Analytical model for abrupt HBTs with application to InPInGaAs type

Solid-State Electronics Vol. 41, No. 9, pp. 1277-1283, 1997 Published by Elsevier Science Ltd Printed in Great Britain P I I : S0038-1101(97)00067-1 0038-il01/97 $17.00 + 0.00

Pergamon

ANALYTICAL MODEL FOR ABRUPT HBTs WITH APPLICATION TO InP/InGaAs TYPE JUAN M. LOPEZ-GONZ,A, LEZ, PAU GARCIAS-SALV~, and LLUIS PRAT Departament d'Enginyeria Electrbnica, Universitat Polit/~cnica de Catalunya, Campus Nord, C/Jordi Girona, 1-3, 08034 Barcelona, Spain (Received 31 October 1996; in revised form 4 February 1997)

Heterojunction bipolar transistors fabricated using an InP/InGaAs configuration show a high performance. In order to understand the physical mechanisms which control the behaviour of the above transistors, an analytical model was developed that takes into account Fermi-Dirac statistics as well as an arbitrary injection level. This general model is simplified in several levels of approximation, thus obtaining a very simple model that enables a clear physical interpretation of the collector current behaviour. Published by Elsevier Science Ltd. Abstract

l. INTRODUCTION

Heterojunction bipolar transistors (HBTs) play an important role in high-speed electronic device applications[l,2]. Recently HBTs based on the InP/InGaAs configuration have reached a transition frequency (fr) up to 186 GHz, which is one of the highest values ever reported for this type of device[1 ]. Development of new models for these devices is needed for both a better understanding of their physical behaviour and a better optimization of their design. Usually abrupt n-p-n HBTs show a spike in the conduction band at the emitter-base interface. Since this spike can have a significant effect on the collector current, conduction through this interface has been a topic of research for the last years[3]. The early work of Marty et a/.[4] based on drift-diffusion transport through the spike was thoroughly revised by Grinberg et a/.[5] and Lundstrom[6], who proposed thermionic emission and tunnelling transport through the interface. Analytical models for these transistors were developed by equating the interface current with the bulk current. These models assume a Boltzmann distribution and a low-level injection condition[5,6]. However, the high base doping level used in modern HBTs demands the use of Fermi statistics. In addition, some transistors can operate under high-level injection condition due to the low doping level at the emitter. Recently Grinberg et a/.[7] proposed a new analytical model that overcomes some of these limitations. Nevertheless, this model still considers the Boltzmann approximation at the interface and does not take into account the effect of electron saturation velocity in the neutral base region, which can be significant in modern HBTs.

Another topic that has important effects on abrupt HBTs is band gap narrowing (BGN) caused by heavy doping[8]. This phenomenon reduces the normal band gap value and also affects the discontinuities of the Ec and Ev level at the interface and, consequently, the device currents. However, the heavy doping effects on Ec and Ev were not taken into account in most studies on these devices. Here, an analytical model that includes FermiDirac statistics and an arbitrary injection level is developed. In addition, the effects of electron saturation velocity on the neutral base region and the influence of heavy doping on the discontinuities of Ec and Ev at the interface were considered. This general model simplifies the Lundstrom[6] and Grinberg[7] models when usual approximations are applied. In order to understand the physical mechanisms which determine the collector current, a very simple model is developed. All these analytical models are compared with the results obtained from a numerical model previously developed by the authors[9]. 2. ANALYTICAL MODEL FOR ABRUPT HBTs

An analytical model for abrupt HBTs will be developed in this section. As it will be seen, the Fermi function, F, i2, and its inverse, Fi~g, appear in this model. These functions make it impossible to obtain closed form solutions for the transistor currents in spite of using analytical approximations in the Fermi expressions. Closed form solutions are only obtained when the Fermi function is approximated by a Boltzmann distribution. The proposed model is described in four subsections: HBTs band structure, current in the neutral base region, extended boundary conditions at both edges of the emitter-base depletion region and simplifications of the general model.

1277

1278

J. M. L6pez-Gonzfilezet al. Table 1. Expressions for parameters in eqn (3)

2.1. Band structure ,]'or abrupt H B T s

A typical band structure for an abrupt HBT is shown in Fig. 1. Band gap energy and electronic affinity are modified when the doping level is very high. These values then become ( E ~ - AEbg") and (qx + qAx). Consequently, the discontinuities of E¢ and E~ at the emitte~base interface change with heavy doping effects as; AE~ = Ec(O ) - EdO +) = aEco - [qAz(0 ) - qAz(0+)]

- qAz(0+)] - [qAEbg"(0 ) - qAE~g°(0+)], (2) where AEcoand AE,o are the discontinuities of Ec and E, at the interface for low doping level. Currents through the interface depend on the discontinuities of Ec and E~, as it will be shown later. Therefore, these currents are also affected by the band gap narrowing and the variation of the electronic affinity. In silicon devices, the main parameter modified by heavy doping is the minority carrier density. This parameter is evaluated through intrinsic carrier density, nj, which depends on the band gap. For that reason, heavy doping effects are modelled by an effective BGN. However, in abrupt HBTs this parameter is not sufficient to model the influence of a high doping level on the electrical characteristics of the device, because the currents also depend on the changes suffered by the electronic affinity of the material. Most of the studies about heavy doping effects consider only band gap narrowing and not information about electronic affinity modification. In relation to different criteria usually applied to divide the estimated amount of BGN between the energy bands, some numerical simulators consider that heavy doping modifies only the majority carrier band edge level (i.e. Ec level for n-type, or E, level for p-type semiconductors)[10], while others consider the

:

\

: qXB

C2

p-type

3 4

4 3

0.2126 eNt.'

0.36307m%25 e ....

0.0186

0.0186m,~,

N b . ( ~ m ~ ) °~

N ~ , ~

c~

0.36307m°~2,s

0.2126A

c,

0.00843(mh, + m,t)

0.0186

Nb.EOSm~

Nbp(~mdh )0 5

(1)

AEv = E,(0 +) - E,(0-) = AE,o + [qAz(0-)

13

c,

n-type

same but for minority carriers[Ill and finally, some of them allow the user to define the distribution of band gap narrowing between Ec and E,. Although heavy doping effects are expected to have a significant influence on HBTs based on InP, InGaAs, A1GaAs and SiGe materials, not much emphasis on these effects was observed in the models found in the literature. Besides the lack of information about heavy doping effects on Eg and Z, there also exist important uncertainties in the quantification of other physical parameters related to materials used in the manufacture of these kinds of HBTs[12-15]. In this work, Jain-Roulston's model[16] was used to model heavy doping effects on the band structure of the device. This model makes it possible to calculate the band gap narrowing and to distribute it between both Ec and E, levels. It was applied successfully to different semiconductors as Si, Ge, SiGe and p-GaAs, and it could also be used with other compounds such as InGaAs and InP. This model assumes that Arc and Nv are not modified by high doping effects, which seems acceptable for p-type semiconductors and also for n-silicon. This might not be valid for III-V n-semiconductors, but, as n-regions in modern n-p-n abrupt HBTs have low doping levels (usually lower than 10tScm-3), the errors introduced by this model will be assumed to be small. According to Jain-Roulston's model[16], the BGN can be distributed between the majority and minority carrier band edges. From this model, the shift of and E¢ and E, levels can be written as;

[- N 1"" J- N ]'" C, LT6. j + C,L1-- j i

\

'h

i i

-Xn

i

0 Xp

Fig. 1. Energy band diagram of an abrupt heterojunction bipolar transistor.

NT" [ N]'" aces.: C,FLT6N +

(3)

where ~, fl, C,, G , G , C4 are given in Table 1. Parameters in Table ! have their usual meaning[16], and the values used in this work for InP/InGaAs HBTs are given in Table 2.

Analytical model for abrupt HBTs

qV~ =

2.2. Current in the neutral base region The electron current density in the neutral base region is supposed to be caused by drift-diffusion mechanisms. Uniform doping will be considered in the base. Under these conditions, this current density can be expressed as [17];

J.(xp) = -q.v..n(xp),

Eo(O + )

KT F,12[--Z-r] (10)

where,

(4)

VN + Vp = Vhi- VBE; Ve = ks'(Vb,- VBE); kB =

EENE E~NE + EBN~"

(1 1)

The electron current density through the interface at x = 0 is the result of thermionic emission and tunnelling mechanisms. When Fermi-Dirac statistics are taken into account, it can be expressed by;

~ LV~

with v~ being the electron saturation velocity, D,s the diffusion coefficient, wB the width of the neutral base region and 7 an injection related factor that can be expressed as; 7 = [ 2 - n~xBp)ln(m + -n(xp)']l -~B/J"

Eo(x.) -

=

where n(xp) is the electron density at the edge of the emitter-base depletion region and vB is the effective electron velocity in the base, given by;

vB

1279

N~

J.(O) = - q v , , 4

_

n(O )

(6) _

/

aEc

-- N~1",,,2~-~-~ +--,/2~ N~ / / ]

On the assumption that the injection level is low (i.e., n(xp)<
with,

~ - K T (1 v,e/= ~/2g'm* +P')

In order to calculate the carrier densities at the edges of the emitter-base depletion region (that is to say n(xp), p(xp), n ( - x,), p(x,)), the electron and hole quasi-Fermi levels Ey, and Etp will be assumed to be constant from - x , to 0- and from 0 ÷ to xp. Both levels show a discontinuity at x = 0118]. The high doping level used in the base makes it impossible to use the Boltzmann statistics, so;

J.(0) = J.(xp) + J,,sB

n = N¢'Fja

=~ E~ = Et. -- K T F~/2

(7)

p = N~'F,/2~)

::~ E~ = Er. + KT'F~J2

• (8)

As it can be seen in Fig. 1;

(14)

where J.(xp) is given by (4) and J.R~ is the recombination current density inside the region comprised between x = 0 + and xp. Note that for each applied VBE, expressions (I 1) give VN and lie. Then, expressions (9)-(14) together with (6) make possible the formulation of a system of three equations with four unknowns, namely n ( - x . ) , n(O-), n(0 +) and n(xp). In case of low-level injection at - x . , it could be stated that n ( - x . ) = Ne and the system could be solved. However, in general, high-level injection can occur in the emitter. In this case, n ( - x . ) is related to p ( - x . ) by the charge neutrality condition at - x . :

n(-x.)-no(-X.)=p(-x.)-po(-X.)

qVu = E~(O-) - E ~ ( - x . )

(15)

where subindex "o" denotes densities at thermal equilibrium. The system of equations then becomes a system of four equations with five unknowns. In

(9)

Table 2. Numericalvalues for the parametersused in Table 1 for InP/InGaAsHBTs n-lnP p-lnos~Ga047As

(13)

where p~ is the tunnelling factor[5] and m* the effective electron mass[19]. Note that when p, = 0, expression (12) gives the thermionic emission current through the spike. Current continuity requires that;

2.3. Extended boundary conditions

= KT.F?/~I~I-KT.F~[~]

(12)

Nb

~

md~

mdh

mhh

mh~

A

1 2

12.4 13.5

0.08 0.041

0.869 0.61

0.85 0.6

0.089 0.05

1 0.75

1280

J. M. L6pez-GonzAlezet al.

order to solve it, p ( - x , , ) must be related to p(0-), p(0 +) and p(xp) through the following new set of equations: qV~ = E~(0-) - E ~ ( - x , , )

p(-:_.) (16) qVe = Ev(xp) -- E~(O+)

_ .....

Fp(x )l_ Kr.F 3

Fp(o+)l,,,._ (17)

LN: l

pressions proposed by Lundstrom[6] and Grinberg et al.[7] are obtained. Under these conditions: F,2(x) = ex Fi~,](y) = l n y

(22)

In this section three approximation levels will be considered: Case A, Grinberg model. Fermi statistics are used only for majority carrier at - x , and xp; Case B, Lundstrom model. Boltzmann statistics are valid in all device regions and the transistor works in low-level injection; Case C, in addition to the approximations of case B, recombination current is neglected in eqn (14). 2.4.1. Case A. If Boltzmann's approximation is valid for n(Xp) and n(0+); n(O +) = n(xp)e v.lv~

(23)

and, if it is also valid for n(0-), eqn (12) becomes;

Jp (0) = qv~rlp(O- )

Ec,(O- ) - E!,(0 +) E

/

J,(0) = K T . l n I l + q v , e r n ( O ÷)e_AE,/xr]

AE,.

Jp(0) = J p ( - x , ) + JRe

(19)

where the hole current at x = 0 is also produced by thermionic emission caused by the discontinuity in the E~ level. This system of equations could be solved by assuming low-level injection at Xp (i.e., p(xp) = NB)). But in a more general case, the system has to be completed with a charge neutrality equation at xp:

(24)

At the same time, by equating eqns (10) and (17) for Vp, and assuming a Boltzmann distribution for p(0 +); n (0 +)'p(O + ) _ ~_, {p(x.)'~ In N,8.n(xp) - " ~:2!, N~ ]

(25)

and taking into account that; E:,(O- ) - Etp(O +) = qVnE

(26)

the following expression is obtained; p(xp) - po(Xp = n(x,) - no(xp).

(20)

Electron and hole densities at points xp, 0 +, 0- and - x , are obtained by solving the simultaneous eqns (9)-(20), applying simple iterative techniques with the thermal equilibrium solution as an initial guess. In this work, the direct and reverse Fermi functions were approximated[20, 21] by: F,,2(x) = 1

1,12 n(Xp) = ~ s e-Fi-~(~"J/Ubele"(°+)-E,~,(0)l/XreV,E/vT; N'v

n~ = N,S.N~e-E, ,/~r.

(27)

Equations (23)-(27) are identical to those proposed by Grinberg et a/.[7]. 2.4.2. Case B. Ifeqn (22) is used in expression (27) it follows that; n? n(xp) = =7-~-s~~ e V"E'vT

3x/~/2 e - ' + [x + 2.13 + (Ix - 2.13l z4 + 9.6)'/zq '5

1 , (28) J,(0) 1+ qv,~rn(O +)e-At'/xr

and making use of eqn (23), we conclude that; Fg~ (x) = ~

In x

+

(2x/~.x/4) 2/3

J,(O ) = qv,,seVPe-~E,/XZ[no(xp)e v.E/vT - n(xp)]. (29)

1 + [0.24 + 1.08-(3V/~/4)2/3] -~ (21) The error of this approximation is less than 0.53%. 2.4. Simplifications o f the general model

As already mentioned, the use of Fermi-Dirac statistics in eqns (9)-(20) makes it impossible to obtain a closed form solution for the transistor currents. When the doping profile allows the Fermi function to be approximated by a Boltzmann distribution, the general model described in the previous section can be simplified, and the ex-

This last expression is the equation derived by Lundstrom[6]. 2.4.3. Case C. If J,(xp) in eqn (4) is approximated by J,(0) of eqn (29)---once J, RB was neglected in eqn (14)---the following expression is obtain; n(xp) =

n°(xp)eVBt/Vr - n°(xp)eVB'/v~, 1 + v a elan_ ~Vpl/xr 1+ A

(30)

1)he!

where, A - v.~BefAE,_uv,rxr. Une!

(31)

1281

Analytical model for abrupt HBTs When eqn (30) is used in eqn (4), the collector current density J, can be expressed by; 1

Jc = J,,(xp) = - q v B ~-+--~ no(xp)e v"~'~v~.

i

i

i

i

I

I

I

I

-4 -4.5

(32)

-s

Equations (31) and (32) provide a very simple physical interpretation of the mechanisms determining the collector current. The value of Jc for A << 1, called J,,, in this work, is the current through the neutral base when n(xp) is not limited by the spike; J,e = - qvBno( xp )e VBE/v~.

~

-5.5 -6 i000 2000 3000 4000 position (Amstrongs)

(33)

The value of J¢ when A >> 1, hereafter called J,s, is;

5000

Fig. 2. Energy band diagram for the InP/InOaAs HBT under analysis.

£,s = - qv,ere -taE' - qVpl/rrno( xp )e v,~/v~ Ny

_

= -qv,e1-~c n(O ).

(34)

Note that J,s is the current through the interface because it is given by the product of the effective electron density at x = 0- and their own velocity, v,~r. Parameter A in eqn (31) can also be expressed by; (35)

A - J,B

J~s'

and eqn (32) can be formulated in the following way; 1

1

1

Jc - J~s + ~,B"

(36)

This last expression shows that the collector current density is caused by two competing mechanisms: the current density through the spike, J,s, and the current density through the neutral base, J , , . A similar formulation was developed by Marty et al.[4] and it is also implicit in the work of Lundstrom[6]. The first transport mechanism is the thermionic emission and tunnelling through the interface, while the second one is the drift-diffusion through the neutral base. The collector current is mainly determined by the most restrictive current. If J,s<> 1 and J~ ~. J,s. On the contrary, if J.. <
Heavy doping effects on the energy band structure of the transistor are very important. If they were neglected, the discontinuities of Ec and Ev at the emitter-base interface would be of 0.212 eV and 0.388 eV, respectively. When these effects are taken into account the values obtained are 0.199 eV and 0.311 eV. These differences have a significant effect on the collector current, as it can be seen in Fig. 3. The current calculated without considering heavy doping effects on the spike is almost 10 times smaller than when they are taken into account. These currents were calculated using the general analytical model presented in the previous section. The electron saturation velocity, vs, also has significant effects on the electron current in the base. This parameter was taken as 8 x 106 cm s-~[13] in this work. In the analysed transistor D , , / w a was set to 7.8 x 106cms-L Consequently, as it can be seen from eqn (5), taking vs into account can reduce this current by a factor of 2 and, hence, it should not be neglected. Figure 4 shows the electron current densities calculated by using the numerical model, the general analytical model, Grinberg model (case A) and Lundstrom model (case B). The same energy band structure was used in all models. The differences observed between the models are due to the

3. APPLICATION TO lnP/InGaAs ABRUPT HBTs i

The models described in Section 2 were applied to calculate the collector current of an InP/InGaAs HBT. The transistor being simulated had doping levels of 10 '8, 1019and 5 x 10 ~6cm -3 in the emitter, the base and the collector, respectively. The width of these regions were set to 0.2, 0.05 and 0.25 #m, respectively. First of all, the HBT was simulated using a numerical model for abrupt HBT[9]. This model numerically solves Poisson's equation consistently with the electron and hole continuity equations, taking into account the Fermi-Dirac statistics and the effect of heavy doping on Ec and Ev. The dependence of electron mobility with electric field was taken from [22]. Figure 2 shows the thermal equilibrium energy band diagram.

i

i

i

le+06

"g =

i00000 10000 1000 100

// ,/

.~

10 0.5

I 0.6

I 0.7

i 0.8

emitter-base voltage

I 0.9

{v)

Fig. 3. Collector current density vs emitter-base voltage for the InP/InGaAs HBT under analysis. (Solid line, heavy doping effects on the energy band diagram included; dashed line, heavy doping effects not included.)

1282

J. M. L6pez-Gonzfilezet al. t

)

i

i

le+06

,~!....

100000

10000

./" /"'

'c' 'd'

..... .....

.... iii;iiii

1000

As the bias voltage is increased, the current increments become progressively smaller in the numerical model (curve 'a') than in the analytical models (curves 'b', 'c' and 'd'). This behaviour may be explained by the fact that the analytical models do not take into account the voltage drops in the neutral regions of the transistor produced by high-level injection. The electric field in the neutral base region under high-level injection can be expressed by;

~ o

100 0.5

""

I

I

I

Jp K T dn/dx E - q#p(NB + n) + q N, + n'

I

0.6 0.7 0.8 emitter-base voltage

0.9

Fig. 4. Electron current densities vs VsE bias voltage calculated from: (a) numerical model; (b) Lundstrom model; (c) Grinberg model; (d) model described in this work and (e) the same model including voltage drop in the neutral regions.

where all parameters have their usual meaning. The voltage drop is obtained by integrating this electric field along the corresponding region;

AV, = influence of Fermi-Dirac statistics, electron saturation velocity and voltage drops at high-level injection. The differences between the three considered analytical models (curves 'b', 'c' and 'd') can be explained in the following way. Electron current depends on electron densities in the depletion region and, particularly, it depends on the electron concentration at point 0 , as stated in eqn (34). Electron concentration at this point is approximated in our model by;

NE

,

VrJ

(37)

while the Grinberg model proposes; ,

(40)

(V)

NE

n(0-) =NcE exp[F~z(~-~cE)-- VNI'vrJ

(38)

and the Lundstrom model; n(0 ) = NeE exp[ln(N~E) - ~--~ur].

(39)

f

~"+"" "Iv dx p q#p(N, + n) 1+ + K---TIn q

n(x,)'] NB ,]" (41)

A similar expression can be obtained for the neutral emitter region. When voltage drops in the neutral emitter and base regions are included in the analytical model (curve 'e' in Fig. 4) a very good agreement with the numerical model is obtained. For this transistor the major term in (41) is the ohmic voltage drop in the neutral emitter. Finally, the simplest analytical model (case C) was applied to the HBT. Figure 5 depicts the electron currents obtained using this model and Lundstrom's model (the former model is a simplified version of the latter). Clearly, both models are in good agreement for the whole range of applied biases. This simple model is able to reveal the physical mechanism that dominates the collector current. Equation (33) gives the electron current when it is limited by diffusion through the neutral base, J,,, while eqn (34) gives the electron current when it is limited by transport through the spike, J,s. The smallest one determines the collector current. Both c~

When the bias voltages are small (e.g., VDe= 0.6 V), expressions (37) and (38) become equivalent because the argument lies in a range where F~,2 can be fairly approximated by the exponential function. However, the argument in (39) is smaller, giving as a result, a smaller value for the electron density at 0-. When the bias voltage is increased the value of VN goes down, so the argument in (37) and (38) grows. In this case, the Fermi function returns smaller values than the exponential approximation, and (37) approaches expression (39). The small differences for low bias voltages between Grinberg's model and the general analytical model are justified by the influence on the electron saturation velocity, which is not taken into account in Grinberg's model.

~ E ~ ~ v ~ ~

le+06 100000 10000 1000 100 10 1

fJ"'" .7" ...-""

'a' 'b'

'i -..........

0.1

~o ~ ~

0.01 0.001

i 0.2

0.3 0.4, 0 . 5 0.6 0.7 emitter-base voltage

0.8 (V)

0,9

Fig. 5. Electron current densities versus VBE voltage calculated from; (a) Lundstrom model; (b) the simplest model described in this work (case C).

1283

Analytical model for abrupt HBTs le+06

i

i

i

i

i

/

y

'Jn'

--

i

mation a very simple model was developed which agrees with the Lundstrom model. This model shows that the transport through the spike is the mechanism responsible for the limitation of the collector current at high bias voltages while the drift-diffusion current through the neutral base is the mechanism responsible at low bias voltages.

I

I00000 i0000 i000

,;;;,;,'"

i00 rn

i0 /~:'"

i

1 " v

,c///

0.i

' JnS ......

!

Acknowledgements--This work was partially supported by CICYT under TIC 96-1058 contract.

0.01 0.001

.2

0.3 0.4 0.5 emitter-base

0.6 0.7 0.8 voltage (V)

O.

Fig. 6. Electron current densities Jns, J,s, and the collector current as defined by eqns (33), (34) and (36). currents and the total electron current have been represented in Fig. 6, where we can see that the smallest current at high voltages is Jns whilst for lower voltages is J~B. For VBE voltages under 0.9 V, p, is greater than unity, which means that the current through the spike is dominated by the tunnelling mechanism. Figure 6 shows how the slope of Jns(VBe) decreases with bias voltage. The main reason for this behaviour is that the tunnelling factor becomes smaller as VBe increases (p, = 83 for VBE= 0.55 V; p, = 1.2 for VBe = 0.9 V).

4.

CONCLUSIONS

A general analytical model for abrupt HBTs should: (a) include high-doping effects on Ec and Ev discontinuities at the emitter-base interface; (b) use Fermi-Dirac statistics for carriers; (c) consider arbitrary injection level at the edges of emitter-base depletion region; and (d) take into account the effects of the saturation velocity on the current through the neutral base. When this model is applied to an I n P / I n G a A s HBT of typical doping profile, a difference of almost one order of magnitude in the collector current is observed if high-doping effects on Ec and Ev discontinuities are not included. The general analytical model proposed here agrees with the Grinberg model for low bias voltages, but gives smaller values for higher voltages. Using the Boltzmann approxi-

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