Description of non-equilibrium phenomena in MIS device under linear voltage ramp bias

Mid-Stare Hectronics Vol. 35, No. IO, pp. 1497-1502, Printed in Great Britain. All rights reserved

DESCRIPTION MIS DEVICE

1992 Copyright

OF NON-EQUILIBRIUM PHENOMENA IN UNDER LINEAR VOLTAGE RAMP BIAS XIUMIAO

Department

0038-1101/92 55.00 + 0.00 0 1992 Pergamon Press Ltd

ZHANC

of Electronic Engineering, Hangzhou Zhejiang 310028, People’s Republic

(Received 22 August

1991; in revised form

University, of China 3 January

Hangzhou,

1992)

Abstract-A new theoretical method to describe the non-equilibrium characteristics of a metal&insulator-semiconductor (MIS) device under a linear voltage ramp bias is presented. Unlike previous theoretical analyses the effect of time-dependent carrier generation of bulk traps on these characteristics has been considered and an improved model for generation width has been applied. A differential equation is formed to describe the change of depletion region width with time during transients. The high frequency MIS capacitance and gate current are related to depletion region width. In this way the non-equilibrium capacitance-voltage (C-V) and current-voltage (1-V) characteristics can be described.

NOTATION

fast voltage ramp. Recently, the non-equilibrium behavior was analyzed by Okeke and Balland[q. The basic assumption in all these works is that the generation process of bulk traps can be characterized by a unique time constant. However, the problem is clearly related to a non-steady-state process. The effect of non-steady-state generation of bulk traps on C-t transients in MIS devices under linear voltage ramp bias has been investigated by the author[8]. Assuming a time-dependent bulk trap generation rate, Allman[9, lo] presented an approximate method (a special iterative scheme) to describe non-equilibrium characteristics of the MIS device under a linear voltage ramp bias. In this paper, instead of the special iterative scheme, an analytical description of non-equilibrium transient processes in MIS devices under linear voltage ramp bias will be presented. In the description a time-dependent bulk trap generation rate[l] is taken into consideration and an improved model for generation width[l2] is developed.

high frequency MIS capacitance per unit area insulator capacitance per unit area initial value of high frequency MIS capacitance per unit area emission probability for electrons emission probability for holes hole generation rate per unit volume in generation region 11 hole generation rate per unit volume in generation region Ill gate current per unit area Boltzmann’s constant extrinsic Debye length doping concentration bulk trap density intrinsic carrier concentration inversion charge per unit area initial value of inversion charge per unit area electronic charge absolute temperature time flat band voltage gate voltage initial value of gate voltage width of depletion region initial value of width of depletion region width of generation region 11 width of generation region Ill voltage sweep rate permittivity of semiconductor

2. THEORY

1. INTRODUCTION

For a fast voltage ramp applied to the gate of a metal-insulator-semiconductor (MIS) device the non-equilibrium Z-V characteristics were first observed by Kuhn and Nicollian[l] and later analyzed by Board and Simmons[2]. A number of methods[34] were developed to extract the physical parameters from the non-equilibrium capacitance-time (C-f) transient response to a sufficiently .ssE 35,1&H

An MIS capacitor with an n-type substrate is considered to be initially biased into strong inversion. This effectively eliminates interface trap emission and generation. When a depletion linear voltage ramp (see Fig. 1) is applied to the device the depletion region width on the semiconductor side increases with time. The device is in a non-equilibrium deep depletion mode. The equilibrium and non-equilibrium semiconductor energy band diagrams are shown in Fig. 2. According to the theory of Simmons and Wei[l I], we can distinguish two interesting regions in the semiconductor bulk, region II and region III (see Fig. 2). The traps in region II were initially below the

1497

Xwhn.40

1498

ZHANG

Fig. 1. The depletion linear voltage ramp applied on the gate electrode of an MIS device with an n-type substrate. majority carrier Fermi level EFn, and therefore effectively full of electrons. The traps in region III were initially above EFn, and so effectively empty of electrons. As has been assumed[ 12,131, the generation region is defined as a part of depletion region where both concentrations of electrons and holes are lower than the intrinsic carrier concentration. From Fig. 2 it is composed of two parts: a generation region II and an effective generation region III. Owing to the different initial conditions the generation rates of the bulk traps in these two regions are different. From Ref. [8], at time t the net hole generation rates per unit volume in part of generation region II and III, which is created at time 7, are given in eqns (1) and (2) respectively

where the charge variation at interface states during transients has been neglected. Afterwards we will also assume that the change of flat-band voltage Vr,, during the transients is negligible. At t = 0, both generation region II and effective generation region III are not yet formed. Hence, we have W,,(O) = W,,,(O) = 0. Inserting gp I,, and g,],,, from eqns (1) and (2) into eqn (3) and using partial integration, then eqn (3) is

dep _ -- qe,eA dt

e,+ ep

x w-N-(en + e,)tl

x

S[ ' 0

W,(7)

-;

W,,,(7)

exp[(e, + e,)7] d7 .

&I,,(,,

gpI,,,(~,

7)

=

epenNt -

(1

-

exp[(e, + e,)(r

e, + ep

7)

=

n

On the other hand, from Fig. 2 the width generation region II can be expressed as

-t)l

1 +zexp[(e,fe,)(i

yg

P 1

w,,=

As holes build-up at the semiconductor surface during the transient and result in rising to the inversion charge, we have

dt -

w-w,.

1

(2)

%_

(4)

1

(1)

- t)]},

According to depletion expressed as

approximation,

Wo=LnJ~

’ gpI,,,@>

7)dW,,(7)

CT, 2-

Wi/2-LbIn

of

(5) W, can be

(6)

where L, is L, = J2kTc,/(q2N,,).

+

1

(7)

Equation (5) indicates that W,, is just the increment of the depletion region width in the transients. According to Ref. [12], the width of effective generation region III can be written as

1

(3)

IVg,-p w2 5

2

>

q2N:, L2,

-

q2N2, Wz

- w + (1 - l/Jz)W,.

(8)

Transient phenomena in MIS devices

1499

to) Depletion

region

Region

Region

(W.

1

I

1

III

I

1 I

tb)

Depletion

I

I

I W1

region

( generation

Region

I

III

\

Generation

region

III

CW,)

Fig. 2. Equilibrium and non-equilibrium semiconductor energy band diagrams: (a) equilibrium (b) non-equilibrium. From eqns (5) and (8), IV,,, can be expressed as a function of W,,. Before the linear voltage ramp is applied to the device, the initial gate voltage Vso, which biases the device into strong inversion, can he written as ,v&&+4%~+~~nW~ tin tin

-+

IVrlll.

(9)

2%

If a depletion linear voltage ramp with voltage sweep rate a is applied to the device at t = 0, the gate voltage V, at time t is IV81= IV,1 + Ialt.

Combining

@n G w2 + 4NnG 7 ‘i c,

+Qp-Qpo=O,

(12)

where C, is the initial value of MIS capacitance per unit area (the value in equilibrium strong inverstion). Here a well-known relation +;+!5 0

(13) m

es

is used. Solving algebraic equation (12), we obtain

(10) 0

I

G

‘I

w,, = :

,‘,,_QpqNDw+qNDw2 -+ IVFBI. tin

w

-Ci”lalt

Similarly, V, can he written as

8

eqns (9), (10) and (1 l), yields

2G

(11)

x

(Ci,lalt-Qp+Qpo>-l

1

* (14)

XIUMIAOZHANG

1500

Differentiating

dw,, -= dt

eqn (14) with respect

to time, yields

Inserting

dW,,,/dt

of

eqn

(19)

into

eqn

(18)

qNoci~(~,,+,,lco)[~i”l”l-~]. (‘5) yie1ds

6s

+

qNo(W,, + G/CO)

k+eJlaI-

F(W,, 1”

-e(W,f W,,)’ s 1

+ W,,,)

1

(20)

2

Ct, l~al+lal* Wo+ W,,)2-

Differentiating

Wi/2-

eqn (15) with respect

d2W,,

-

LLIn

q2NhW’o

+

W,J2

(21)

q*N; Lz,

to time, yields

Finally, we obtain the following for W,,

differential

equation

Es

-=

dt*

d* -= W,, dt2

qNoG(W,, + ~/co)

X _P-d'Q dt'

1

_

W,, + G/CO -W,,>tl~+G(W,,,t)

(‘6) Differentiating

dt*

where

eqn (4) yields

d2Qp= qe,e,N,

d W,,, dr

c, + eP

(22)

F( W,,, t) = e, + ep +

+ (e, + e,)

t,e,N,[Wo + WI, - X(@‘U, l)l N,,G(W,,

+ &o)X(W,,>

t) (23)

x(W,,+

W,,,)

-(e_+eP)z

(17)

I

and

where eqn (4) has been used once again. Inserting d2Q,/dt2 of eqn (17) and dQ,/dt eqn (15) into eqn (16) we obtain

d2W,, -= dt2

1

_

Obviously,

(‘8)

WI,,-dw,, .

w,, + w,,, + woiJ5

.

(24)

is

dt

(25)

Because W,, = 0 and dQ,/dt = 0 at t = 0, from eqn (15) another initial condition may be obtained

From eqns (5) and (8) obviously, d W,,, /dt in eqn (18) can in principle be accurately expressed as a function of W,, and its derivative. It is noted that the last term in the radical of eqn (8), a small revision, varies slowly (those variables are involved in natural logarithm). In the calculation of the time derivative of WI,,, therefore, approximately we may regard it as an invariable for simplicity. Thus from eqns (8) and (5) we have

dt

[A’( W,,, t) - W,/,,h]

W,,(O) = 0.

1

-

(e,+e,)lal

one of the initial conditions

d*

- (e,+e,)lal

‘iv’b’o

+ c&o)

In

!f!fG( w,,+ w,,,) No(W,, + G/CO) Gin

(1 dw,,, -=

6s

d’,(W,,

-f@

dw,,,

v,N

- NoG(W,, + ~/co)

4

G(W,,, *I =

-(en+e,)T

W,, + W’o

-

of

Cola1

w;,(o)=-. Let d W,, /dt = y, we have

w,=y

! I

(27)

1

Y* - QW,,, y = - w,, + ~JCO

Correspondingly,

t)y + G(W,,, t).

the initial conditions

WI,(0) = 0 (19)

(26)

qN,

y(O)=-.

ColaI @J,

become

Transient

phenomena

in MIS devices

1501

The relation between high frequency MIS capacitance per unit area C and W,, is similar to eqn (13)

or

tin

c=

l+

cin(W,

+

(30)

Wll)/%.

In a manner similar to Ref. [9], the gate current per unit area ZBas a function of W,, can be written as lZ*l = Cinlcll

=CinltYl

w>y -@+vo+ bv,,)Z.

(31)

Solving the differential equation (22) under initial conditions of eqns (25) and (26) [or the set of differential equations (27) under initial conditions of eqn (28)], and combining it with eqns (30), (31) and (lo), we can obtain non-equilibrium C-V and Z-V characteristics. AND DISCUSSIONS

Figures 3 and 4 show a set of normalized non-equilibrium C-V and Z-V characteristics of an MIS device under linear voltage ramp biases with various sweep rates, respectively. The set of differential equations is solved by the Runge-Kutta method. Of course, other numerical solution methods for ordinary differential equations can also be used. The corresponding computer programs, which are standard and ready-made, can be found in books on numerical analysis. The parameters used in computation are: V#, = -4V, C,, = 1.57 x lo-’ Fcm-2, No = 1.0 x lOI cmm3, e, = eP = 41.52 s-‘, N, = 5.14 x 10” cmm3. From Fig. 3, we can see that in the transient processes high frequency MIS capacitance monotonically decreases. It is corresponding to gradual expansion of depletion region during the transients. The higher voltage sweep rate, the faster decrease of high frequency MIS capacitance. At lower voltage sweep rate, as can be seen from Fig. 3, high frequency MIS

1

I.1 = 5oOV/r

I -12

I -11

I -10

I -9 Gate

I

I

-7

-6 voltage

V,

1

-6

B

E

I

-5

-40

t

(volts1

Fig. 3. Normalized non-equilibrium C-V characteristics of an MIS device biased by various

linear voltage

I -11

1 -10

/ -9 Gate

I

-6

voltage

I -7 V,

Fig. 4. Normalized non-equilibrium an MIS device biased by various

-F

3. RESULTS

I -12

ramps.

I

-6

(volts)

lo

I

-5

-4

.5 !

P

I-V characteristics of linear voltage ramps.

capacitance tends to “saturation” in the later stage of the transients. The phenomenon of MIS capacitance saturation has been observed in experiments[3]. Figure 4 shows that a low voltage sweep rate (e.g. 0.5 V s-l) the gate current monotonically increases during the transients. On the contrary, at a high voltage sweep rate (e.g. 50 and 500 ss’) it monotonically decreases. At a medium voltage sweep rate (e.g. 5 V SK’) it initially undergoes a fall, and then monotonically rises. These results may be understood from the components of gate current ZB. As the depletion region expands rapidly at the high voltage sweep rate, the main source of gate current comes from expansion of the depletion region instead of from carrier generation. Obviously, this current component is dependent on the expansion rate of the depletion region rather than on the size of depletion region. With the expansion of the depletion region the carrier generation grows, so that for a certain voltage sweep rate the depletion region expands more and more slowly. This implies that gate current will decrease in such a transient. On the other hand, at low voltage sweep rate the main component of gate current is generation current of minority carriers, which is related to the width of the depletion region. The wider the depletion region, the higher the generation current. Therefore, the gate current will increase in this transient. 4. CONCLUSIONS

Instead of previous special iterative schemes, a new theoretical method is formulated to describe nonequilibrium characteristics of an MIS device under linear voltage ramp bias. A set of differential equations describing depletion region variation is formed, in which the non-steady-state generation effect of bulk traps is considered and an improved model for generation width is developed. Combining them with the dependence of high frequency MIS capacitance and gate current on the change of depletion region, we can describe non-equilibrium C-V and Z-V characteristics. The accurate description of non-equilibrium characteristics is important to probe bulk trap

XIUMIAOZHANG

1502

properties. Of course, further experiments are necessary in order to extract bulk trap position and density. Finally, it is worthwhile to notice, as indicated in Ref. [8], that in those devices with low doping concentration and high density of traps the nonsteady-state generation effect of bulk traps on nonequilibrium characteristics is significant. Therefore, for those devices the carrier generation rate of bulk traps must be time-dependents as has been treated in this paper. REFERENCES 1.M. Kuhn and E. H. Nicollian, 370 (1971).

J. Electrochem.

Sot. 118,

2. K. Board and J. G. Simmons, Solid-St. Electron. 20,859 (1977). 3. R. F. Pierret, IEEE Trans. E/e&on Deu. ED-19, 869 (1972). 4. R. F. Pierret and D. W. Small, IEEE Trans. Electron Dev. ED-22, 1051 (1975). Solid-St. Electron. 21, 1057 (1978). 5. K. Taniguchi, Solid-St. Electron. 21, 6. P. Kuper and C. A. Grimbergen, 549 (1978). 7. M. Okeke and B. Balland, Solid-St. Electron. 27, 601 (1984). 8. X. Zhang, Solid-St. Electron. 34, 43 (1991). 9. P. G. C. Allman, Solid-St. Elecfron. 25, 241 (1982). 121 (1982). 10. P. G. C. Allman, IEE Proc. Pt Il29, 11. J. G. Simmons and L. S. Wei, Solid-St. Electron. 19, 153 (1976). 12. X. Zhang, Solid-St. Electron. 33, 1139 (1990). 13. K. S. Rabbani and D. R. Lamb, Solid-St. Electron. 21, 1171 (1978).