- Email: [email protected]
GaAs heterojunction bipolar transistors
Solid-State Electronics Vol. 35, No. 11, pp. 1567-1512, 1992 Printed
0038-I101/92 85.00+ 0.00 Copyright 0 1992Pergamon Press Ltd
in Great Britain. All rights reserved
EFFECTIVE LATERAL DIFFUSION LENGTH IN Npn AlGaAs/GaAs HETEROJUNCTION BIPOLAR TRANSISTORS WILLIAM LIU Central Research Laboratories, Texas Instruments, P.O. Box 655936, MIS 134, Dallas, TX 75265, U.S.A. (Received
12 March
1992; in revised form
I8 April 1992)
Abstract-The effective lateral diffusion length for the base minority carriers in HBT is calculated. This effective lateral diffusion length is a measure of the spreading distance toward the base contacts for the base minority carriers injected from the emitter before reaching the collector. With knowledge of this effective lateral diffusion length, the extrinsic base surface recombination current for any device geometries, epitaxial designs and operating conditions is easily determined. This quantitative determination of surface recombination current allows one to directly correlate one’s experimental results on extrinsic base surface recombination with theoretical values.
1. INTRODUCTION
AlGaAs/GaAs heterojunction bipolar transistors (HBTs) fabricated with typical etching techniques have exposed extrinsic base surfaces. Since the free GaAs surface is characterized by a high surface recombination velocity and a large number of surface states[ 11,significant extrinsic base surface recombination in HBTs occurs. The resulting extrinsic base surface recombination current, Zsurfa,is especially dramatic when the device size is small such that the periphery-to-area ratio is large[2]. Several experimental studies on the effects of extrinsic base surface recombination current were published[2-91. While these experimental results provide direct evidence that the extrinsic base surface recombination substantially degrades the device current gain and l/f noise performance, there was no accompanied theoretical calculation to examine quantitatively the effects of the surface recombination. The reason for the lack of a theoretical calculation of Iaurfais the following. Unlike the base bulk recombination current which occurs primarily in the intrinsic base region (under the emitter mesa) and is readily determined from a one-dimensional analysis, the extrinsic base surface recombination current occurs on a plane normal to the device current flow (Fig. 1). Therefore, a theoretical determination of fsurra requires solving a twodimensional current continuity equation in both the extrinsic and intrinsic base regions with complex boundary conditions on the base surfaces. Even though an analytical solution for Zsurfawas demonstrated[lO,l I], the solution was in a formidable form and the numerical calculation required non-trivial computer programming. Such difficulties in calculating Isulfa impede one from quantitatively determining the magnitude of IJulra to compare with his experimental results.
In an attempt to facilitate the determination of I surfarthis paper calculates the effective lateral diffusion length, Ld, which was first proposed by Fonger [12]. This effective lateral diffusion length is a measure of the spreading distance toward the base contacts for the base minority carriers injected from the emitter before reaching the collector. The dependence of Ld on collector current (I,), emitter width (24), base doping concentration (N,_), emitter-base contact spacing (d2 - d,), and base thickness ( YO) is calculated for typical Npn HBT structures. These calculated results demonstrate that L, varies strongly with base thickness, but is relatively insensitive to all other variables. The physical reasons for such characteristics are explained. With knowledge of Ld, Zsudacan then be easily determined from a simple calculation for arbitrary device geometries, epitaxial designs, and operating conditions. 2. THEOREITCAL BACKGROUND
The schematic diagram of a Npn HBT under consideration is shown in Fig. 1, from which the extrinsic base surface recombination current is given as, 4 I sura= q&h
I4
(1)
n(~,y)ly=,dx
where h is the emitter length, S, is the extrinsic base surface recombination velocity, and d2 - d, is the spacing between base-emitter contacts. The base minority electron concentration, n(x, y), satisfies the 2-dimensional current continuity equation in the base (both the extrinsic and intrinsic regions)
1567
vz+
y)
+fan("YY) ---
ay
WY)
0”
=
0
(2)
1568
WILLIAM
LIU Xo2-
dl_*
d 2 +
Base Contact
i
i I
Passivation Ledge
Base Contact
0
Collector Fig. 1. A schematic diagram of a Npn HBT under consideration.
where f is a base electric field factor, equal to the base electric field divided by the thermal voltage, and t, and D, are the minority electron lifetime and diffusion coefficient, respectively. A straightforward determination of Zsurfais not possible since the two dimensional electron concentration of eqn (2) cannot be easily solved for with the set of complicated boundary conditions inherent in the HBT structure shown in Fig. 1 (see Fig. l(b) of Ref. [lo]). Nevertheless, a complex analytical solution for n (x, y), and thus Zsurra 9does exist[lO], and Is,,,+can be expressed as
zsurfa= -qS,h
t
$,,,K,,, sinh(y,Y,,)
m=, X
sin I,,, d2 - sin I,,, d,
(3)
AlI
where A,,,, Y,,,, B,,, and K,,, are related to the m th Eigen coefficients of the associated Eigen solutions when one applies the technique of separation of variables to solve eqn (2). $,,,, in contrast, is specifically related to the boundary conditions of a given device structure.
The expression of $,,, involves finding the inverse of a 3n x 3n matrix, where n is the maximum number of the terms used in approximating the infinite series of eqn (3). n is typically taken to be 90 for reasonable accuracy[lO]. Therefore, the determination of requires non-trivial computer + ,,,, and thus ZsUKaa, programming. Consequently, while many experimental results pointed out the undesirable effects of ZsUtia [2-91, there was no attempt to compare those experimental results to a quantitative calculation of ZSUm such as that given in eqn (1). A simplistic way to allow such comparison is to make available a tabulation of ZsUli.calculated from eqn (1) as functions of various parameters. However, such a direct tabulation is impractical since IsUlfais a strong function of many parameters. Some of these parameters relate to device geometry (such as d,), some relate to epitaxial design (such as N&, and some even relate to operating condition (such as I,). Such strong variations of IsUrnrender the tabulation of IsUrfsinconvenient to use. An alternative is to tabulate a normalized parameter which closely relates to Zsumin a simple fashion but has the property that it is relatively independent of variations in device geometries,
Diffusion length in heterojunction transistors
1569
epitaxial layer designs, and operating conditions. A parameter which fits this requirement is the effective lateral diffusion length, 4. It is a characteristic length for the exponential decrease of electron concentration on the extrinsic base surface n(x,JJ)l,.=,=
no *ev[ - 6 - 4 &I
(4)
where no is the uniform electron concentration under in the intrinsic base region (under the emitter mesa). Physically, L, is a convenient measure of the spreading distance toward the base contacts for the base minority carriers injected from the emitter before reaching the collector. Its magnitude compared to the base thickness thus largely determines whether the extrinsic base surface recombination current is significant or not. One notes that the concentration no can be determined from the collector current, Z,, which can be written as q ’ d, . h .D. *h/Y,. Moreover, as will be demonstrated later, d2 - d, % Ld in typical device geometries. Therefore, the relation between Ld and Za.tiacan be expressed as Zsutia=
sayol,
+VLdno = m.
n
(5)
Ld.
I
From the same detailed two-dimensional analysis which derives the analytical solution of ZSUfla of eqn (3), an analytical solution of Z, is given as[lO] I,=
-qD.he”Ya”~1I~K~~~~. m
(6) tn
Combining eqns (3), (5) and (6), one then can write an analytical expression for Ld Ld
d
=
_!_
e/‘Yo/2
yo - sin 1, d, cIn ti,KmsinW, Yo)sin A,,,d2An
m
L,
(7)
’
~Jr,K,y,+ nm
(A,
Fig. 3. Calculation of L,, as a function of base-emitter contacts spacing. d, = 1 pm, N,_ = 2 x lOI crtm3, and Y,=800a.
It should be noted that eqn (7), though complex, is analytical and is derived from eqn (2) without approximation. A numerical program is used to calculate the values of L, for various device geometries, epitaxial layer designs and operating conditions. It will be shown that L, is relatively insensitive to the variations in most parameters, thus demonstrating the usefulness of tabulating Ld rather than ZSUm directly. Once Ld is tabulated as shown in Figs 2-4, ISUrf,, is easily determined for any device from eqn (5). This would greatly simplify the attempt to correlate experimental results to quantitative theoretical values of Zsurfa,without the need to write a non-trivial numerical program[ lo]. Note that the above theoretical analysis bases on the solution to the 2-dimensional current continuity equation in the base layer[ lo]. Because the base layer is a quasi-neutral region, this analysis does not involve solving the Poisson equation as would be necessary for space charge region such as the base-emitter junction. A computationally much more
100
70
-
10’9
device
90
7
80
y
70
:
60
-
50
:
”
I,
9
,
”
I
”
I,
”
1020 N bcuc (one’)
40
Fig. 2. Calculation of L,, as a function of base doping concentration. 4 = 1 pm (hence, as shown in Fig. 1, the actual
f’
emitter width is 2/rm), d,-d,=3000~.
Y. = 800 A,
and
0
L~l’ll*‘lll’lll”” 400
800 y,,
1200
1600
2000
(A)
Fig. 4. Calculation of L,, as a function of base thickness. d,=lpm, Nbuc= 2 x lOi9 cme3, and d2 - d, = 3000 A.
1570
WILLIAM
involved study which considers the Poisson equation was carried out[l3]. The study demonstrated that in the quasi-neutral base region, the surface recombination current has an ideality factor close to unity because the surface recombination rate is limited by the availability of minority carriers and the quasiFermi level does not remain flat. While this computationally more involved study generates the voltage distribution in addition to the desired current calculation, the present analysis naturally leads to a unity ideality factor for IsUrfsand is adequate for the purpose of determination of the effective diffusion length. The fact that the present analysis leads to unity ideality factor for Z6UtiB can be understood from eqn (5), where ZsUfi,,is shown to be directly proportional to Z,, with a proportional constant dependent on the device geometry. Since Z, is known to have a unity ideality factor, Zstia also has a unity ideality factor, as determined by the computationally more involved study. As mentioned, this theoretical analysis is based on an analysis which determined the magnitudes of various base current components, and consequently, the current gain for several device geometries and device epitaxial structures[lO]. Experimental current gains measured from passivated and unpassivated HBTs with different base doping levels were shown to agree with the published analysis[lO]. Furthermore, Fig. 3 of the published analysis demonstrated that with a d2 - d, = 1500 8, depleted AlGaAs ledge to passivate the extrinsic base surface of the HBTs, Zsurra was virtually eliminated and current gain saturated at its highest values. This published result is again supported by a separate experimental study which demonstrated that a passivation ledge (achieved by self-alignment) with dz - d, = 2000 8, was long enough to passivate the extrinsic base surface and achieve maximum current gain[6]. (Yet another experimental work demonstrated partial passivation with a passivation ledge having dz -d, = 1 pm; however, its authors acknowledged that there was significant misalignment tolerance of at least 0.5pm[3].) Since the present analysis to determine the effective diffusion length originates from the experimentally verified analysis[ lO],‘it is expected that Isure calculated from the L, derived in this analysis will agree with its experimental values.
3. CALCULATIONS
AND DISCUSSIONS
The numerical program used to calculate L, from eqn (7) is a modification of the program used in Ref. [lo]. The Einstein relationship is used to relate D, to the minority electron mobility, p,,. In the calculations, the material parameters p,, and r, as function of the base doping concentration, Nbase,are taken from Ref. [14]. The surface recombination velocity in an unpassivated surface is assumed to be S, = 1 x 107cm/s[15]. For the base contacts, the
LIIJ
surface recombination velocity is taken to be 2 x lO’cm/s, which is half of the average free electron thermal velocity. This is a result of assuming that there is no reflection for electrons striking the base contact interface[l6]. The HBT structures under consideration have a uniform base layer, thus the base quasi-electric field is O(f = 0). L, as a function of Z, is calculated for a typical self-aligned microwave HBT with d, = 1 pm (hence, as shown in Fig. 1, the actual device emitter width is 2 pm), N,_ = 2 x lOi cm-), dz - d, = 3000 A, and Y0= 8OOA. In general, IsUrf. which scales linearly with no is directly proportional to Z,. In contrast, the calculation shows that Ld remains - 72 A for all the operating current levels. This constancy of L, results because, despite the increase in n,, the general characteristics of the minority carrier profile on the surface remain unmodified. In addition, L, is calculated for the same HBT structure but with varying emitter width (2d,). Again, the calculated result is independent of the current levels. More importantly, this calculation demonstrates that L,, calculated from eqn (7) is also independent of d, ; Ld is found to be 72 j, 1 A for d, varying from 0.5 to 5 pm. Such finding agrees with the intuitive view that increasing the emitter width only increases the total number of electron carriers flowing through the emitter mesa, but does not affect how far the carriers near the mesa periphery diffuse laterally toward the base contacts. As will be demonstrated, Ld is an ideal normalized parameter, being insensitive not just to Z, and 4, but also to d2- d, , and N,_. It is sensitive, however, to the value of Y, (as it should be). Therefore, one can then use Ld = 72 A along with eqn (5) to quantitatively determine the value of Zsurfa for any HBT with Y0= 800 A. However, if one were to directly tabulate Zrurrainstead of L,, then one would necessarily tabulate Zsurra against additional parameters such as Z,, d,, and N bilse,rather than just Y,,. Note that for this HBT, the carrier concentration at the extrinsic base surface is well characterized by an effective lateral diffusion length roughly equal to 72 A. This distance is indeed much smaller than the ledge distance d, - d,, as assumed in the derivation of eqn (5). Figure 2 shows the values of L, as a function of the base doping concentration for a typical selfaligned HBT with d, = 1 /*m, dz - d, = 3OOOA, and Y0= 800 A. As Nbaw increases, both D,, and ‘t, decrease, causing more base minority carriers to recombine in the bulk base layer. However, such material parameter modifications in the bulk base region do not strongly affect the extrinsic base surface carrier concentration, which is primarily determined by the surface recombination velocity. Therefore, the exponential decrease profile for the surface carrier concentration is not significantly modified. As shown in Fig. 2, Ld does not change dramatically with Nb_. For 1 x 10’9cm-3 < N baxc 1 x 10zocrn3, Ld varies between 73 and 71 A, representing a 3% variation.
Diffusion length in hetcrojunction transistors
In contrast, the magnitude of Zstia depends on N sase much more strongly (from eqn (5), Zsulr. directly depends on D,, which changes -7% from Nbase= 1 x lOI9cmm3 to Nt_= 1 x 1020cm-3[14]). Note that this relatively constant value of Ld again supports characterizing the extrinsic base surface recombination by L,. Figure 3 illustrates the variation of Ld as a function of base-emitter contacts spacing for a typical selfaligned HBT with Nbase= 2 x 1Or9cmm3, d, = 1 p, and Y,,= 800 A. As shown in Fig. 3, L,, is small (58 A) when the base-emitter contact spacing is only 500 A. However, L, increases rapidly as d2 -d, increases, saturating at a value of -72 A as d2 - d, reaches 1500 A. This variation of Ld agrees with an intuitive view. That is, when d2 - d, is very short (< 1500 A), the laterally diffused carriers can pass by the extrinsic base surface and physically reach the base contact interface[3,11]. This result in an appreciable amount of base contact recombination and a lesser amount of extrinsic base surface recombination. Since L,, is defined to characterize only the extrinsic base surface recombination, it starts out having smaller values. However, as d2 -d, approaches 1500 A, the base contact recombination current becomes negligible. Any additional increase in d2 - d, does not modify the surface carrier concentration profile since the carrier concentration has exponentially decreased to nearly zero at x - d, = 1500 A. Since typical selfalignment techniques in HBTs rely on a certain under-cut to prevent base-emitter contact shorting and such an undercut typically has a minimum distance of 2000 8, (> 1500 A)[3,6,17], Ld is again considered to be constant for all practical baseemitter contacts spacings. Figure 4 illustrates the values of L, as a function of the base thickness for a typical self-aligned HBT with Nbax = 2 x 10’9cm-3, d, = 1 pm, and d2 - d, = 3000 A. In contrast to prior calculations which demonstrate that L,, remains relatively constant as Z,, Nbase, d, , or d2 - d, vary, Fig. 4 shows that L, is a strong function of Y,. It varies from 48 A at Y, = 300 A to 94 A at Y,,= 2000 A. This strong variation results because as Y,,increases, the injected carriers must diffuse laterally through a greater distance to reach the collector. Consequently, a larger portion of the carriers can diffuse laterally toward the base contacts and recombine on the extrinsic base surface. It is clear from Fig. 4 that decreasing Y, decreases Ld, hence reducing Zsurfa.Furthermore, such decreases in Y0also reduces bulk base recombination current[l8]. Therefore, current gain is expected to increase significantly with a thin base structure. However, this advantage is counteracted by an increase in base resistance associated with the thinner base, which degrades device high frequency performance. From these calculated results, one observes that Ld is a strong function of Y0 but it has very weak dependence in all other geometric parameters, epitaxial layer designs and operating conditions. Conse-
L,
1571
(A,
300 I 100
-
30 _ ,,,,,,, I, ,,,,,,,,, 102
10”
,,,,,,,,,
104
105
‘ ,,,,,,,, 10”
/,,,,,
u 10’
So ( cmlsec )
Fig. 5. Calculation of L,, as a function of surface recombination velocity. d, = 1 pm, N,_ = 2 x lOI cxre3, d,-d,=3000& and Y,=8OOA.
quently, one only needs to use Fig. 4 and eqn (5) to determine quantitatively the magnitude of Zsurra for all Npn HBTs. In contrast, Zstia exhibits strong dependences on various parameters. Therefore, quantifying the extrinsic base surface recombination with the normalized parameter, Ld, is much more convenient than tabulating Zsyrfadirectly. It is interesting to see how Ld varies with the assumed value of surface recombination velocity. Figure 5 shows L,, as a function of S, for a typical self-aligned HBT with Nt_ = 2 x lOI9cme3, d, = 1 pm, d, - d, = 3000 A, and Y0= 800 A. When the exposed extrinsic base surface is near ideal with S, = lo3 cm/s, Ld remains at a large value of N 220 A. However, L,, decreases rapidly when S, is greater than lO’cm/s. This rapid decrement of L, indicates that the surface carrier profile shows an even more rapid exponential decrease. This is expected since, as S, increases, the carriers which are injected from the emitter and which diffuse laterally toward the base contacts would survive only for a short distance before being recombined. As mentioned, L, decreases to N 72 A when S, = 10’ cm/s, which is the experimentally determined surface recombination velocity for exposed GaAs surface[l5]. In addition, Fig. 5 demonstrates that once the extrinsic base surface recombination velocity can be reduced below 10’ cm/s, the surface carrier profile will exponentially decrease in a fashion unaffected by the extrinsic base surface recombination. This shows that the extrinsic base surface needs not to be perfect (S, = 0) in order for the base bulk recombination current to dominate the total base current. 4. SUMMARY The quantitative determination of the extrinsic base surface recombination current for arbitrary HBT geometry, epitaxial design, and operating
WILLlAM
1572
dition is achieved with the calculation of a normalized parameter, Ld . This effective lateral diffusion length for the base minority carriers is found to vary strongly with the base thickness, but is relatively insensitive to all other parameters. The calculation of I s,,rfD thus involves first identifying the 15, value corresponding to a particular base thickness from Fig. 4, and then calculating ZsUfi8from eqn (5). This easy allows one to directly correlate determination of ZJUria one’s experimental results on extrinsic base surface recombination with theoretical values.
LIU 4. R. N. Nottenburg, C. J. Sandroff, D. A. Humphrey, T. H. Hollenbeck and R. Bhat, Appl. Phys. L&t. 52, 218 (1988). 5. S. Shikata, H. Okada and H. Hayashi, J. Vat. Sci. Technof. B 9, 2479 (1991). 6. N. Hayama and K. Honjo, IEEE Electron Deu. L&t. 11, 388 (1990).
7. W. Liu, Electron. L.&t. 28, 379 (1992). 8. D. Costa, W. Liu and J. S. Harris Jr, IEEE Deuice Res. Conf. naner VIA-3 (1991). 9. D. Co&a and J. S.‘Harhs Jr, Int. Symp. GaAs and Related Compounds (1991). 10. W. Liu, D. Costa and J. S. Harris Jr, Solid-State Electron.
34, 1119 (1991).
11. D. P. Kennedy and P. C. Murley, Solid-State
discussions with D. Costa and A. Khatibzadeh on the need to quantify the extrinsic base surface recombination and on I/fnoise are greatly appreciAcknowledgements-The
ated. The helpful suggestions by C. Dai and J. Harris, and the technical support in computation by E. Ziem are also acknowledged.
Electron.
3, 215 (1961).
12. W. Fonger, Transistors I, RCA Laboratories, Princeton, NJ (1956). 13. S. Tiwari, D. J. Frank and S. L. Wright, J. appl. Phys. 64, 5009 (1988).
14. S. Tiwari and S. L. Wright, Appl.
Phys.
Lett.
56,
563 (1990).
REFERENCES
1. D. E. Aspnes, SurJ Sci. 132, 406 (1983). 2. 0. Nakajima, K. Nagata, H. Ito, T. Ishibashi and T. Sugeta, Japan. J. appl. Phys. 24, L596 (1985). 3. W. S. Lee, D. Ueda, T. Ma, Y. C. Pao and J. S. Harris Jr, IEEE Electron Devices L.&t. 10, 200 (1989).
15. P. E. Dodd, T. B. Stellwag, M. R. Melloch and M. S. Lundstrom, IEEE Trans. Electron Devices 38, 1253 (1991). 16. J. P. McKelvey, Solid State and Semiconductor Physics, p. 350. Harper & Row, New York (1960). 17. K. Nagata, 0. Nakajima, Y. Yamauchi, T. Nittono and H. Ito, IEEE Trans. Electron Devices 35, 2 (1988). 18. see for example, S. Sze, Physics of Semiconductor Devices, p. 141. Wiley, New York (1981).
0038-I101/92 85.00+ 0.00 Copyright 0 1992Pergamon Press Ltd
in Great Britain. All rights reserved
EFFECTIVE LATERAL DIFFUSION LENGTH IN Npn AlGaAs/GaAs HETEROJUNCTION BIPOLAR TRANSISTORS WILLIAM LIU Central Research Laboratories, Texas Instruments, P.O. Box 655936, MIS 134, Dallas, TX 75265, U.S.A. (Received
12 March
1992; in revised form
I8 April 1992)
Abstract-The effective lateral diffusion length for the base minority carriers in HBT is calculated. This effective lateral diffusion length is a measure of the spreading distance toward the base contacts for the base minority carriers injected from the emitter before reaching the collector. With knowledge of this effective lateral diffusion length, the extrinsic base surface recombination current for any device geometries, epitaxial designs and operating conditions is easily determined. This quantitative determination of surface recombination current allows one to directly correlate one’s experimental results on extrinsic base surface recombination with theoretical values.
1. INTRODUCTION
AlGaAs/GaAs heterojunction bipolar transistors (HBTs) fabricated with typical etching techniques have exposed extrinsic base surfaces. Since the free GaAs surface is characterized by a high surface recombination velocity and a large number of surface states[ 11,significant extrinsic base surface recombination in HBTs occurs. The resulting extrinsic base surface recombination current, Zsurfa,is especially dramatic when the device size is small such that the periphery-to-area ratio is large[2]. Several experimental studies on the effects of extrinsic base surface recombination current were published[2-91. While these experimental results provide direct evidence that the extrinsic base surface recombination substantially degrades the device current gain and l/f noise performance, there was no accompanied theoretical calculation to examine quantitatively the effects of the surface recombination. The reason for the lack of a theoretical calculation of Iaurfais the following. Unlike the base bulk recombination current which occurs primarily in the intrinsic base region (under the emitter mesa) and is readily determined from a one-dimensional analysis, the extrinsic base surface recombination current occurs on a plane normal to the device current flow (Fig. 1). Therefore, a theoretical determination of fsurra requires solving a twodimensional current continuity equation in both the extrinsic and intrinsic base regions with complex boundary conditions on the base surfaces. Even though an analytical solution for Zsurfawas demonstrated[lO,l I], the solution was in a formidable form and the numerical calculation required non-trivial computer programming. Such difficulties in calculating Isulfa impede one from quantitatively determining the magnitude of IJulra to compare with his experimental results.
In an attempt to facilitate the determination of I surfarthis paper calculates the effective lateral diffusion length, Ld, which was first proposed by Fonger [12]. This effective lateral diffusion length is a measure of the spreading distance toward the base contacts for the base minority carriers injected from the emitter before reaching the collector. The dependence of Ld on collector current (I,), emitter width (24), base doping concentration (N,_), emitter-base contact spacing (d2 - d,), and base thickness ( YO) is calculated for typical Npn HBT structures. These calculated results demonstrate that L, varies strongly with base thickness, but is relatively insensitive to all other variables. The physical reasons for such characteristics are explained. With knowledge of Ld, Zsudacan then be easily determined from a simple calculation for arbitrary device geometries, epitaxial designs, and operating conditions. 2. THEOREITCAL BACKGROUND
The schematic diagram of a Npn HBT under consideration is shown in Fig. 1, from which the extrinsic base surface recombination current is given as, 4 I sura= q&h
I4
(1)
n(~,y)ly=,dx
where h is the emitter length, S, is the extrinsic base surface recombination velocity, and d2 - d, is the spacing between base-emitter contacts. The base minority electron concentration, n(x, y), satisfies the 2-dimensional current continuity equation in the base (both the extrinsic and intrinsic regions)
1567
vz+
y)
+fan("YY) ---
ay
WY)
0”
=
0
(2)
1568
WILLIAM
LIU Xo2-
dl_*
d 2 +
Base Contact
i
i I
Passivation Ledge
Base Contact
0
Collector Fig. 1. A schematic diagram of a Npn HBT under consideration.
where f is a base electric field factor, equal to the base electric field divided by the thermal voltage, and t, and D, are the minority electron lifetime and diffusion coefficient, respectively. A straightforward determination of Zsurfais not possible since the two dimensional electron concentration of eqn (2) cannot be easily solved for with the set of complicated boundary conditions inherent in the HBT structure shown in Fig. 1 (see Fig. l(b) of Ref. [lo]). Nevertheless, a complex analytical solution for n (x, y), and thus Zsurra 9does exist[lO], and Is,,,+can be expressed as
zsurfa= -qS,h
t
$,,,K,,, sinh(y,Y,,)
m=, X
sin I,,, d2 - sin I,,, d,
(3)
AlI
where A,,,, Y,,,, B,,, and K,,, are related to the m th Eigen coefficients of the associated Eigen solutions when one applies the technique of separation of variables to solve eqn (2). $,,,, in contrast, is specifically related to the boundary conditions of a given device structure.
The expression of $,,, involves finding the inverse of a 3n x 3n matrix, where n is the maximum number of the terms used in approximating the infinite series of eqn (3). n is typically taken to be 90 for reasonable accuracy[lO]. Therefore, the determination of requires non-trivial computer + ,,,, and thus ZsUKaa, programming. Consequently, while many experimental results pointed out the undesirable effects of ZsUtia [2-91, there was no attempt to compare those experimental results to a quantitative calculation of ZSUm such as that given in eqn (1). A simplistic way to allow such comparison is to make available a tabulation of ZsUli.calculated from eqn (1) as functions of various parameters. However, such a direct tabulation is impractical since IsUlfais a strong function of many parameters. Some of these parameters relate to device geometry (such as d,), some relate to epitaxial design (such as N&, and some even relate to operating condition (such as I,). Such strong variations of IsUrnrender the tabulation of IsUrfsinconvenient to use. An alternative is to tabulate a normalized parameter which closely relates to Zsumin a simple fashion but has the property that it is relatively independent of variations in device geometries,
Diffusion length in heterojunction transistors
1569
epitaxial layer designs, and operating conditions. A parameter which fits this requirement is the effective lateral diffusion length, 4. It is a characteristic length for the exponential decrease of electron concentration on the extrinsic base surface n(x,JJ)l,.=,=
no *ev[ - 6 - 4 &I
(4)
where no is the uniform electron concentration under in the intrinsic base region (under the emitter mesa). Physically, L, is a convenient measure of the spreading distance toward the base contacts for the base minority carriers injected from the emitter before reaching the collector. Its magnitude compared to the base thickness thus largely determines whether the extrinsic base surface recombination current is significant or not. One notes that the concentration no can be determined from the collector current, Z,, which can be written as q ’ d, . h .D. *h/Y,. Moreover, as will be demonstrated later, d2 - d, % Ld in typical device geometries. Therefore, the relation between Ld and Za.tiacan be expressed as Zsutia=
sayol,
+VLdno = m.
n
(5)
Ld.
I
From the same detailed two-dimensional analysis which derives the analytical solution of ZSUfla of eqn (3), an analytical solution of Z, is given as[lO] I,=
-qD.he”Ya”~1I~K~~~~. m
(6) tn
Combining eqns (3), (5) and (6), one then can write an analytical expression for Ld Ld
d
=
_!_
e/‘Yo/2
yo - sin 1, d, cIn ti,KmsinW, Yo)sin A,,,d2An
m
L,
(7)
’
~Jr,K,y,+ nm
(A,
Fig. 3. Calculation of L,, as a function of base-emitter contacts spacing. d, = 1 pm, N,_ = 2 x lOI crtm3, and Y,=800a.
It should be noted that eqn (7), though complex, is analytical and is derived from eqn (2) without approximation. A numerical program is used to calculate the values of L, for various device geometries, epitaxial layer designs and operating conditions. It will be shown that L, is relatively insensitive to the variations in most parameters, thus demonstrating the usefulness of tabulating Ld rather than ZSUm directly. Once Ld is tabulated as shown in Figs 2-4, ISUrf,, is easily determined for any device from eqn (5). This would greatly simplify the attempt to correlate experimental results to quantitative theoretical values of Zsurfa,without the need to write a non-trivial numerical program[ lo]. Note that the above theoretical analysis bases on the solution to the 2-dimensional current continuity equation in the base layer[ lo]. Because the base layer is a quasi-neutral region, this analysis does not involve solving the Poisson equation as would be necessary for space charge region such as the base-emitter junction. A computationally much more
100
70
-
10’9
device
90
7
80
y
70
:
60
-
50
:
”
I,
9
,
”
I
”
I,
”
1020 N bcuc (one’)
40
Fig. 2. Calculation of L,, as a function of base doping concentration. 4 = 1 pm (hence, as shown in Fig. 1, the actual
f’
emitter width is 2/rm), d,-d,=3000~.
Y. = 800 A,
and
0
L~l’ll*‘lll’lll”” 400
800 y,,
1200
1600
2000
(A)
Fig. 4. Calculation of L,, as a function of base thickness. d,=lpm, Nbuc= 2 x lOi9 cme3, and d2 - d, = 3000 A.
1570
WILLIAM
involved study which considers the Poisson equation was carried out[l3]. The study demonstrated that in the quasi-neutral base region, the surface recombination current has an ideality factor close to unity because the surface recombination rate is limited by the availability of minority carriers and the quasiFermi level does not remain flat. While this computationally more involved study generates the voltage distribution in addition to the desired current calculation, the present analysis naturally leads to a unity ideality factor for IsUrfsand is adequate for the purpose of determination of the effective diffusion length. The fact that the present analysis leads to unity ideality factor for Z6UtiB can be understood from eqn (5), where ZsUfi,,is shown to be directly proportional to Z,, with a proportional constant dependent on the device geometry. Since Z, is known to have a unity ideality factor, Zstia also has a unity ideality factor, as determined by the computationally more involved study. As mentioned, this theoretical analysis is based on an analysis which determined the magnitudes of various base current components, and consequently, the current gain for several device geometries and device epitaxial structures[lO]. Experimental current gains measured from passivated and unpassivated HBTs with different base doping levels were shown to agree with the published analysis[lO]. Furthermore, Fig. 3 of the published analysis demonstrated that with a d2 - d, = 1500 8, depleted AlGaAs ledge to passivate the extrinsic base surface of the HBTs, Zsurra was virtually eliminated and current gain saturated at its highest values. This published result is again supported by a separate experimental study which demonstrated that a passivation ledge (achieved by self-alignment) with dz - d, = 2000 8, was long enough to passivate the extrinsic base surface and achieve maximum current gain[6]. (Yet another experimental work demonstrated partial passivation with a passivation ledge having dz -d, = 1 pm; however, its authors acknowledged that there was significant misalignment tolerance of at least 0.5pm[3].) Since the present analysis to determine the effective diffusion length originates from the experimentally verified analysis[ lO],‘it is expected that Isure calculated from the L, derived in this analysis will agree with its experimental values.
3. CALCULATIONS
AND DISCUSSIONS
The numerical program used to calculate L, from eqn (7) is a modification of the program used in Ref. [lo]. The Einstein relationship is used to relate D, to the minority electron mobility, p,,. In the calculations, the material parameters p,, and r, as function of the base doping concentration, Nbase,are taken from Ref. [14]. The surface recombination velocity in an unpassivated surface is assumed to be S, = 1 x 107cm/s[15]. For the base contacts, the
LIIJ
surface recombination velocity is taken to be 2 x lO’cm/s, which is half of the average free electron thermal velocity. This is a result of assuming that there is no reflection for electrons striking the base contact interface[l6]. The HBT structures under consideration have a uniform base layer, thus the base quasi-electric field is O(f = 0). L, as a function of Z, is calculated for a typical self-aligned microwave HBT with d, = 1 pm (hence, as shown in Fig. 1, the actual device emitter width is 2 pm), N,_ = 2 x lOi cm-), dz - d, = 3000 A, and Y0= 8OOA. In general, IsUrf. which scales linearly with no is directly proportional to Z,. In contrast, the calculation shows that Ld remains - 72 A for all the operating current levels. This constancy of L, results because, despite the increase in n,, the general characteristics of the minority carrier profile on the surface remain unmodified. In addition, L, is calculated for the same HBT structure but with varying emitter width (2d,). Again, the calculated result is independent of the current levels. More importantly, this calculation demonstrates that L,, calculated from eqn (7) is also independent of d, ; Ld is found to be 72 j, 1 A for d, varying from 0.5 to 5 pm. Such finding agrees with the intuitive view that increasing the emitter width only increases the total number of electron carriers flowing through the emitter mesa, but does not affect how far the carriers near the mesa periphery diffuse laterally toward the base contacts. As will be demonstrated, Ld is an ideal normalized parameter, being insensitive not just to Z, and 4, but also to d2- d, , and N,_. It is sensitive, however, to the value of Y, (as it should be). Therefore, one can then use Ld = 72 A along with eqn (5) to quantitatively determine the value of Zsurfa for any HBT with Y0= 800 A. However, if one were to directly tabulate Zrurrainstead of L,, then one would necessarily tabulate Zsurra against additional parameters such as Z,, d,, and N bilse,rather than just Y,,. Note that for this HBT, the carrier concentration at the extrinsic base surface is well characterized by an effective lateral diffusion length roughly equal to 72 A. This distance is indeed much smaller than the ledge distance d, - d,, as assumed in the derivation of eqn (5). Figure 2 shows the values of L, as a function of the base doping concentration for a typical selfaligned HBT with d, = 1 /*m, dz - d, = 3OOOA, and Y0= 800 A. As Nbaw increases, both D,, and ‘t, decrease, causing more base minority carriers to recombine in the bulk base layer. However, such material parameter modifications in the bulk base region do not strongly affect the extrinsic base surface carrier concentration, which is primarily determined by the surface recombination velocity. Therefore, the exponential decrease profile for the surface carrier concentration is not significantly modified. As shown in Fig. 2, Ld does not change dramatically with Nb_. For 1 x 10’9cm-3 < N baxc 1 x 10zocrn3, Ld varies between 73 and 71 A, representing a 3% variation.
Diffusion length in hetcrojunction transistors
In contrast, the magnitude of Zstia depends on N sase much more strongly (from eqn (5), Zsulr. directly depends on D,, which changes -7% from Nbase= 1 x lOI9cmm3 to Nt_= 1 x 1020cm-3[14]). Note that this relatively constant value of Ld again supports characterizing the extrinsic base surface recombination by L,. Figure 3 illustrates the variation of Ld as a function of base-emitter contacts spacing for a typical selfaligned HBT with Nbase= 2 x 1Or9cmm3, d, = 1 p, and Y,,= 800 A. As shown in Fig. 3, L,, is small (58 A) when the base-emitter contact spacing is only 500 A. However, L, increases rapidly as d2 -d, increases, saturating at a value of -72 A as d2 - d, reaches 1500 A. This variation of Ld agrees with an intuitive view. That is, when d2 - d, is very short (< 1500 A), the laterally diffused carriers can pass by the extrinsic base surface and physically reach the base contact interface[3,11]. This result in an appreciable amount of base contact recombination and a lesser amount of extrinsic base surface recombination. Since L,, is defined to characterize only the extrinsic base surface recombination, it starts out having smaller values. However, as d2 -d, approaches 1500 A, the base contact recombination current becomes negligible. Any additional increase in d2 - d, does not modify the surface carrier concentration profile since the carrier concentration has exponentially decreased to nearly zero at x - d, = 1500 A. Since typical selfalignment techniques in HBTs rely on a certain under-cut to prevent base-emitter contact shorting and such an undercut typically has a minimum distance of 2000 8, (> 1500 A)[3,6,17], Ld is again considered to be constant for all practical baseemitter contacts spacings. Figure 4 illustrates the values of L, as a function of the base thickness for a typical self-aligned HBT with Nbax = 2 x 10’9cm-3, d, = 1 pm, and d2 - d, = 3000 A. In contrast to prior calculations which demonstrate that L,, remains relatively constant as Z,, Nbase, d, , or d2 - d, vary, Fig. 4 shows that L, is a strong function of Y,. It varies from 48 A at Y, = 300 A to 94 A at Y,,= 2000 A. This strong variation results because as Y,,increases, the injected carriers must diffuse laterally through a greater distance to reach the collector. Consequently, a larger portion of the carriers can diffuse laterally toward the base contacts and recombine on the extrinsic base surface. It is clear from Fig. 4 that decreasing Y, decreases Ld, hence reducing Zsurfa.Furthermore, such decreases in Y0also reduces bulk base recombination current[l8]. Therefore, current gain is expected to increase significantly with a thin base structure. However, this advantage is counteracted by an increase in base resistance associated with the thinner base, which degrades device high frequency performance. From these calculated results, one observes that Ld is a strong function of Y0 but it has very weak dependence in all other geometric parameters, epitaxial layer designs and operating conditions. Conse-
L,
1571
(A,
300 I 100
-
30 _ ,,,,,,, I, ,,,,,,,,, 102
10”
,,,,,,,,,
104
105
‘ ,,,,,,,, 10”
/,,,,,
u 10’
So ( cmlsec )
Fig. 5. Calculation of L,, as a function of surface recombination velocity. d, = 1 pm, N,_ = 2 x lOI cxre3, d,-d,=3000& and Y,=8OOA.
quently, one only needs to use Fig. 4 and eqn (5) to determine quantitatively the magnitude of Zsurra for all Npn HBTs. In contrast, Zstia exhibits strong dependences on various parameters. Therefore, quantifying the extrinsic base surface recombination with the normalized parameter, Ld, is much more convenient than tabulating Zsyrfadirectly. It is interesting to see how Ld varies with the assumed value of surface recombination velocity. Figure 5 shows L,, as a function of S, for a typical self-aligned HBT with Nt_ = 2 x lOI9cme3, d, = 1 pm, d, - d, = 3000 A, and Y0= 800 A. When the exposed extrinsic base surface is near ideal with S, = lo3 cm/s, Ld remains at a large value of N 220 A. However, L,, decreases rapidly when S, is greater than lO’cm/s. This rapid decrement of L, indicates that the surface carrier profile shows an even more rapid exponential decrease. This is expected since, as S, increases, the carriers which are injected from the emitter and which diffuse laterally toward the base contacts would survive only for a short distance before being recombined. As mentioned, L, decreases to N 72 A when S, = 10’ cm/s, which is the experimentally determined surface recombination velocity for exposed GaAs surface[l5]. In addition, Fig. 5 demonstrates that once the extrinsic base surface recombination velocity can be reduced below 10’ cm/s, the surface carrier profile will exponentially decrease in a fashion unaffected by the extrinsic base surface recombination. This shows that the extrinsic base surface needs not to be perfect (S, = 0) in order for the base bulk recombination current to dominate the total base current. 4. SUMMARY The quantitative determination of the extrinsic base surface recombination current for arbitrary HBT geometry, epitaxial design, and operating
WILLlAM
1572
dition is achieved with the calculation of a normalized parameter, Ld . This effective lateral diffusion length for the base minority carriers is found to vary strongly with the base thickness, but is relatively insensitive to all other parameters. The calculation of I s,,rfD thus involves first identifying the 15, value corresponding to a particular base thickness from Fig. 4, and then calculating ZsUfi8from eqn (5). This easy allows one to directly correlate determination of ZJUria one’s experimental results on extrinsic base surface recombination with theoretical values.
LIU 4. R. N. Nottenburg, C. J. Sandroff, D. A. Humphrey, T. H. Hollenbeck and R. Bhat, Appl. Phys. L&t. 52, 218 (1988). 5. S. Shikata, H. Okada and H. Hayashi, J. Vat. Sci. Technof. B 9, 2479 (1991). 6. N. Hayama and K. Honjo, IEEE Electron Deu. L&t. 11, 388 (1990).
7. W. Liu, Electron. L.&t. 28, 379 (1992). 8. D. Costa, W. Liu and J. S. Harris Jr, IEEE Deuice Res. Conf. naner VIA-3 (1991). 9. D. Co&a and J. S.‘Harhs Jr, Int. Symp. GaAs and Related Compounds (1991). 10. W. Liu, D. Costa and J. S. Harris Jr, Solid-State Electron.
34, 1119 (1991).
11. D. P. Kennedy and P. C. Murley, Solid-State
discussions with D. Costa and A. Khatibzadeh on the need to quantify the extrinsic base surface recombination and on I/fnoise are greatly appreciAcknowledgements-The
ated. The helpful suggestions by C. Dai and J. Harris, and the technical support in computation by E. Ziem are also acknowledged.
Electron.
3, 215 (1961).
12. W. Fonger, Transistors I, RCA Laboratories, Princeton, NJ (1956). 13. S. Tiwari, D. J. Frank and S. L. Wright, J. appl. Phys. 64, 5009 (1988).
14. S. Tiwari and S. L. Wright, Appl.
Phys.
Lett.
56,
563 (1990).
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15. P. E. Dodd, T. B. Stellwag, M. R. Melloch and M. S. Lundstrom, IEEE Trans. Electron Devices 38, 1253 (1991). 16. J. P. McKelvey, Solid State and Semiconductor Physics, p. 350. Harper & Row, New York (1960). 17. K. Nagata, 0. Nakajima, Y. Yamauchi, T. Nittono and H. Ito, IEEE Trans. Electron Devices 35, 2 (1988). 18. see for example, S. Sze, Physics of Semiconductor Devices, p. 141. Wiley, New York (1981).