correcting separation errors related to contact resistance measurement

ELSEVIER S C I E N C E

Microelectronics Journal 29 (1998) 21-30 © 1997 Published by Elsevier Science Limited

Printed in Great Britain. All rights reserved 0026-2692/98/$19.00

iiiii ̧̧̧ ¸̧ ~ ~i!iii ii!iiiiiiiiiii~!i!ii!i!~!iiii!ii

p i i : S 0 0 2 6 - 2 6 9 2 ( 9 7 ) 0 0 0 2 8 - 1

Correcting separation errors related to contact resistance measurement Yao Li 1% H. Barry Harrison 1 and Geoffrey K. Reeves2 1School of Microelectronic Engineering, Grifl~th University, Brisbane, Queensland 4111, Australia 2The Royal Melbourne Institute of Technology, Melbourne, Victoria 3001, Australia

A two-layer transmission line model has been proposed to model the sheet resistance of a semiconductor layer of finite thickness outside the ohmic contact region of a transmis- sion line model test pattem. It is shown that the effective sheet resistance is a function of the separation between the contacts and that there is a pivot point in the curve of the total resistance between the contacts versus their separation. The work reveals an error in the measurement of Re using the standard tim for small contact spacing. The results are in agreement with simulation results obtained from the boundary element method. © 1997 Published by Elsevier Science Ltd.

1. Introduction

T he transmission line model, introduced by Shockley [1] and refined by Berger [2],

has largely been used[ to characterize electrically the ohmic contact metallization systems o f

*Author to whom all con-espondence should be addressed. Tel: (+61 7) 3875 3625. Fax:(61 7) 3875 5384. E-mail: Y.Li@me.gu.edu.au.

major interest in semiconductor technology [3]. In particular, the contact property o f alloyed contact on compound semiconductor has been extensively studied as a function o f the deposi- tion characteristics and annealing procedures. By using this model, the specific contact resis- tance Pc is extracted from the contact resistance Re, while Re is obtained by measuring the total resistance between two contacts, and Rc is taken as the resistance which current flow encounters after passing under the leading edge o f the contact. For contacts o f identical geome- trical and electrical characteristics, as shown in Fig. 1, the following expression holds for each pair o f contacts:

Rtot~ = 2Re + Rsnl /w (1)

where Rtotal is the total resistance measured between the two contacts at a separation l, and Rsn is the sheet resistance outside the contact region, while w is the width o f the contacts.

21

Y. Li et al./Correcting separation errors

Rc L

Rtotal

I R c

I

Rsjw

Fig. 1. C o m p o n e n t o f total resistance be tween two ident i - cal contacts.

alloyed contact on GaAs and a Au-In/HgTe/ HgCdTe heterojunction contact. However, the TLTLM still assumed zero thickness in contact layers.

In this paper, a two-layer structure is proposed in order to model the sheet resistance outside the ohmic contact region and thus complement the theory and practice of the TLTLM. The results calculated from the model are compared with the simulation results using the boundary element method and are shown to be in good agreement.

The transmission line model is based on two fundamental assumptions: (a) the interface region is uniform; and (b) the semiconductor layer underneath the contact as well as that outside the contact region have zero thick- ness, i.e. the horizontal voltage on the semi- conductor and the vertical voltage across the interface. These hypotheses are challenged due to the down-scaling in the dimensions of semiconductor devices, especially when the length of the contact and the separation between the contacts can be of like dimen- sions to the thickness of the semiconductor layer.

A trilayer transmission line model (TLTLM) [4] has been proposed to present a contact with three layers in the contact region, such as an

2. The two-layer transmission line model for sheet resistance between ohmic contacts

The current flow beneath the horizontal planar contact is non-uniform. Because of the non- zero thickness of the semiconductor layer, this non-uniform current density may extend to the region immediately outside the contact. In parti- cular, the current may crowd in the upper part of the layer near the leading edge of the contact, as shown in Fig. 2. The current flowing from the upper part of the leading edge of the contact will have a different effective length from that flowing from the lower part of the leading edge. Thus, the resistance each portion encounters from the leading edge of one contact to that of another is different.

t

l ..K.

io | ..... V~ !

,k

.~i.:.:.:.: ~5~::::~:~.®,:::~i::::~,~ ~ ............. ; .....................

I i ........ |

........ 1 i !}[ = i

Fig. 2. Cur ren t distribution in the semiconductor be tween two contacts.

22

Microelectronics Journal Vol. 29, Nos 1-2

To describe exactly the current inside the semiconductor layer with finite thickness, a numeral method, such as the boundary element and the finite element methods, should be used. To represent approximately the finite thickness using an analytical model, the semiconductor is divided into two layers. The thickness of the layers, tl and t2, depends on how the current crowds near the contact, for instance. Suppose Rhl and Rh2 are the sheet resistances that the horizontal component of the current encounters in the lower and upper layers, respectively. If the current in the semiconductor is uniform, i.e. the vertical current component is zero, then

Rsh = R h l / / R h 2

Where the current distribution is not uniform, then the vertical component will encounter another resistance represented by Pvertical, which has the same: unit as the contact resis- tivity Pc in the interface between two different layers in the contact region. If the material is homogenous, then Pvertic,1, Rhl and Rh2 are related through the material resistivity. It is important to note tlhat Pvertical is not an inter- face-specific resistance but rather the vertical component of the distributed resistance of a semiconductor sheet with finite thickness.

Therefore, the semiconductor can be repre- sented by a two-layer network, as shown in Fig. 3. The current, say io, entering from one end will exit the other end. A fraction 3q of total entering current will enter the lower layer while a fraction f2 of the total current will exit the lower layer. By makingjq#f2, the model can be applied to the general case of non-identical contacts electrically connecting to the conducting layer. Thus, at the left end in Fig. 3, X=0, i1(0)=3qi0 and at the right end, x=l, il(l)=f2io.

By considering one element of the network, we can write down the voltage across Pvertic~(wdx) and the voltage in the loop, respectively,

V - Pvertical dil wdx

and

Rh2 dx Rhl dx V-+- i 2 - - -- (V- t - dV) + i I - -

w w

Then the expression for the current in the lower layer is

. ['Rsh . i, = (f, sinh(l- -Eh, J s-i-h0/-

+(f2 Rsh'~ sinh(x/7)'~ - R-~hL] sinh(//7) J

(2)

x=O

(1 :/~) ~o

II

X

x=l

6 Rh:&/w b

I I

. . . . , I I Rhl&/w

V(x+dx)

(I-A) ~o

f2 io

~raotl&~

Fig. 3. A network for calculation of resistance between contacts.

23

Y. Li et al./Correcting separation errors

where

y _ P.¢~ic~ (3) Rhl + Rh2

RhlRh2 (4) Rsh - - Rh l Jr" Rh2

Note that Pverti~a and Rhl or Rh2 are related through the material electrical properties.

Equation (2) may be plotted to illustrate how the current in the lower layer is distributed along the sheet length. For a semiconductor sheet of

J~=0.65,3~=0.55 and R,h/Rhl=0.3, Fig. 4 shows the current distribution in the lower layer for various values of I/7. It is worth noting that J~ andJ~ are determined by the contact parameters at both ends of the semiconductor layer [4]. Although the parameters Rsh/Rhl and y seem to depend on the sheet only, they are actually determined by both the sheet resistance and the contacts at each end because the thickness of the lower layer depends on how the current

crowds near the contact. From Fig. 4, it can be seen that the lower layer current at both ends is equal to J~i0 and J2i0 as required by the boundary conditions and then approaches i0R,h/ Rhl inside the lower layer. The parameter 7 determines how fast equilibrium is approached.

3. Sheet resistance and total resistance

3.1. Sheet resistance In the standard transmission line model, the contact resistance is given by R&Vdio, where Vc is the voltage drop across the contact interface at the leading edge of the contact and i0 is the current collected by the contact. Because of non- uniform current in the finite thickness of the semi- conductor sheet, the lower and upper parts of the sheet at the leading edge of the contact may not have the same potential. If V~ is taken as the voltage drop across the contact interface at the lower part of the leading edge of the contact when defining R&Vdio, as shown in Fig. 2, then the voltage drop from the left to right hand ends

8

=.

E O Z

1.0

0.8

0.6

f1=0.65 r2=o 55 R=~/Rh1=0.3

\ \

- - b ' ~ = l

----- U~=5 . . . . uv=10 - - - - U~=20

\ '" \ / ' / / . /

0.4 \ \ \ ~ " . t / " / / ./

o.2 1 0.0 i I i i

0.0 0.2 0.4 0.6 0.8 1.0

Normalized length, #7

Fig. 4. Current distribution in lower layer (il) for various values ofl/'y.

24

Microelectronics Journal, Vol. 29, Nos 1-2

of the lower layer, V~a~oss, can be used to define the effective sheet " reslstan,ce, Rsh, that is,

l #

Vacross = io --Rsh (5) W

By such a definition, eq. (1) will still hold and Fig. 1 is a valid representation of the total resistance between the contacts ifR~h is replaced by R~h'.

The voltage drop from left to right ends can be obtained by integrating the voltage across Rhl dx/w in Fig. 3:

/0 Rhl il I V~oss= - - d x = i o - W W

(.)q +A r~ _ e ,h ) cosh(l/y) - 1_ / { R , h + 2 k ~ . - ~ h , (l/r)sinh(I/~,) J

In comparison with eq. (5), we have

, ( f l c o s h ( / / } , ) - 1 " ~ Rsh = Rsh 1 + (I/y)sinh(l/y),] (6)

where

f l = 2(3q +j2Rhl2 Rsh 1) (7)

The effective sheet resistance is no longer a constant but a function of contact spacing I when fl~0, as shown in Fig. 5. R'~h has a large deviation from Rsh and is very sensitive to l if l<y (an unlikely situation), and R'~h~R~h if

3.2. Total resistance The total resistance between the contacts is one of the directly measurable parameters and has been commonly used to obtain R~ and R,h [5]. In the standard transmission line model, the total resistance is assumed to be proportional to the separation of the contacts. In reality, however, it may become a complex function of the contact separation, particular for small separation if the semiconductor layer can not be assumed to have zero thickness.

2.0

1.5 - ~

=, ............................... ~--7277L- 77::.z 7 -.-: t Y --

1.0

r,- t '" 1 / / / / / . . . . . 13=0 0.5 _ t .......... 15=+0.5

0 . 0 I I t

0 2 4 6 8 10 lh,

Fig. 5. Sheet resistance variation with I/',/. The sheet resistance is a constant when fl=0. In other cases, the deviation is pro- portional to 3. R'~h approaches R,h as I approaches infinity.

25

Y. Li et al./Correcting separation errors

From eqs. (1) and (5), the total resistance between contacts is

l !

R = 2R~ + - - R s h = 2R~ + al + b[Fn] W

(8)

where

Rsh a = (9) W

b - - ashflY (10) W

[Fn] - cosh(x) - 1 sinh(x) - t anh (2 )andx = (11)

The function [Fn], which was introduced when buried contacts were discussed [6], now describes the dependence of Rtot~ on the separa- tion. It can be estimated in two regions as follows:

[ g n ] : t a n h ( 2 ) ~ ( x ; 2 x~lX<l)

Then the total resistance,

~ 2 R ~ + ( a + ~ ) l i f l < ~ }

Rt°tal= L 2(Rc+~)+al i f y < l < o o

(12)

Comparing eq. (12) with eq. (1), it can be concluded that, when using the traditional measurement as described by eq. (1), an error for Rc will exist if l>>y, while an error for Rsh will exist if l<y. In the former case, 6R,:=b/2; in the latter case, 6Rsh=bw/(2y).

Figures 6 and 7 are plots of the total resistance as a function of the contact separation of various values of y, Rsh and b. There is a pivot point in each curve, which approximately locates at

As R~ increases from 150Q/sq to 300QJsq .......... • ~ . . - -

o c

i . Q

O ¢ -

" bS"

'<C 2R c (a

o

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Distance between contacts, l(p,m)

Fig. 6. The total resistance as a fimcdon of the separation between the contacts for various R,h and 7- The parameters for the solid curve are b=1.5 f~, 7=0.01 #m, and R~=150 fl/sq. The dot curves show the dependence of the total resistance on R,h, where b=1.5 ~, 7=0.01 #m and Rsh=200, 250, 300fl/sq, while the dashed curves show the dependence of the total resis-

tance on 7, where b=1.5 fl, R,h=150 fl/sq and 7=0.05, 0.02, 0.01 ~m.

26

Microelectronics Journal, Vol. 29, Nos 1-2

Ipivot~2.9y, as shown in Fig. 6. O n the left-hand side of the pivot point, the curves approach nearly linearly to 2Re at I=0. At the right-hand side of the pivot point, the curve becomes a straight line, ofwhicl i the slope is dependent on R~h only, moving down in parallel with the decrease of b, as shown in Fig. 7. When b=0, the whole curve is shnply a straight line with its slope equal to R~w, approaching 2Re at/=0.

It is worth noting that, although there are four parameters to be determined when measuring the total resistance, they independently control the different parts of the Rtotal versus l curve. If the measurement data cover a broad range of l, which includes the pivot point and some portion of its left-hand side, all parameters are easily obtained. What is difficult is that the location of the pivot point /pivot may be too close to /=0. The fibrication of test patterns

with some of the contact separation smaller than /pivot (for example /<0.1/~m) may not be feasible.

4. Comparison with simulation by the boundary element method

Alloyed ohmic contacts to GaAs were simu- lated using the boundary element method. The length of the contact was assumed to be d=l #m and other parameters of geometry were chosen referring to [7]: width w=100#m and thickness T=0.3jum. In the contact region, there are three layers: the metal layer, the alloyed layer and the unreacted semiconductor layer. The semiconductor layer beneath the contact has the same conductivity as the semi- conductor outside the contact region, so that the sheet resistances for these regions are different from each other because of their

As R~ increases from 150Q/sq to 300D./sq .. ...........

. . . . . . " f . f .

"~) t/~J~ ~ _ ~ _ ~ AS b decreases from 1.5~ to 0

I--

. . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . .

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Distance between contacts,/(pro)

Fig. 7. The total resistance as a function of the separation between the contacts for various Rsh and b. The parameters for the solid curve are b=1.5 ~], 7=0.01/zm and R~h=150 f~/sq. The dot curves show the dependence of the total resistance on R,h, where "7=0.01 #m, b=1.5 fl and R~h=200, 250, 300 ft/sq, while the dashed curves show the dependence of the total resis-

tance on b, where 7=0.01 #rn, Rsh=150~/sq and b=l.0, 0.5, 0fL

27

Y. Li et al./Correcting separation errors

different thicknesses. For a semiconductor layer of 0.3#m thickness with sheet resistance Rsh=284~2/sq [6], the equivalent conductivity can be calculated as 0.2=117 (~cm) -1. Its corre- sponding impurity concentration at 300K is about 1.7x 10 iv cm -3 [8].

Various thicknesses and conductivities of the alloyed layer, t and 0.1, have been simulated with the separation between identical contacts from /--0.025/1m to l=0.6/lm. This range of contact separation was chosen because the pivot point is located here. As discussed in the previous section, the simulated data covering the pivot point are necessary in order to obtain all parameters of our model, i.e. the parameters in eq. (8).

Figure 8 shows the simulation results and the comparison with the calculation from the model. The geometrical and electrical para- meters as shown in Fig. 2 are the following:

d=l #m, T=0.3 #m, t=0.15 #m, a2=117 (~cm) -1, a1=2, 117 and 300(~2cm) -1. The triangle and circular points in Fig. 8 are the simulation data with three values of conductivity of the alloyed layer. By carefully choosing the parameters, Re, Rsh, b and 7 in eq. (8), the calculation curve can be matched to the simulation results very well. In the case of0.1=2 (~cm) -1, the pivot point can be seen clearly. Because the pivot point is close to I=0, all practical measurement of i~tota 1 to extract Rc using the standard transmission line model, in which the separation is usually greater than 1 #m, would result in an error 6 for R~. For example, if 01=2 (~cm) -1, then R~=7.2 ~2, b=1.56 ~, and &=10.8%. When 0.1 increases, Rc will decrease and so will b, but the error will become larger. For example, if 0.1=300 (~cm) -1, then Rc=0.14~2, b=0.21 ~, and a measure error &=75%.

Figure 9 shows the simulation results with t=0.025#m and the other parameters are the

n¢"

o

i,- m 10

. Q

d=ll.tm t=0.15pro T=0.31/m o2=117(£Zcm) -~

• o~=2(£zcm) ~ Rc=7.2~, R==284D../sq b=1.56fl, ~=0.0175gm

• o]=117(Qcm) 1 Rc=0.31~, R==257£~/sq b=0.282£4, ~f=0.035grn

• m=300(~cm) -~ Re=0.14E4, Rth=245~./sq b=0.21Q, 1=0.042pm

. . . . I ' ' ' ' I . . . . I ' ' ' ' ] . . . . I . . . . I ~ ' ' ~

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

D i s~n~ be~een Conta~s, l (p.m)

Fig. 8. A comparison between the calculation from the model and the simulation data using the boundary element method, where the thickness of the alloyed layer is 0.15#m. The triangle and circular points are the simulation data and the curves

are the calculation results.

28

Microelectronics Journal, 1/ol. 29, Nos 1-2

8

u~ ~5

6 ¢ -

0 ¢J t- .

8

2 re 2

J • ol=2(f~crn) "1

R¢=3.36fL R==278~q/sq I0=0.13~, ~=0.03pm

• or1=117(flcm) "1 Rc=0.31 f~, R,h=257fZ/s q b=0.282t3, y=0.0351u'n

• ot=300(f~m) "t Re=0.185fL R=~=227~sq

d= 111m t=O.O25gm T=O.31.tm o2=117(Ocm) "1

b=0.31~, y=0.051tm

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Distance between Contacts, l (l~m)

Fig. 9. A comparison between the calculation from the model and the simulation data using the boundary element method, where the thickness of the alloyed layer is 0.025 #m. The triangle and circular points are the simulation data and the curves

are the calculation results.

same as those in Fig. 8. It should be noted that, contrary to the case in Fig. 8, b increases with an increase of o l. When oq=2(f~cm) -1, ~=1.9%, smaller than the counterpart of Fig. 8. However, when O" 1=300 (~"~cm)- 1, ~=84%, greater than that in Fig. 8. The reason for b increasing with al is that, when the alloyed layer is very thin, the; higher the conductivity of the layer, the more crowding of the current there is in the leading edge of the contact.

5. Conclusion

A new technique ha:; been proposed for model- ling the finite thickness of the semiconductor layer in transmission line model test patterns. The technique introduces a modified value of the sheet resistance between the ohmic contacts. It is show~a that the effective sheet resistance is a function of the separation between the contacts and that there is a pivot point in the curve of the total resistance

between the contacts versus their separation. The model has also quantified the errors in the measurement of Re using the standard transmis- sion line model.

The boundary element method has been used to simulate the contact patterns, and this confirms the pivot point in the total resistance curve. By matching the calculated curve with the simulation data, all parameters of the model can be obtained. Test patterns are currently being fabricated and results from them will be compared with the model.

Acknowledgements

The support of the Australian Research Council (ARC) through the large grants scheme is great- fully acknowledged. Y. Li would like to acknowledge Dr Jun w . Lu for providing simu- lation software for this work.

29

Y. Li et al./Correcting separation errors

References

[1] Shockley, W. Research and investigation of inverse epitaxial UHF power transistors, Report No. AI-TOR- 64-207, Air Force Atomic Laboratory, Wright-Pater- son Air Force Base, Ohio, 1964.

[2] Berger, H. Models for contacts to planar devices, Solid-State Electron., 15 (1972) 145

[3] Scorzoni, A. and Finetti, M. Meta!/Semiconductor Contact Resistivity and its Determination from Contaa Resistance Measurements, North-Holland, Amsterdam, 1988.

[4] Reeves, G.K. and Harrison, H.B. An analytical model for alloyed ohmic contacts using a trilayer transmis- sion line model, IEEE Trans. Electron Devices, 42 (1995) 1536.

[5] Reeves, G.K. and Harrison, H.B. Obtaining the specific contact resistance from transmission line model measurements, IEEE Electron Device Lett., EDL-3 (1982) 111.

[6] Reeves, G.K. and Harrison, H.B. Determination of contact parameters of interconnecting layers in VLSI circuits, IEEE Electron Devices, 3 (1986) 328.

[7] Henry, H.G. Characterization of alloyed AuGe/Ni/ Au ohmic contacts to n-doped GaAs by measure- ment of transfer length and under the contact sheet resistance, IEEE Trans. Electron Devices, 36 (1989) 1390.

[8] Sze, S.M. Semiconductor Devices: Physics and Technology, John Wiley, New York, 1985.

30