IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 35. NO. 2. FEBRUARY 1988 133

A Two-Layer Magneto-TLM Contact Resistance Model: Application to Modulation-Doped FET

Structures

Abstruct-The resistance for a two-layer structure with planar con- tacts is derived, with both the contacts and the region between the con- tacts modeled as transmission lines. Magnetic-field effects are in- cluded, which yields additional information on mobilities and carrier concentrations. The theory can be fitted to the usual resistance versus contact-separation data, as a function of magnetic-field strength. Fit- ted parameters include mobilities and carrier concentrations in each layer, both under the contact and in the bulk material between the contacts, as well as specific contact resistivities for the contact bar- riers. Experimental results are obtained for an AIGaAs/InGaAs mod- ulation-doped FET structure.

I. INTRODUCTION HE PROBLEMS of contact resistance have become T more acute in recent years with the advent of submi-

crometer devices, which can have very small intrinsic re- sistances [ 1 3 . Thus, much effort has been expended in an attempt to improve ohmic contacts, and also to improve the techniques by which they are measured. For planar devices, such as metal-semiconductor field-effect transis- tors (MESFET’s), the transmission-line model (TLM) has been widely used to extract the intrinsic metal-semicon- ductor (M/S) barrier resistance, in a form called the spe- cific contact resistivity p,, which is a figure of merit for the ohmic contacts [2]. This model, used in conjunction with the test pattern of Fig. l(a), gives two parameters: r,, the sheet resistance of the bulk material between the contacts, and pc R,,, where R, is the sheet resistance of the material under the contacts. To determine p, it is usually assumed that R, = r,, which may be true if the contacting materials do not diffuse or alloy appreciably (Fig. l(b)), but is not true in general (Fig. l(c)). It is sometimes pos- sible, by means of an additional measurement, to deter- mine the “end resistance,” which gives the correct R , and thus p, [3].

Recently, we have introduced a new technique based on the magneto-transmission line model (MTLM), which, besides r, and pcR,, also gives the bulk mobility p , the bulk sheet carrier concentation ns, and the mobility of the material under the contact pc [4]. The determinations of

Manuscript received June 15, 1987; revised August 28, 1987. This work was performed at the Avionics Laboratory, Wright-Patterson Air Force Base, under Contract F33615-86-C-1062.

The author is with the University Research Center, Wright State Uni- versity, Dayton, OH 45435.

IEEE Log Number 8718016.

p and n, make an additional Hall-effect measurement un- necessary. Also, if p, = p , then it can be safely assumed that R, = r,y and pc can be accurately calculated. If pc # p , it still may be possible to estimate R,y from the value of P c .

Unfortunately, all the techniques discussed above work only for single-layer devices such as MESFET’s, and are not applicable to heterostructure devices, in particular modulation-doped FET’s (MODFET’s) shown in Fig. l(d) and (e). Recently, Feuer [5] has developed a two- layer model in which the space between the contacts (bulk) is treated as a distributed resistance, but the contacts themselves are evidently treated as equipotential vertical surfaces, through both layers. Thus, no information con- cerning the material beneath the contact metallization can be obtained. In this paper, we treat both the bulk and con- tact regions as two-layer distributed resistances, and also include magnetic-field effects. The model is shown to fit MODFET data very well, with reasonable parameters, over a wide range of magnetic-field strengths (0-1 8 kG) and temperatures (5-300 K) .

11. TWO-LAYER TLM Both the one-layer TLM and one-layer MTLM have

been discussed in detail elsewhere [2], [4]. Thus, we will not rederive these models but will show later how they fit in as limiting cases of the two-layer model. The basic two- layer circuit considered here, shown in Fig. 2, consists of two contacts of length I, separated by bulk semiconductor material of length 1. In practice, the contacting materials may diffuse through one or both layers [ 11, [6] so that R, , # rr l , and R\* # rY2. For example, in the MODFET structure shown in Fig. 2, if the contacts are the com- monly used Au/Ge/Ni, then it is well knFwn that all three materials can diffuse more than 1000 A under typ- ical anneal conditions [6]. The specific contact resistivity p < , denotes the M / S barrier resistance between layer 1 and the metal, while p( denotes the barrier resistance be- tween layers l and 2, both in the contact region and in the bulk. If a GaAs cap layer has been added in order to de- crease the overall parasitic resistance, then the two-layer model does not strictly apply, although p c 2 could then be considered to represent the barrier between layer 2 and the GaAs cap, i.e., the barrier through all of the AlGaAs

0018-9383/88/0200-0133$01 .OO @ 1988 IEEE

134 IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 35, NO. 2. FEBRUARY 1988

..--P Planar Contact Diffused Contact

Fig. I . (a) The TLM pattern used in this study. The nominal dimensions of I , , 12. I,, I,, l'., and w are 2 , 4, 6, 10, 40, and 65 pm, respectively. (b) Planar and (c) diffused contacts in MESFET material. (d) Planar and (e) diffused contacts in MODFET material.

- V - -

Con I a c I Diffusion

I1 -----c---c-c

x o e t X

f c

Fig. 2. A two-layer distributed-resistance circuit applicable to MODFET material. Layer 2 is assumed to be mainly due to the 2D electron gas, while layer 1 is due to parallel conduction in the doping layer.

material. The cap would normally not add a third layer in the material between the contacts since in this region it would be depleted by the surface potential.

Consider first the contact region, 0 5 x I I, in Fig. 2. The relevant differential equations at a point x can be writ- ten

where V, and Zj denote the potential and conventional (positive) current in layer i , and w is the device width. Here, the left contact is held at potential V . Equations (la)-( Id) can then be transformed into two second-order

differential equations

% + k;Vl - k;V2 du2

= o , 0 5 x 5 1 , (2b)

where k: = R , , / p C l , k i = Rs2/pc2, and k:2 = R,$I/pc2. A convenient way to solve systems of coupled differential equations is by means of Laplace transforms [7], i .e., to find functionsfi(s) and&(s) such that VI = L { f i ( s ) } , and V2 = L { f 2 ( s ) } . It turns out thatf l (s) andf2(s) both have the following form:

( 3 ) polynomial of order s3

J ( s ) = (s2 - k:)($2 - k i ) - k:2S2'

Unfortunately, the k:2s2 term in the denominator makes an analytical solution intractable. Since it is far more in- structive and useful to work with a closed-form solution, we will confine ourselves to cases for which k:2 << k: , or pc2 >> p c l . Then, since V2 < V , it is also true that k:2V2 << k:V, and therefore both k t 2 terms drop out of (2a). In the end, when fitting a particular set of data, it will be necessary to see whether the fit gives pc2 >> pel;

otherwise, the theory will not be applicable. Fortunately, literature values [ 11, [5] suggest that we can expect pc2 =

and pcl = IOp6 Q 1 cm2, so that p c 2 / p c . , = 10. The Laplace-transform solutions of (2a) and (2b) give

expressions for Vl(x) and V 2 ( x ) in terms of V,(O), V2(0) , I,(O), and 12(0), which constitute the four unknowns in the problem. Equations for I , ( x ) and 1 2 ( x ) then follow from ( la ) and (lb), respectively. The geometry of the circuit in Fig. 2 requires that I 1 ( I c ) = Z2(1(.) = 0, giving two equations that relate the unknowns to V . However, two more equations are necessary because of the four un- knowns.

LOOK: TWO-LAYER MAGNETO-TLM CONTACT RESISTANCE MODEL 13s

The other equations can be obtained by solving the dis- tributed-resistance problem in the bulk region, -1 5 x I 0 in Fig. 2. The relevant differential equations are

where K : E r r l /pc2 and K ; = rs2/pL2. The solution to (4) and ( 5 ) involves a quantity defined by k, where k2 = K : + K; = ( r s l + rs2)/pc2. Note that we have assumed that p c 2 in the bulk is equal to the value in the contact region, not out of necessity, but because it is a reasonable assumption and avoids adding another fitting parameter to the final equation. After solving for V l ( x ) and V 2 ( x ) in terms of the previous four unknowns, we can get the final two necessary equations from the relationships V - VI( 0) = VI( -1 ), and V - V 2 ( 0 ) = V2( -1 ), which must hold because of symmetry. Then, from the four equations and four unknowns, ZI( 0 ) and Z2( 0 ) can be solved in terms of V , and the total circuit resistance obtained from R = V/Z = V/[Z l (0 ) + Z2(0)]. The answer can be written as a function of I, sinh (kllc), cosh (klZc), sinh (k21C), cosh ( kZ1, ), sinh (kl ), and cosh (kl ). However, the nature of the actual TLM pattern in Fig. 1, as compared to the cir- cuit in Fig. 2, requires a further restriction. The problem arises with the boundary conditions Z l ( l c ) = Z2(1,) = 0, which are not precisely correct for the actual two-layer TLM structure, because unlike the single-layer case, there can be parallel current paths involving contacts to the left and right of the two being measured at any given time. The correct boundary condition would be Zl(lC) + Z 2 ( l c ) = 0, since only two contacts are current sinks in the cir- cuit at any one time. Fortunately, however, we are aided by the fact that the contacts are almost always “electri- cally long,” i.e., kllc, k21, 5 2, so that effectively, I l ( l c ) = Z2( 1, ) = 0. In this approximation, sinh (kllc) = cosh (k, 1, ), which allows simplification and elimination of these terms. However, it is not necessary to impose any restrictions on the size of kl, even though it will turn out in the case presented here that kl 2 2 for all of the 1 ’ s in the TLM pattern. The final result can be written in the familiar form

where the Ci’s are two-layer “correction” factors given by

(7)

(9)

whereF(k1) = s inh(k l ) / [ l + cosh(kl)] . Notethatfor kl 5 1, F ( k l ) = k1/2, while for kl L 4, F(k1) = 1. Thus, we immediately see two very important properties of two-layer contact-resistance measurements: 1) R does not become truly linear in 1 until 1 2 4k-’, and thus the usual extrapolation of R versus 1 to get R, must be ap- proached with caution; and 2) the slope of R versus 1 de- pends only on the parallel sheet-resistance combination of rsl and rS2, the bulk sheet resistances, as long as 1 2 k- ‘ , which usually is easy to satisfy.

We next consider several limiting cases of (6) . 1) 1 + 0. In this case, C2 + 1 and

4, 1 R ( l + 0 ) = 2 1 + - [ Rs2 1 + k,/k2] 2 (lo)

which would always be the extrapolated value of R versus I for small enough I( I << k - ‘ ) , and which therefore gives the “true” two-layer contact resistance. However, it might not be technically possible to make 1 << k - l in every case.

2) rs2 -+ co. Here, the bulk is one-layer, and it can be shown that C1 C2 + 1, giving

Le., the classical one-layer result. The reason that contact layer 2 does not enter at all is that we previously assumed that p r 2 >> p C I in order to get closed-form solutions to the differential equations. Thus, in this approximation, current entering contact layer 1, from bulk layer 1, flows directly into the contact metal without mixing into contact layer 2.

3 ) rsl -+ 00. In this case, C,C2 + klRs2/k2RsI, and thus

also a classical one-layer result. Note that R,y, and pcl do not enter (12), again because of the original assumption pc2 >> p c l , i.e., the barrier resistance from contact-layer 1 to the metal is considered small compared to that from contact layer 2 to contact layer 1. Case 3 would apply if a surface or gate potential fully depleted bulk layer 1.

4) Rs2 + 00. Here, the result is

W S 2 + 4

136

This example illustrates well how the F ( kl ) term can be part of the “slope” (if kl 5 1 so that F(kE) - I ) , or part of the “intercept” (if kl 5 4 so that F ( k l ) is inde- pendent of I ).

5 ) pc2 -, 03. For pc2 -+ 00, both k and k2 + 0, and it can be shown that (6) reduces to the one-layer case, (1 1). This is, of course, the expected result since, if pc2 = 00, no current can flow from layer 2 to layer 1, in either the bulk or contact regions.

111. TWO-LAYER MTLM Equation (6) has six unknown parameters, r, , , R s l , rs2,

Rs2, p e l , and pc2. The independent variables, I and w, are rather hard to vary since they are often fixed by a single available test pattern, such as the one in Fig. l(a). By adding a perpendicular magnetic field, however, two ad- vantages are obtained: 1) it is possible to continuously vary the relative as well as absolute sizes of the terms in (6); and 2) mobility information is obtained directly. The relevant magnetoresistance formulas are

r,, = YSIO( 1 + d B 2 ) = ( 1 + PL12B2)/ensrPr ( 1 4 4

Rsi = Rsio( 1 + pLfiB2)

= ( 1 + P 3 2 ) / e n , c , C L c r (14b) where p e l and n,,, , respectively, denote the mobility and sheet carrier concentration in layer i under the contact. Equations (14a) and (14b) are valid under quite general conditions, as discussed elsewhere [8]; in particular, for degenerate electrons, usually the case for MESFET’s and MODFET’s, they hold for arbitrary B , and the mobilities are true “conductivity” mobilities (no differences due to energy averaging). The number of fitting parameters is now 10 (n,,, nsc,, p , , pel, and per f o r i = 1 , 2 ) , and the data must be fitted with a computer. For the data pre- sented below (Fig. 3), a least squares fit to 20 points, on a DEC8800 computer, typically required less than 2 s/temperature. However, as expected with such a large number of fitting parameters, local minima were abun- dant, and it was necessary to try several different sets of starting values in order to determine the final uniqueness of each parameter. As seen in Table I , the accuracy is good for some of them and poor for others. However, for single-layer MTLM problems, described by ( 1 1) and (14), graphical techniques give easily obtained, unique values for nT1, p l , and pel, as shown before 141. In that case, if R,, can be estimated by comparing pl and p c l , then pel can also be accurately calculated.

IV. RESULTS The MODFET material reported here had the following

structure, from bottom to top: seomi-insulating substrate, ld-pm p--GaAs buffer layer, 200-A p--Ino 15Gao 8sAs, 30- A p--Alo lsGao s5As, 350-A n+-Alo 15Gao 8 5 A ~ , and fi- nally, a cap layer, 200-A nf-GaAs. Much information has already been reported on this particular structure [9],

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 2, FEBRUARY 1988

O L 0 2 4 6 8 1 0

Contact Separation (p)

Fig. 3. Resistance versus contact spacing for MODFET material at 77 K . The solid line is a two-layer fit , using (7) and ( IO) , and the dashed line is a one-layer fit, using (8) and (loa).

[lo], and the details are not important here. We may con- sider it as a generic MODFET structure for testing our two-layer model.

The TLM pattern used for this study had contact sepa- rations, as measured by an optical microscope, of 1.5, 3.6, 5.9, and 9.8 pm. The voltage used was 20 mV, and magnetic field strengths B ranged from 0 to 18 kG. The R versus I data are shown in Fig. 3 along with fits using one-layer and two-layer MTLM theories. The two-layer model fits very well whereas the one-layer model does not. (In this case, the one-layer model was fitted at B = 0 for all 1, and at B = 18 kG for I = 1.5 pm, somewhat arbitrarily. However, other combinations looked just as bad. ) Also, the one-layer TLM theory (no magnetic field) was fitted and was very close to the B = 0 curve. Thus, each model fits the B = 0 data fairly well.

The fitted parameters are given in Table 1 for all three models. For the bulk material, the two-layer MTLM sug- gests that the AlGaAs retains about 2.3 X 10l2 elec- trons/cm2, with 296-K mobility of 1500 cm2/V * s , and the 2DEG about 1.2 X 10l2 electrons/cm2 with a mobil- ity of 7200 cm2/V * s . The latter data are in good agree- ment with Hall-effect measurements 191, when account is taken of the parallel conduction. The one-layer MTLM predicts too low a value of 2DEG mobility, again because of the parallel conduction. Note that the n+-GaAs cap layer should be largely depleted due to the surface poten- tial, but, if not, then it would effectively constitute a small part of layer 1 in the bulk region.

The contact region, however, is probably much more complicated due to contact-element diffusion. The 296-K results in Table I suggest that layer 2 consists of a large concentration of electrons ( - 1 x loi3 cm-2) of lower mobility ( -4000 cm2/V s ) than that of the 2DEG elec- trons in the bulk. Also, since the 77-K layer 2 mobility

LOOK TWO-LAYER MAGNETO-TLM CONTACT RESISTANCE MODEL 137

TABLE 1 BULK A N D CONTACT ELECTRICAL PARAMETERS FROM VARIOUS TLM MODELS

b u l k c o n t a c t

I l C l pc2 - - - sc2 nscl “2 ~ - - - u l ’ b Model T’( K) I n - n . - ’ S L - S I . . - - - 10 I 'ern-'

Tvo-layer(a) 300 1.2t0.1 2 . 3 t 0 . 4 (mag. f i e l d )

77 1.3tO.1 2.7t0.3

One-layer(b) 300 1.8k0.1 (mag. f i e l d )

77 1.3tO. I

104cm2/~-sec 10 12cm-2 104cm2/~-sec

0.72t0.02 0.15t0.05 l l t 2 0.7’0.2 0.4t0.1 0.12t0.07

2.73t0.02 0.11?0.02 8f5 0.4f0.3 0.6fO.1 0.06t0.03

One-layer(c) 300 r = ( 6 . 1 ? 0 . 1)x102Cl/0 (no mag. f i e l d )

77 r =(z .o*o. I ) ~ I o ~ ~ / o

0.58f0.02 1.8tO.1 0.56k0.02

2.3’0.1 1.350. I 1.3t0.1

R =r (assumed) s s

R =r (assumed) s s

10-62-cm2

9+4 1.0k0.5

7f4 1.0t0.5

0 . 9 7 t 0 . 0 2

2 . 4 ’ 0 . 1

0.99k0.02

2 . 5 ’ 0 . 1

~ ~ - ~ ~ ~ ~ p ~~ ~

a) C a l c u l a t e d from least-squares f i t of Equat ions 6 and 1 4 .

b)

c ) Calcula ted from Equation l l a t B=O. Assumption t h a t Rs=rs.

C a l c u l a t e d from f i t to Equations lland 14 at low B. Assumption that ”bulk= ncOntact

under the contacts is much lower than that in the bulk, it appears that the 2DEG under the contacts contains excess ionized impurities. It is probable that the Ge, at least, has diffused through the 2DEG region, as has been observed by RBS [6] and other techniques. However, the full tem- perature dependence of the mobility, to be published else- where, suggests that the 2DEG is at least partially main- tained under the contact.

Contact layer 1, on the other hand, evidently can be approximated by about 1 X 10l2 electrons/cm-2 with mobility about 1200 cm2/V s . The specific contact re- sistivity from layer 1 into the metal is pc, = 1 x lop6 Q * cm2, a typical value measured by others [ l ] , whereas the current transfer from layer 2 to layer 1 is represented by p c 2 = 1 X lop5 Q cm2, which is also typical of measured and calculated values for Al, $ao ,As /GaAs heterostructure barriers [ 5 ] , [ 111. Fortunately, our fit gives pc >> p c ,, which is necessary for the applicability of the theory. The lack of a significant temperature dependence for either pcl or pc2 suggests tunneling mechanisms for each. Note from Table I that the single-layer models would both have predicted a much higher temperature de- pendence for pcl , and the tunneling nature might thus have been hidden.

In conclusion, it appears that the two-layer MTLM technique will be a useful tool for both bulk and contact studies in MODFET materials, as the one-layer MTLM has been for MESFET materials. It is the first technique that yields mobility information on the material under- neath the contacts. However, it is clear that more exhaus- tive studies need to be carried out in order to interpret some of the parameters in particular cases, especially pc2 and nAcZ. It would be very helpful to have a TLM test pattern with one or more submicrometer values of 1 in order to get information in the region kl s 1.

ACKNOWLEDGMENT We would like to thank T. Cooper for the electrical

measurements, W. Theis for a data-fitting routine, H. Morkoc for the sample, A. Ezis for the TLM test pattern and helpful discussions, and P. Schwenke for typing the manuscript.

REFERENCES N. Braslau, “Contact and metallization problems in GaAs integrated circuits,” J . Vac. Sci. Technol., vol. A4, pp. 3085-3090, 1986. H. H. Berger, “Models for contacts to planar devices,” Solid-Srarr Electron.. vol. 15. pp. 145-158, 1972. G. K. Reeves and H. B. Harrison, “Obtaining the specific contact resistance from transmission line model measurements,” IEEE Elec- tron Device Lett., vol. EDL-3, pp. 111-113, 1982. D. C. Look, “Mobility measurements with a standard contact resis- tance pattern,” IEEE Elecrron Device Le f t . , vol. EDL-8, pp. 162- 164, 1987. M. D. Feuer, “Two-layer model for source resistance in selectively- doped heterojunction transistors,” IEEE Trans. Electron Devices, vol .

D. D. Cohen, T. S . Kalkur, G . J . Sutherland, and A. G. Nassibian, “Backscattering analysis of AuGe-Ni ohmic contacts of n-GaAs,” J . Appl. Phys., vol. 60, pp. 3100-3104, 1986. R. V. Churchill, Operational Mathematics. New York: McGraw- Hill, 1958, ch. 2. D. C. Look and G. B. Norris, “Classical magnetoresistance mea- surements in Al,Ga, _,As/GaAs MODFET structures: Determina- tion of mobilities,’’ Solid-State Elecfron., vol. 29, pp. 159-165. 1986. A. Ketterson, W. T. Masselink, J . S . Gedymin, J . Klem, V’. Kopp, and H. MorkoG, “Characterization of InGaAs/AIGaAs pseudo- morphic modulation-doped field-effect transistors,” IEEE Trans. Electron Devices, vol. ED-33, p. 564, 1986. D. C. Look, T. Henderson, C. K. Peng, and H. MorkoC, “Mobility and parasitic resistance measurements in AlGaAs/GaAs and Al- GaAs/InGaAs MODFET structures,” in Gallium Arsenide and Re- lated Compounds, W. T . Lindley, Ed. Bristol: IOP, 1986, pp. 557- 562. S . J . Lee and C. R. Crowell, “An analysis of low source resistance HEMT with multiple cap layer,” presented at the Device Research Conf., paper IIIA-6, 1984.

ED-32, pp. 7-11, 1985.

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138 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 2. FEBRUARY 19x8

David C. Look (M’87) received the B. of Physics and M.S. degrees from the University of Minne- sota in 1960 and 1962, respectively, and the Ph.D. degree in physics from the University of Pitts- burgh in 1965. His thesis work introduced the concept of studying low-frequency molecular mo- tions by analyzing nuclear spin-lattice relaxation in the rotating reference frame, a commonly used technique today.

From 1966 to 1969, he was an Air Force officer at Wright-Patterson AFB, Dayton, OH, and car-

ried out nuclear magnetic resonance studies in 11-IV semiconducting com- pounds. From 1969 to 1971, he was Research Physicist, and then from

1971 to 1980, Senior Research Physicist, at the University of Dayton, spe- cializing in NMR and transport studies of 11-VI and 111-V semiconducting compounds. Since 1980, he has been Senior Research Physicist and Re- search Professor of Physics at Wright State University, Dayton, OH, car- rying out transport measurements in 111-V materials and devices. Most of this research is conducted at the Avionics Laboratory, Wright-Patterson AFB, where he is Principal Investigator of an on-site research contract, sponsored by Wright State University. Recent interests include new tech- niques to measure basic electrical parameters in actual device structures, such as MESFET’s and MODFET’s, and the identification of point defects in GaAs.

Dr. Look is a fellow of the American Physical Society and a member of the Electrochemical Society and the American Scientific Affiliation.