Determination of semiconductor resistance under a contact

Determination of semiconductor resistance under a contact

M. Ahmad a,b,*, A.P. Shah c, D.K. Sharma b, N.R. Roy a, B.M. Arora c

a Department of Physics, Ranchi University, Ranchi 834008, Indiab Department of Electrical Engineering, B-61, IIT, Powai, Mumbai 400076, India

c Tata Institute of Fundamental Research, Colaba, Bombay 400005, India

Received 28 February 2001; received in revised form 28 March 2001; accepted 20 August 2001

Abstract

In this work the expression for end resistance has been modified for the case of finite metal resistance. It has been

shown that using this approach the contact resistivity for alloyed contacts can be determined even in case of small

transfer length and large contact length. This is contrary to the case where metal resistance is assumed to be zero.

The changes required for different probe positions have also been considered. � 2002 Elsevier Science Ltd. All rights

reserved.

1. Introduction

Contacts to thin semiconductor layers on non-con-

ducting substrates are generally evaluated by using

either a transmission line method (TLM) or Kelvin re-

sistor method. Despite the advantages of the Kelvin

resistor method, the TLM [1] is still widely used [2] for

evaluating contacts to III–V compound semiconductors.

This is because it is difficult to fabricate a passivation

layer on III–V compound semiconductors, which is re-

quired in the Kelvin resistor method. In comparison,

passivation is a common part of Si technology. Apart

from the classical TLM method, another technique

called trilayer TLM has been reported in the literature

[3] for separately determining contact resistivities of the

metal-alloyed layer interface and the alloyed layer-

semiconductor interface. Although the trilayer TLM

provides this important information about the inter-

faces, the overall contact resistivity as seen by the device

is still given by the standard TLM. In short, the TLM

remains a popular method for quick evaluation of the

contact resistivity [4–6]. In the analysis of measurements

made by using TLM, some assumptions are usually

made: (i) metal resistance Rsm is zero, (ii) the sheet re-

sistance of the semiconductor Rsc under the contact

is equal to the semiconductor sheet resistance Rss be-tween the contacts. Neither of these assumptions is

correct in general. It is known that Rsc can be estimatedexperimentally by measuring the end resistance Re asshown schematically in Fig. 1 by passing a constant

current i0 between two contact pads and measuring thevoltage Ve between one of these pads and a third adja-cent pad, Re ¼ Ve=i0 [7]. In this paper we provide an

expression for the end resistance, which is an improve-

ment compared to that of Refs. [7–9], taking into ac-

count the finite value of the metal resistance. Since the

end resistance and the contact resistance measurements

can depend on the placement of the electrodes, partic-

ularly in the case when the metal film has significant

resistance, we provide expressions for the effect of shift

in the position of the current probe and the voltage

probe on these measurements. Note that the effect of

shift of the voltage probe position alone has been con-

sidered earlier in Refs. [10,11]. Next, we show that the

semiconductor sheet resistance Rsc can be evaluated byan alternative method in which the contact resistance

measurements are made on rows of TLM patterns

of varying width and length. Finally we give a few

experimental example cases to illustrate the use of end

resistance measurement in determination of contact

resistivity.

Solid-State Electronics 46 (2002) 505–512

*Corresponding author. Present address: Department of

Electrical Engineering, B-61, IIT, Powai, Mumbai 400076,

India.

E-mail address: [email protected] (M. Ahmad).

0038-1101/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0038-1101 (01 )00286-6

2. Theory

2.1. End resistance method

2.1.1. Case of zero metal resistance (Rsm ¼ 0)

In the standard TLM, a number of contact pads of

rectangular geometry are fabricated on the semicon-

ductor with variable spacing L between the pads shown

schematically in Fig. 1(a). In this case the total resistance

R measured between any two contact pads (Fig. 2(a)) is

twice the contact resistance Rc, plus the resistance of thesemiconductor layer [1].

R ¼ 2Rc þ RssLW

ð1aÞ

Under the assumption of zero metal resistance, the

contact resistance Rc is given by

Rc ¼RscLtW

cothdLt

ð1bÞ

where Lt is the transfer length defined as Lt ¼ ðqc=RscÞ1=2,

qc is the contact resistivity, Rsc and Rss are the semi-conductor sheet resistance values below the contact and

between the pads respectively, W and d are the width

Nomenclature

qc contact resistivity

W , d contact width and contact length, respec-

tively

Rc contact resistance

Re end resistance

Lt transfer length

a modified transfer length

Rsm sheet resistance of the contact metal

Rsc sheet resistance of the semiconductor under

the contact

Rss sheet resistance of the semiconductor out-

side the contact

Dx shift of the current probe from the contact

edge

Dx1 shift of the voltage probe from the current

probe

d1 contact length minus Dxq0c apparent contact resistivity obtained with-

out end resistance correction

Fig. 1. (a) Experimental arrangement for end resistance measurement; (b) double transmission line model for rectangular geometry,

where x ¼ 0 is the current probe position. The lower bank of resistors relate to the sheet resistance of the semiconductor under the

metal ððRsc dxÞ=W Þ and away from the metal ððRss=W ÞLÞ. The top bank of resistors relate to the sheet resistance of the metal

ððRsm dxÞ=W Þ. The connecting resistors qc= ðW dxÞ denote the interface resistance between the metal and the semiconductor. End re-sistance is measured by the probe arrangement shown.

506 M. Ahmad et al. / Solid-State Electronics 46 (2002) 505–512

and the length of the contact pad and L is the spacing

between the contact pads. In Eq. (1a) R and the geo-

metrical factors are measured quantities and Rss is aknown material parameter. Therefore Rc can be foundby curve fitting the data of R vs. contact spacing L. The

transfer length Lt can be found from Eq. (1b) provided

Rsc is known, which in turn leads to the determination ofthe specific contact resistivity qc from

qc ¼ RscL2t ð2Þ

Since Rsc is generally not known, an extra measure-ment of the end resistance Re can be used to separatelyfind Lt and Rsc. For Rsm ¼ 0, the expression for the end

resistance [7] is given by

Re ¼LtRsc

W sinh dLt

¼ Rccosh d

Lt

ð3Þ

Various parameters in this expression have been de-

fined above. We can see that the value of Re is measur-able reliably only if the contact length is of the order of

the transfer length. For example, consider the case of

transmission line measurement in which the contact

length d is equal to 3Lt. In this case the value of endresistance will be about 10% of the contact resistance.

2.1.2. Case of non-zero metal resistance (Rsm 6¼ 0)

The case of the contact resistance for Rsm 6¼ 0 has

been treated earlier in Refs. [12,13]. Following the

method of Marlow and Das [12] the double transmission

line equivalent circuit for the contacts with non-zero Rsmis shown in Fig. 1(b). Using the notations given in Sec-

tion 2.1.1, the expression for the contact resistance Rc forcontacts with length d P 4a is given in a symmetrical

form as follows:

Rc ¼RscRsmd þ ðR2sc þ R2smÞa

ðRsc þ RsmÞWð4aÞ

where,

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qcRsc þ Rsm

rð4bÞ

a is the modified transfer length and reduces to Lt forRsm¼ 0. The above relation has also been derived in Ref.[13] without the above restriction on the contact length

in which case one obtains

Rc ¼1

ðRsc þ RsmÞWðR2sc

"þ R2smÞa coth

da

þ RscRsm d

(þ 2asinh d

a

)#ð5Þ

The corresponding end resistance is given by

Re ¼aRsc

W sinh da

þ aRsmW

cothda

ð6Þ

This expression can be obtained from Eq. (8), derived by

us later in Section 2.2. In this case the current i2 reducesto i0ðRsc þ eRsmÞ=ðeðRsc þ RsmÞÞ at x ¼ a for d � a as

compared to i0=e at the transfer length for Rsm¼ 0.Now, one can have an idea of the effect of the metal

sheet resistance Rsm on the values of Re. (i) According toEq. (3) with Rsm¼ 0 the end resistance is extremely smallfor large d. It decreases to zero exponentially with in-creasing d. (ii) According to Eq. (6) however, with Rsm6¼ 0 we see that a significant difference occurs. A finite

value of end resistance, larger by orders of magnitude, is

found even in the case of (a) small transfer lengths, (b)

small Rsm and (c) large d. For example, consider a typicalcase of d, W ¼ 50 lm, Rsc¼ 22 X, qc ¼ 1:5� 106

X cm2. Re ffi 2� 109 X for Rsm¼ 0; while Re ffi 0:5 mXfor Rsm ¼ 0.01 X, and 5 mX for Rsm¼ 0.1 X. Corre-sponding Rc ffi 1 X in all the three cases. (iii) According

to Eq. (6) the end resistance decreases exponentially with

increasing value of d, but it reaches a finite asymptoticvalue of Rsma=W for large d. From the ratio of Eqs. (4a)

and (6), for small Rsm and large d, Re=Rc ¼ Rsm=Rsc.

2.2. Probe position corrections

In Eqs. (4a) and (5) the current and voltage probes

are assumed to be at the extreme ends as shown in Fig.

2(a). Correction due to shift in the voltage probe posi-

tion from the pad end has been reported earlier [10] for

d P 4a contact geometry, but there is no mention of theeffect of shift in the current probe position. In practice,

neither of the probes are positioned at exact edges of the

pads as shown in Fig. 2(a) or 3(a). Probe position related

corrections to the above equations for Rc and Re becomeparticularly important in the case of finite metal resis-

tance. These are considered next.

Fig. 2. Probe positions in contact resistance measurement.

M. Ahmad et al. / Solid-State Electronics 46 (2002) 505–512 507

2.2.1. Current and voltage probes shifted from the contact

edge

Consider that both current and voltage probes are

shifted from the pad edge by the same amount Dx asshown by the dashed line in Fig. 2(b). For this case, we

have derived the expressions for Rc and Re using theboundary conditions i1ðd1Þ ¼ 0, i2ðd1Þ ¼ i0, i2ð0Þ ¼ i02ð0Þ,and

d

dxi2ðxÞ ¼

d

dxi02ðxÞ at x ¼ 0

as follows (the details being given in Appendix A):

Rc ¼1

ðRsc þ RsmÞW

�

RscRsm d1 þ2a

sinh d1a þ cosh d1

a tanhDxa

( )þ

ðR2sc þ R2smÞa cosh d1a þ R2sca sinh

d1a tanh

Dxa

sinh d1a þ cosh d1

a tanhDxa

266666664

377777775

ð7Þ

where d1 ¼ d Dx. This equation is similar to that ob-tained by Suzuki et al. [14,15] for the case of source

drain contact resistance of silicided thin film SOI

MOSFET’s. Corresponding expression for end resis-

tance is given by

Re ¼

RscRsm

þ coshd1a

sinhd1aþ cosh

d1atanh

Dxa

aRsmW

� 1

"

Rsc 1 exp Dxa

� �2ðRsc þ RsmÞ 1þ exp 2 Dx

a

� �#

ð8Þ

The above equation reduces to Eq. (6) for Dx ¼ 0. Now,

using Eqs. (5) or (7) and Eqs. (6) or (8) for d, d16 4a, asimultaneous solution for a and Rsc can be obtained,

which in turn can be used to get qc using Eq. (4a)

2.2.2. Voltage probe position between 0 and d1If the voltage probe is shifted to the right of the

current probe on pad 1 by an amount Dx1, and to the lefton pad 2 by the same amount, as shown by the solid line

in Fig. 2(b), there is an additional correction term for Rcby an amount DRc, to be subtracted from the RHS of

Eq. (7), as discussed in Appendix A. It is given by

DRc ¼ 0 for Dx1 ¼ 0; and,

DRc ¼RscRsmDx1

ðRsc þ RsmÞW

þ R2smaðRsc þ RsmÞW

1 cosh Dx1a

�sinh d

a

RscRsm

�"þ exp

� d

a

��

þ 1 exp

� Dx1

a

�#for Dx ¼ 0 ð9bÞ

For d1 P 4a, this is simplified to

DRc ¼RscRsm

ðRsc þ RsmÞWDx1

�þ a

RsmRsc

1

� exp

� Dx1

a

��ð9cÞ

Similarly, in the case of end resistance measurement,

when the voltage probe is shifted by a distance Dx1 to theright of the current probe on pad 2, as shown in Fig. 3(c),

the term DRe to be subtracted from the value of Re givenin Eq. (8) is the same as given in Eqs. (9a)–(9c), i.e.,

DRe ¼ DRc ð9dÞ

We may note that the position of voltage probes on

pad 1 is not important as no current is flowing in this

pad. Similarly the position of current probe on pad 2 is

not important since the voltage is measured only be-

tween the other two pads.

2.2.3. Voltage probe position between 0 and DxNow, if the voltage probe is placed to the left of the

current probe on pad 1 and to the right on pad 2, the

term to be subtracted from the RHS of Eq. (7), for

the shift of the voltage probe by Dx1 from the current

probe, i.e., the x ¼ 0 position, can be shown to be as

follows (detail given in Appendix A)

DRc ¼R2sma

ðRsc þ RsmÞW

RscRsm

þ cosh d1a

sinh d1a þ cosh d1

a tanhDxa

�1 exp Dx1

a

� �þ exp 2Dx

a

�1 exp Dx1

a

� �1þ exp 2Dx

a

�ð10aÞ

The term DRc reduces to zero for Dx1 ¼ 0. In the case of

Re measurement also the correction term to be sub-

tracted for a similar shift of the voltage probe by Dx1 asshown in Fig. 3(d) is again given by Eq. (10a), i.e.

DRc ¼RscRsmDx1

ðRsc þ RsmÞW

þ R2smaðRsc þ RsmÞW

1 cosh Dx1a sinh Dx1

a tanh Dxa

�þ cosh d1

a exp Dx1a

�sinh d1

a exp d1a

�cosh Dx1

a

sinh d1a þ cosh d1

a tanhDxa

" #ð9aÞ


DRe ¼ DRc ð10bÞ

2.3. Variable contact size method

The variation of contact length is not of much use in

end resistance measurement, since for small transfer

length the values of Rsc and the contact resistivity be-come geometry dependent and stabilise only for d of

about 50 lm [3]. We may note that, in case of d P 4a,the variation of d can also be used to estimate the

modified resultant semiconductor resistance under a

contact and hence the correct value of contact resistivity.

This is discussed in the following:

Let the contact resistance determined for a particular

row of transmission line model with contact pads of

length d 0 and varying contact spacing be Rc1 as foundfrom Eq. (1a) and Rc2 for another row of TLM pattern

of contact length d 00. Then using Eq. (5) for the case of

finite metal resistance one obtains

DRcjW ¼ Rc1 Rc2 ¼RscRsm

W ðRsc þ RsmÞðd 0 d 00Þ ð11Þ

Thus we see that the value of Rsc can be found fromEq. (9a) as other parameters can be measured experi-

mentally. DRc can be optimised using a suitable range oftest structures with variation in d and W. The accuracy

of Rsc determination is dependent on the measurement ofDRc and Rsm. Therefore, the value of Rsc has to be ob-tained from a set of different d values to improve the

accuracy. The proposed method works better when the

metal sheet resistance is high and comparable to Rsc.

3. Experimental detail

In order to explore the effect of metal sheet resistance

on end resistance and to find the contact resistance re-

lated parameters a and Rsc, the following samples wereused. Two samples are p type GaAs (i) A (p � 4:8� 1017

cm3) and B (p � 1:6� 1017 cm3), with a 2 lm thick

epitaxial layer grown on semi-insulating (SI) GaAs with

contacts of AuZn. Sheet resistance of the contact metal

was deliberately enhanced by using about 200 A thick

Au–Zn film. The other three samples are n type GaAs

(n � 8� 1016 cm3) of 1 lm thick epitaxial layer grown

on SI GaAs with different contacts of about (i) 2000 �AAthick AuGeNi/Au (sample C), (ii) 2000 �AA AuGeNi /Pd/

Au, (sample D) and (iii) 1000 �AA AuGeNi (sample E).

Samples C and D were aged at 450 �C for 3.5 h and

sample E was aged at 400 �C for about 1.5 h before

measurements. All samples were prepared by organic

cleaning followed by defining a mesa photolithographi-

cally, etching the epilayer in orthophosphoric acid:hy-

drogen peroxide:DI water::3:1:50 and rinsing in DI

water. TLM geometry contacts were formed by lift-off

technique, patterning with positive photoresist followed

by vacuum evaporation of the contact metal. The TLM

pattern typically has a pad length d of 170 lm and a

spacing between the pads varying from 25 to 125 lm. Toevaluate metal sheet resistance, contact materials were

also deposited on reference SI GaAs. Subsequent to

deposition, contacts were formed by rapid thermal an-

nealing at 430 �C in flowing nitrogen on a graphite boat

for 30 s for p-GaAs and at 445 �C for 2 min on a tan-

talum boat heater for n-GaAs. The reference samples

were also subjected to the same heat treatment as the

samples under test.

The contact resistance was measured using a Karl

Suss prober AP 4. A constant current of 0.1 mA was

passed through two contact pads using a Keithley 220

programmable current source and voltage was measured

between the same pads using a Keithley 2000 digital

multimeter. Four probes were used, two for passing

current and two for measuring voltage. The current and

voltage probes were kept collinear so as to minimise the

errors which could be introduced by the two dimen-

sional flow of current. The end resistance was measured

by passing current between two pads and measuring

voltage between one of these and a third adjacent probe

as shown in Figs. 1 and 3. The sheet resistance of the

metal was measured by four probe technique on the

blanket metallised and alloyed SI GaAs sample for

AuZn contact. For AuGeNi contacts the resistance Rsmwas measured on a line contact in the SI region of GaAs

Fig. 3. Probe positions in end resistance measurement.


by passing current between a length and measuring

voltage using two separate probes.

4. Results and discussion

Next we consider the experimental results of end re-

sistance and contact resistance measurements. The val-

ues of Rc found from the R–L plots are given in Table 1together with the measured end resistance values for

different contacts. Using these values and the known

contact parameters d, W and metal sheet resistance Rsm,the values of Rsc, a and contact resistivity are obtainedby solving Eqs. (5) and (6) or Eqs. (7) and (8) simulta-

neously. The corrections due to shift in the probe posi-

tions from the pad end have also been accounted in the

calculations presented in Table 1.

We first discuss the results of AuZn contacts to p-

GaAs, i.e. the samples A and B. The values of Re arehigh as expected because of the high values of Rsm of

these samples. If one uses Eqs. (3) and (5), no solution

for Rsc can be found. The obvious reason being that thevalue of Re expected from Eq. (3) is smaller than the

experimental value by about more than two orders of

magnitude. When the expression including the effect of

metal resistance, i.e., Eq. (8) is used, reasonable values

of a and Rsc are found. Eq. (4b) is then used to find thecontact resistivity qc. Thus we see for large values of Rsmthat the results of end resistance experiment cannot be

used in the analysis of contact resistivity unless the

modified expression including the metal sheet resistance

is used.

We find results similar to the above for the aged

contacts on n type samples. In samples C, D and E, the

metal sheet resistance is relatively smaller. Still, using

measured Rc and Re one cannot find a solution for theparameters a and Rsc if one uses the expressions whichneglect Rsm. On the other hand these parameters are

readily obtained using the expression which includes

Rsm. The values of qc obtained for these contacts werehigh because the contacts were aged as specified above.

We also note here that the Rsc values obtained by ex-periments are considerably different from the original

semiconductor sheet resistance Rss. If one had used

Rsc ¼ Rss, one would get apparent contact resistivity

values q0c which are also listed in Table 1 and are quite

low but very misleading.

The experiments conducted above are somewhat

contrived to enhance the end resistance and demonstrate

its effectiveness in the determination of realistic qc. Evenin the cases where the actual contact resistivity and Rsmare small, it is important to measure the end resistance

Re which could be in the milliohm range in realistic cases

where qc could be as small as 106 X cm2 and Rsm ffi 0:1–

0.01 X/sq. This is considerably larger than the nano-ohmvalue calculated if Rsm is neglected altogether.

Finally for non-zero Rsm, the correction required forthe displacement of probes from the exact edges is

considerable, specially for Rsm values of a few ohm or

more in the above cases, i.e. for sample A, B and E. To

avoid errors and more complex formula it is better to

keep the probes as near to the edges of the pads as

possible. In case Rsm is made nearly zero by using a thickoverlayer or plating then the formula for Rc becomesmuch simpler and DRc, DRe tends to be negligible. Onlythe expression for Re as given by Eq. (6) for Dx ¼ 0 or

Eq. (8) Dx 6¼ 0, remains significantly different from the

case given by Eq. (3), when Rsm is exactly zero.

5. Conclusion

In this work a modified expression for end resistance

has been presented for the case of finite metal resistance.

It has been shown that the end resistance does not de-

crease exponentially with increasing contact length but

has a finite asymptotic value, once the metal resistance,

however small it may be, is taken into account. Further,

it is shown that the end resistance is easily measurable

even in case of small transfer length, large contact length

and small metal resistance. Several cases are presented

which underline the importance of using the modified

formula to include the end resistance correction for es-

timating the actual value of contact resistivity. In the

case of alloyed contacts, the correction can be fairly

large because of the modification of the semiconductor

resistance Rsc from the nominal value of Rss. It has beenshown that the experimental data on contacts with finite

Rsm and large contact length can be explained only whenthe resistive effect of metal resistance is included in the

Table 1

Calculation of contact resistivity qc from the measured values of the end resistance, Re and contact resistance Rc

Sample Rss (X/sq) Rsm (X/sq) d, W (lm ) Rc (X) Re (X) Rsc (X/sq) a (lm) qc (X cm2) q0

c (X cm2)

A 258 39 170, 73 45 18 38 46 1:6� 103 9� 106

B 775 39 170, 73 80 28 70 57 3:7� 103 1:0� 104

C 445 2.2 127, 1050 2.5 1.1 28 81 1:9� 103 1:3� 104

D 445 2.8 130, 1050 3.5 1.7 34 90 3� 103 2:7� 104

E 142 5.5 170, 670 1.25 0.25 14 40 3:2� 104 9:6� 106


end resistance. An alternative method has also been

proposed to evaluate the resultant modified semicon-

ductor resistance under the contact which uses rou-

tine Rc measurements with a mask set of different

contact lengths or by measuring Rc for different probepositions.

Acknowledgements

The author would like to thank Prof. R. Lal, Prof. J.

Wasi and Mrs. Rani of IIT, Bombay, for their labora-

tory support in electrical measurements.

Appendix A

Following the method used by Marlow and Das a

one dimensional double transmission line model for

rectangular geometry is being used as shown in Fig. 1(b).

In this analysis we have assumed that voltage drops

along the y-axis are negligible compared to drops

along the current flow path which is x-axis. In this

condition one dimensional approximation is appropri-

ate and the formula derived allows us to evaluate the

contact resistivity with minimal error. In practice this

happens when the voltage and current probes are col-

linear.

The distributed resistance is divided into infinitesimal

elements whose values are given in Fig. 1(b). Let us

assume that the current enters or leaves the contact

at a point away from the edge by Dx as shown in

Fig. 1(b). Let the total current entering and leaving the

contact be i0, it is divided into two components i1 andi2 in the region between x ¼ 0 to d1, i1 flowing throughthe metal and i2 through the semiconductor. In the

second region, i.e. between 0 and Dx let the cur-

rent flowing in the semiconductor and the metal be i02and i01 respectively. The resulting potential drops in themetal and semiconductor are V1ðxÞ and V2ðxÞ respec-tively. The following Eqs. (A.1)–(A.3) are written re-

lating the voltage, current and parameter qc, Rsc andRsm.

V2ðxÞ V1ðxÞ ¼ di2qc

W dxðA:1Þ

d

dxV1ðxÞ ¼ i1ðxÞ

RsmW

ðA:2Þ

d

dxV2ðxÞ ¼ i2ðxÞ

RscW

ðA:3Þ

Now, in the region between x ¼ 0 and d1

i1 ¼ i0 i2 ðA:4Þ

Differentiation of Eq. (A.1) and substituting for the

voltages from Eqs. (A.2) and (A.3) results in a second

order inhomogeneous differential equation

d2

dx2i2ðxÞ ðRsc þ RsmÞi2ðxÞ þ

Rsmi0qc

¼ 0 ðA:5Þ

The solution is sum of a general solution obtained

by equating the term without i2ðxÞ to zero, plus a par-ticular solution. Since the third term is a constant,

the particular solution is obtained by assuming i2ðxÞ aconstant.

i2ðxÞ ¼ C1 þ C2 expxa

� �þ C3 exp

� xa

�ðA:6Þ

where,

C1 ¼Rsmi0

Rsc þ RsmðA:7Þ

is the particular solution.

Using Eq. (A.4) we get

i1ðxÞ ¼ C1RscRsm

C2 expxa

� � C3 exp

� xa

�ðA:8Þ

Now, in the region between x ¼ 0 and Dx

i01 ¼ i02 ðA:9Þ

Using Eqs. (A.1)–(A.3) with i2 replaced by i02 and Eq.(9a), the following differential equation is obtained

d2

dx2i02ðxÞ

ðRsc þ RsmÞi02ðxÞqc

¼ 0 ðA:10Þ

The solution for which can be written as

i02ðxÞ ¼ C4 expxa

� �þ C5 exp

� xa

�Using the boundary condition: i02ðxÞ ¼ 0 at x ¼ Dx

the equation reduces to

i02ðxÞ ¼ C4 expxa

� �� exp

� 2Dx

a

�exp

� xa

��ðA:11Þ

The constants C2, C3, C4 are found by applying thefollowing boundary conditions,

i2ð0Þ ¼ i02ð0Þ ðA:12aÞ

d

dxi2ðxÞ ¼

d

dxi02ðxÞ

��at x ¼ 0 ðA:12bÞ

and

i1ðd1Þ ¼ C1RscRsm

C2 expd1a

� � C3 exp

� d1

a

�¼ 0

ðA:12cÞ


It can be shown that

C2 ¼ C1RscRsm

1þ tanh Dxa

�þ exp d1

a

�2 sinh d1

a þ 2 cosh d1a tanh

Dxa

" #ðA:13aÞ

C3 ¼ C1RscRsm

1 tanh Dxa

�þ exp d1

a

�2 sinh d1

a þ 2 cosh d1a tanh

Dxa

" #ðA:13bÞ

C4 ¼C1

1þ exp 2Dxa

� RscRsm

þ cosh d1a

sinh d1a þ cosh d1

a tanhDxa

" #

ðA:13cÞ

According to the definition of end resistance Re givenabove in Section 2.1.2, it is V2ðxÞ V1ðxÞ at the end of thecontact divided by i0, but since the current probe is as-sumed to be shifted from the edge by Dx; the potentialdifference measured by the voltage probes is now ¼ VAC.It has been assumed that one voltage probe is at the

same position as the current probe and the other one is

at the adjacent pad; since no current is flowing in this

pad the voltage measured is same as that at A.

Following the path ABC,

VAC ¼ RscW

Z Dx

0

i02 dxþqcW

d

dxi02

��x¼0

ðA:14aÞ

Substituting i02 from Eq. (A.11), we obtain

Re ¼VACi0

¼ aC4Wi0

" Rsc 1

� exp

� Dx

a

��2

þ qca2

1

�þ exp

� 2Dx

a

��#ðA:14bÞ

The expression for end resistance Re is found by sub-stituting for the constants. The result has been given in

Eq. (4a) of Section 2.1.2.

According to its definition the contact resistance is

the voltage drop at the front edge of the contact divided

by the total current flowing through the contact. Thus, it

is given as V2ðxÞ V1ðxÞ at x ¼ d1, plusR 0d1dV1 divided by

i0, this is ¼ ðVEF þ VFCÞ=i0, assuming that the voltageprobe is also at Dx. Using Eqs. (A.1) and (A.2), we get

V2ðxÞ V1ðxÞjd1 ¼qcWa

C2 expd1a

� �� C3 exp

� d1

a

��ðA:15aÞ

Z d1

0

dV1 ¼RsmW

C1xRscRsm

� C2a exp

xa

� �

þ C3a exp� xa

��d1

0

ðA:15bÞ

On substituting for the constants, adding and simplify-

ing and diving the sum by i0, the expression for Rc isobtained and given in Eq. (4a) of Section 2.1.2.

Now, when the voltage probe position is shifted to a

place between 0 and d1 from current probe, i.e., x ¼ 0

position, by Dx1 then a termR Dx10

dV1 ¼ ðRsm=W ÞR Dx10

i1 dxis to be subtracted from the expression for Rc. Now, whenthe voltage probe position is shifted to a place between 0

andDx from current probe by Dx1 on pad 1 then a termRDx10

dV1 ¼ ðRsm=W ÞRDx10

i01 dx is to be subtracted fromthe expression for Rc. The same result holds for the caseof end resistance.

References

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Determination of semiconductor resistance under a contact

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