hand side of equation (30) instead of I and then integrate over the interval of possible l-values. This complication, however, is not of principal importance. Thus in equation (31) instead of I one has:

I,tt =

of the sign in Re E takes place at w = w,)

(30)

i.e. Irff decreases as &“* and is delayed in the phase against I by 45”. The appropriate change will also be observed in the photo-EMF. Thus if at w then at w > D/l’ 1El m wfnm”‘. In the log IE ]/log w plot intersection of continuations of the straight lines with the slopes m and m - I/? takes place at: w=(r),=7

The change

defined by:

T,

In Fig. 6 plotted are w-dependences of Re E and Im E for typical values 7 = IO-” set and 7, = IO-‘set and I// = 10. Table 2 gives the frequency intervals within which Re E and Im E have simple power dependences on w. For w > wi

Table 2. Intervals

(32)

Let us consider in greater detail a usual case of MIS structure on n-type Si with A = 10~J-lO~” at room temperature, when 7, < 7, so that in the whole frequency range under investigation G, 9 G,, and C’ % C,,. We assume I < X. From Fig. 2 on account of equations (I I) and (30) one has:

of Re E and Im E power dependences on w.

Silicon MIS structure, “bulk” excitation Im E

Re E Interval w

0.7, ’ 71 ‘,(w:, “T)

P w,,t7 co, (0,.w,w

Exponent

’

0 -? -I/? -312

Interval CC)

0.7, I 71 I. WI (0,.W&f

Exponent

I I -312

E=

(33)

ImE'

Photo-EMFof

625

MIS structure\-1

where CO?is the frequency at which the straight lines Re E x w “’ and ImE ^* We’ intersect in the log (Re E. Im E)/log w plot. For the monochromatic light wq = 2w,. The light absorption in a semiconductor is usually determined from the measurements of intensity of light passing through a thin semiconductor wafer. In this case, if the incident light is non-monochromatic, the effective absorption depth established experimentally is given by: Y

/:fl =

In w,

rod / set

Fig. 6. Re E (full strong lines) and Im E (dashed strong lines) as

I

I(/) dl -In

I

I(1) exp (-x/l)

d/’

where x is wafer thickness. It may be shown that left, I&, and IL are linked by the following relations:

functions of o at the “bulk” excitation. The MIS structure on Si. relative units. Weak lines stand for Re E and Im E at the “surface” excitation.

From the preceding sections and directly from equations (33) and (34) and Fig. 6 it follows that 7, can be easily found from the low frequency measurements: in the log (Re E, Im E)/log w plot four straight lines Re E x CO’, Re E c~ C2, Im E 1 co’ and Im E x &’ intersect at 0 = 02 = r,-‘, in addition to Re E(wJ = -Im E(wz). The value D/l’= O, can be a priori known or determined from the high frequency measurements as frequency corresponding to intersection of the straight lines IEl x CL’ and IEl 1 w-“?. If wg is known from the experiments, then with the same assumptions ~‘(07) B I and 7, ti r from equations (35) and (33) one can calculate 7:

7=47,

\

1

/

’

(37)

If the incident light is non-monochromatic and has the spectral density I(I), equations (33) and (34) would be integrated over the interval of possible l-values. Transitions between the characteristic w-intervals presented in Table 2 will be more smooth. Equations (36) and (37) will include instead of I the effective depth:

(38)

J

I(1) dl

If the frequency range (w” + r-‘, w1/2) is rather wide so that the experimental results follow the straight line ReE x w -I”, the I&value (or I&/D) may be easily found from the experimental results:

(39)

The signs of equality correspond light.

to a monochromatic

6.CONCLUSION

Analysis performed in the present paper shows that even in a comparatively simple case of strong inversion the photo-EMF is a complex function of modulation frequency of the incident light. At low frequencies the photo-EMF is constant in its amplitude and mainly real. i.e. it is in the incident light modulation phase. The imaginary photo-EMF component is small at low frequencies and increases proportionally to frequency. Such a situation is valid up to frequencies at which recombination in the bulk, in the space charge region or at the backside of the sample still ensures an equilibrium within the period of alternating signal. At high frequencies, when equilibrium fails to be settled, the imaginary component of the photo-EMF predominates and decreases reciprocally with frequency. The real component here is small and decreases as Wan”‘.Therefore, the photo-EMF modulus decreases reciprocally with frequency, while the phase is delayed against the light modulation phase by 90”. The above conclusion is pertinent in the case when the light is absorbed in a thin semiconductor layer near the interface. For bulk light absorption there appears an additional phase delay resulting from the diffusion of generated carriers from the bulk to the space charge region. This delay is 45” for high-frequencies as a supplement to the above mentioned 90”. The real and imaginary photo-EMF components become equal in their amplitudes and decrease as We”‘. Measurements of the frequency dependence of the photo-EMF permit one to determine some physical

626

R. S. NAKHMANSON

parameters of semiconductor. In particular, making use of the bulk light absorption, one can determine the bulk lifetime of the Si MIS structures, which fact may be emphasized because usual technique of measurement of admittance under inversion regime does not provide such a possibility. Some examples of treatment of the measurement results will be discussed in a subsequent paper. Finally we note that methods of determination of physical parameters of the MIS structures developed in section 4 may be applied to treatment of the results of admittance measurements, especially to Ge MIS structures

REFERENCES I. R. S. Nakhmanson.

2. 3. 4.

5. 6. 7.

Fi:. Tekh. Po~~~pruu.1, X25 (1967): SOP. Ph.vs.-Semicond. 1, 687 (1967). R. S. Nakhmanson. Fiz. Ttwd. Tda 7. 3439 (196.0: Sou. Ph.vs-So/id Sfirrr 7. 2772 (1966). C. G. Garrett. Phm. Ret 107. 478 (1957) A. Goetzberger and E. H. Nicollian, Bell Syst. Tech. .I. 46. 513 (1967). R. S. Nakhmanson and V. G. Erkov. Phyt Sfdus Solidi (~1)2. 627 (1970). B. E. Deal, A. S. Grove, E. H. Snow and C. T. Sah. J. Nectrochem. Sot. 112, 308 (1965). R. S. Nakhmanson. Fi. Tech. Po/uproc. 4. 439 (1970): Sov. Phys.-Semicond. 4. 372 (1970).