Analysis of abrupt and linearly graded heterojunction bipolar transistors with or without a setback layer

J.S. Yuan J.H. Ning

Abstract: Analytical equations of electric field, electrostatic potential, and junction capacitance for abrupt and linearly graded heterojunctions with or without a setback layer have been derived. The equations are general and applicable to both abrupt and linearly graded heterojunction bipolar transistors. Collector and base currents and small- signal parameters such as transconductance and cutoff frequency of the abrupt and linearly graded heterojunction bipolar transistors with or without a setback layer are evaluated. Comparisons between the isothermal model, the numerical model. and the present model including the self- heating effect are demonstrated.

1 Introduction

The use of energy-gap variations beside electric fields to control the forces acting on electrons and holes results in greater design freedom and permits an optimisation of doping levels and geometries, leading to higher hetero- structure performance for microwave and high-speed circuit applications [I ] . For an abrupt n-p-n hetero- junction bipolar transistor (HBT), the conduction-band energ) spike necessitates thermionic emission and tunnel- ling for electron transport from the emitter to base and the valence-band discontinuity suppresses hole injection from the base t o emitter [2]. Compositional grading of the emitter base junction allows one to smooth out a large part of the conduction-band discontinuity so that the bandgap difference at the emitter:base heterojunction is concentrated in the valence band for achieving the mawmum current gain 131. The other approach is to insert a thin layer of intrinsic GaAs (setback layer or spacer) between the emitter and base in abrupt hetero- junction bipolar transistors. The setback layer can alter the potential barrier of the heterojunction and improve the cmitter injection efficiency [4] and to reduce the impurit) diffusion from the heavily doped base to the emitter.

The advances in heterojunction bipolar transistor per-

IEE. 1995 Paper 203OC~ (E3). tirst receibed 30th November 1994 and in revised

The authors are uith the Department of Electrical and Comptuer Engineenng. Vni\ersit? of Central Florida. Orlando. FL i2Xlh . [JSA

254

f O I l n 5th Md) 1995

formance increase the need for models to guide HBT device and circuit design. Theoretical analyses of the HBT modelling have been reported by numerous authors [5-l21. Lundstrom [5, 61 derived boundary conditions for heterojunctions and developed an Ebers-Moll model for HBTs. The model 161 describes both single- and double-heterojunction transistors with or without band spikes and applies to uniform or graded-base HBTs. Grinberg et al. [7] investigated the effect of graded layers and tunnelling on the performance of AlCaAs/GaAs het- erojunction bipolar transistors. Tasselli et d. [SI verified the charge-control model via experiment for AlGaAs/ GaAs heterojunction bipolar transistors by comparing the forward and reverse current behaviours. Grinberg and Luryi [9] derived exact formulas for the current- voltage characteristics for a double-heterojunction HBT, valid for arbitrary levels of injection and base doping, including the degenerate case. Parikh and Lindholm [lo] developed a new charge-control relation for hetero- junction bipolar transistors. The relation is valid for arbitrary doping density profiles and for all levels of injection in the base. Liou et al. [l I ] reported an analyti- cal model including thermal and high current effect for the HBT without the conduction band spike. Liou et al. [I21 also published an analytical model for current trans- port in AlGaAsiGaAs abrupt HBTs with a setback layer. It is our understanding that a comprehensive analytical model which treats the abrupt and linearly graded het- erojunction bipolar transistors with or without a setback layer including the self-heating effect for analysing the currents, transconductance, and cutoff frequency of HBTs in a unified manner has not been reported.

We present a unified analysis of collector and base currents for abrupt and linearly graded HBTs with or without a setback layer using some techniques published in References 11-13. The paper derives the analytical equations of space-charge layer thicknesses, potential barriers. and junction capacitance and extends the analysis to account for current transport including the self-heating effect. In addition to common-emitter current gain, small-signal parameters such as transconductance and cutoff frequency as a function of graded-layer and setback-layer thicknesses of the HBT for isothermal and thermal conditions are evaluated.

This work is supported in part by National Science Foundation, grant ECS-9311775.

I E E Proc.-Circuits Devices SJ'Sf., Vol. 142, No. 4, August 1995

2 Theory

2.1 Electric field The alternative statement of Gauss's law is mathematic- ally expressed by

dD d[e(x)&(x)]

dx _ -

dx =' -

where D is the electric displacement density, E is the per- mittivity. & is the electric field, and p is the charge density. Expanding the derivative of eqn. 1 obtains

d&x) p &(x) dc(x) d x E ( X ) E ( X ) dx

-

Consider an N / p h heterojunction bipolar transistor including a graded layer and a setback layer at the emitter-base heterojunction as shown in Fig. 1. The depletion region is divided into four regions. Region I

graded layer selback layer

E

Fig. 1 Srhernaric o f N p.h helerojuncrion bipolar t r i i n ~ i ~ t o r

( - X , < x <: - X , ) has a constant permittivity, thus d&/ dx = 0. Assuming free-carrier concentrations are much less than donor and acceptor concentrations (i.e. the depletion approximation), eqn. 2 gives

where q is the electron charge, N , and E , are the doping and permittivity in the N-type region, X , is the graded- layer thickness, and X , is the depletion thickness includ- ing the graded layer. The permittivity in the graded region ( - X, < .U < 0), however, is position dependent. Assume a linearly graded heterojunction such that

( 4)

where c2 is the permittivity in the p-type region. Inserting eqn. 4 into eqn. 2 yields

E(.;) = E 2 - ( E l - E 2 ) X / X g

( 5 ) d&,(-x) R9(xX&i - 82) - _ P dx X,dX) E ( X )

-

The solution to the differential equation (eqn. 5 ) is

The permittivity in the setback layer (0 < x < X , ) and region I1 (X, < x < X , ) is independent of position. Using the depletion approximation, we obtain the electric field

SAY) 2 ( x , - x,) for o < x < X , (7)

&,(x) z 6 ( x , - x) for X , < x < X , (8)

E 2

6 2

I E E Proc.-C~rcuirs Derires Sysl. I'ol 142, N o 4. August I995

where N, is the p-type doping concentration, X, is the setback layer thickness, and X , is the depletion thickness including the setback layer.

2.1 Electostatic potential Once the position-dependent electric field is known, the electrostatic potential in each region can be- easily obtained

7 -,(.U) = - &,(U) d.y I-:,

qN,X,(x + X,) - __ - q N D ( X , - X,)2 - __ - 28 I E 2 - E l

where

q N E,(O) = a (X, - X , ) c 2

for 0 < x c X , (11) r x

q N 2E2

= f S ( X S ) - 2 ( 2 X , x - xz - 2 X , x, + X i )

for X, < x < X z (12)

2.3 Depletion capacitance The modulation of space charge in the depletion region with respect to the applied bias results in a depletion capacitance. The depletion capacitance of a hetero- junction is expressed as

where 7 .hz is the junction built-in potential and V is the applied voltage. The junction built-in potential is given

255

where E , is the energy bandgap, E , is the energy level for conduction band, A E , is the energy bandgap difference (€cl - E,,), N , is the density of states for conduction band, N , is the density of states for valence band, ni is the intrinsic concentration, and subscript 1 and 2 refer to the n- and p-type quasineutral regions. Using the bound- ary condition of Dl( -Xg) = Dg( - X g ) . we obtain

(15)

Furthermore, using Ybi - V = - Y , ( X , ) , and eqns. 11 and 15, we have

ND x , = x , + XI - NA

-r, + (r: - 41-,r,)~/~ 2 r 1

x , = (16)

where

Using eqn. 16 in deriving dX,/d(*Y,, - V ) and inserting the resulting equation into eqn. 13 gives the depletion capacitance per unit area

qNA c, = (r: - 4r1r3)1’2 Eqn. 17 is a general solution for the heterojunction with a linearly graded layer and a setback layer. For a linearly graded heterojunction without a setback layer ( X , = 0), eqn. 17 reduces to

qNA c, =

where

For an abrupt heterojunction with a setback layer ( X , = 0) eqn. 17 reduces to

1 112

C j 2(vbi - V X E ~ N D + E 2 N A ) +- xi] qEIE2 N A N D E:

For an abrupt heterojunction without a setback layer ( X , = X , = 0), eqn. 17 reduces to the well known formula [ 2 ] as follows

qEIEZ N A N D

cJ = [ 2 ( V b i - V)(&,ND + E,N,)

3 Model equations

3.1 Base current Consider a heterojunction bipolar transistor with a lin- early graded junction and a setback layer as shown in Fig. 1. The base current of the HBT consists of the emitter recombination, base recombination, emitter-base space-charge region recombination, and surface recomb- ination:

J B = J B E + J B B + JSCRI + JXR, + J S C R S

+ JSCRZ + J B S (18)

The base current due to emitter recombination is expressed as

where D, is the hole diffusion coefficient, N E is the base doping, X, is the emitter depth, A E , is the valence-band discontinuity, and the electrostatic potential from x = - X I to X , is given by

The notation in eqn. 20 is E, the permittivity in the emitter, eB the permittivity in the base, and N E is the emitter doping. Position variables of X I , X, , X , , and X , are defined in Fig. 1.

Using the charge-control model, the base recomb- ination is given by

J B B = d X z X X , - X A / ~ ~ B (21) where n ( X , ) is the electron concentration at the edge of emitter-base space-charge layer in the base, X , - X , is the base width, and tB is the base recombination lifetime. The electron concentration is approximately equal to

where Ybi is the built-in potential in the emitter-base heterojunction, V,, is the base+mitter voltage, and AE, is the conduction-band discontinuity. A more accurate

IEE Proc.-Circuits Devices Syst., Vol. 142, No. 4, August 1995 256

expression of n ( X , ) including effect of emitter-base space- charge region recombination is derived in Section 3.2.

Using the Shockley-Read-Hall recombination, the recombination current in the depletion region, linearly graded junction, and setback layer is expressed as [14]

where n and p are the position-dependent electron and hole concentrations in the space-charge region and T~~

and T " , are the electron and hole recombination lifetimes. The subscripts 1, g, S , and 2 represent the integration from X I to - X g , - X g to 0, 0 to X, , and X , to X , , respectively. Inserting n = n, exp [ (EFn - Ei)/kT], p =

n, exp [ ( E , - E, , ) jkT] , n I = ni exp [(E7 - E , ) / k T ] , and p1 = n, exp [ ( E , - E,.)/kT] into eqn. 23 yields [I31

The electron current density across the heterointerface is found as the difference between two opposing fluxes [2]

J,( - X g ) = qv. yJn( - X ; ) - n( -X:) exp ( - AEc/kT)]

(27) where v, is 1/4 of the mean electron thermal velocity, y. the electron tunnelling coefficient [2], A& is the band discontinuity in the conduction band, and YE, =

- [-V"&X,) - Yg(0)l. and YBgc = -AEc/q - Y & O )

Consider the typical heterojunction bipolar transistor with uniform base doping and bandgap, the electron transport in the base is given by diffusion for base width

= q V . y . { N , exp ( -qY' , , /kT) - n(xA x exp [ 4 ( Y , , + Y'ss + -Y,,C)/kTl}

- Yl( - X g ) , = - [ v Z ( X Z ) - y.5'(XS)1? Y'flS =

+ V1(-Xg).

where

C', = ( E ! , + E,,)/2kT - 0.5 In ( T ~ , / T ~ ~ )

= 0.5AEJx = - X I ) + Ei(x = - X I )

+ EAU = X,) - kT In ( N , / N , )

the intrinsic carrier concentration ni is

n i x ) = niE for - X I < x < - X g

for - X , < x < 0

for0 < x < X ,

for X, < x < X ,

= ni, exp [ -AEG(x)/2kT]

= ni,

= niB and the intrinsic energy level is [13]

Ei(x) 2 - A;z(.x) - AEG(x),'2 - qd -(x)

A.x(x) and AEG(x) are the position-dependent electron affinity and bandgap difference, respectively. The position-dependent electrostatic potential 9 '(x) can be found in eqns. 9-12. For AlCaAs/GaAs system, A;z = -0.6AEG [13]; E, (u) is reduced to O.lAEG(x) - qV(x) .

The surface recombination current of the HBT is due to high surface states at the emitter surface and the extrinsic base. A high surface-state density and a high surface recombination velocity degrade the current gain of the HBT. The surface recombination can be a major component to the overall base current, especially for small-geometry devices where the device area-to-peri- meter ratio is small [15]. The surface recombination current increases exponentially with the base+mitter voltage with an ideality factor which is closer to 1 than 2 (1 < nL < 1.33). as evidenced by the experimental data [16]. The surface recombination current is expressed as

JBS JBSO exp ( q V B d n L k T ) (25) where J,,, is the pre-exponential factor. Note that J,, can be significantly reduced by using surface passivation on top of the GaAs extrinsic base or InGaAs semicon- ductor.

3.2 Collector current The collector current at the edge of emitter-base space- charge layer in the base is

J JXz) = J k X J - ( J S C R ~ + JSCRS + Jx,,)

I E E Proc -Circuits Drirrrs Sysr.. Vol. 142. .Yo. 4. August lYY5

(26)

much less than electron diffusion length. This yields

Combining eqns. 24, 26-28 obtains the electron concen- tration at x = X , including effects of thermionic emission and space-charge recombination

Inserting eqn. 29 into eqn. 28 gives

4". y n N E exp (-qYfll/kT) - ( J S C R g + J S C R S + JSCR2)

1 + V , Y , exp CdY,, + Vas + YBgc) /kT1(X3 - X2)/Dn (30)

Eqn. 30 accounts for thermionic emission across the het- erojunction interface and space-charge recombination in the emitter-base heterojunction for determining the collector current of the HBT.

3.3 Self heating Heterojunction bipolar transistors using GaAs semi- insulating substrate exhibit a self-heating effect owing to low thermal conductivity of GaAs semi-insulating sub- strate [17]. The increment of temperature due to self heating is a function of power dissipation and thermal resistance. Assuming the heat is dissipated throughout the semi-insulating substrate with a lateral diffusion angle and the thermal conductivity is proportional to (T /TJb . The Kirchhoff transformation gives the junction tem- perature as [18]

T = [I/T:-' - (b - l ) R , , o P , / T ~ ] - " b - l (31) where P , is the power dissipation, RthO is the thermal resistance at room temperature To, and b = 1.22 [17].

To account for temperature dependence of collector and base currents, temperature-dependent physical parameters must be used. Equations for doping and

251

temperature-dependent D , and D, and temperature- dependent n , . €(;. and I , for GaAs semiconductor are uted 111. 191.

3.4 Small-signal parameters Once the DC parameters are determined, small-signal parameters such as transconductance and output con- ductance for ditferent heterojunction systems can be readily derived. The transconductance of the HBT is defined as the change of collector current with respect to hik :

In eqn 32 <X2 iV,, is obtained by using eqn 16 and i n ( X , ) ?PHt is solced numerically by using eqn 29 The output conductance describes the base-width modulation effect on the collector current and IS defined ds the change of I , with I:,

il, i l ; , Yo = __

(33)

3.5 Cutoff frequency The cutoff frequency of the HBT is a figure of merit. This parameter is important for analysing the bandwidth of small-signal amplifier and the power gain of power amplifier. The cutoff frequency is given by the well known approximation:

r; 2

I

(34)

where r , is the emitter AC resistance ( r , = kT/ql , ) , C,, is the emitter--base junction capacitance, C,, is the collector-base junction capacitance, X , - X , is the collector-base depletion layer width, D, is the minority- carrier diffusion coefficient in the base, i's is the saturation velocit) in the collector-base depletion region, and R , is the collector resistance. C,, and C,, for different hetero- junction systems are obtained by using eqn. 17.

4 Results and discussions

The analytical equations derived in Sections 2 and 3 are calculated and compared with experimental data or numerical simulation. The N /p heterojunction has N , = 5 x 10'- cm- and N , , = 1 x 1019 cm ~ 3 . The physical parameters used in the calculation include E , , = 1.773 eV. E,,, = 1.423 eV, n,, = 2368.6 cm-', and niL = 2.05 x lo6 cm ', Fig. 2A shows the space-charge layer thickness from the emitter-base metallurgical junction to the quasineutral emitter region (XI) against the emitter- base junction voltage for an abrupt AI,,,Ga, ,As/GaAs heterojunction (X, = 0 A) with the setback-layer thick- ness X , = 0. 50, 100, 150, and 200 A. The space-charge

2%

layer thickness X I decreases with junction voltage. For larger setback-layer thickness, X I is smaller and more

0 8 0 9 1 1 1 1 2 1 3 1 4 1 5

j u n c t ~ o n voltoge, V

U

Fig. 2A X, = 0, 50, 100, 150, and 200 A

X , against junct icn coltage of abrupt hererojunction with

X , = 2 0 0 A ~~ x , = 5 o A

x, = IN) A ~ x,=15oA non x , = o A

sensitive to the change of junction voltage. The space- charge layer thickness from the emitter-base metallurgi- cal junction to the quasineutral base region ( X , ) against the emitter-base junction voltage for different X , is shown in Fig. 2B. For all junction biases, X , increases

.. . ..... . - .-. "u"u 100

C8 0 9 1 1 1 12 1 3 1 4 1 5 junction vo l tage, V

b

Fig. 2 8 X, = 0,50,100,150, and 200 A

X, against junction voltage of abrupt heterojunction with

xS=2wA ~~ X, = 50A ~ x S = i 5 n A ono x,=oA

X , = i M i A

with setback-layer thickness. At X , = 0 8, the space- charge layer thickness X , decreases with the junction voltage. The thickness X , becomes insensitive to junction voltage when X , > X , (a! X , = 0 A. The space-charge layer thickness against junction voltage for a linearly graded heterojunction with X , = 0 8, and X , = 50, 100,

IhE Proc.-Circuits Devices Syst., Vol . 142, N o . 4, August 1995

150. and 200 A is shown in Fig. 3. The upper lines in Fig. 3 represent X I and the lower lines in Fig. 3 represent X, . Both X, and X, are independent of graded-layer thick- ness and decrease with junction voltage.

1 3 8 3 9 . 1 ' 1 2 ' 3 1 L 1 5

jurc t ion voltage, V

Space-charge laier thickness uyainrt junction ioltnge of abrupt Fig. 3 and lineurly graded heterojunctron w t h X, = io. 100. 150. and 200 A

To examine the sensitivity of X , and X , on different heterojunctions, the emitter-base depletion capacitance as a function of setback-layer and graded-layer thick- nesses is shown in Fig. 4. Fig. 4a plots the effect of setback-layer thickness on the heterojunction capacitance at V = -1.0, -0.5, 0, 0.5, and 1.0 volt. In Fig. 4a the squares, lines, and triangles represent the present model and the circles represent the MEDICI simulation [20]. The agreement between the present model predictions and MEDICI device simulation results is very good. The junction capacitance decreases with setback-layer thick- ness for all junction voltages owing to the increase of space-charge layer thickness X, . The sensitivity of C , with respect to X, is more pronounced at higher forward biases. Fig. 4b shows the effect of graded layer thickness on the heterojunction depletion capacitance. The deple- tion capacitance increases with junction voltage, but is independent of graded-layer thickness because XI is insensitive to the change of graded layer thickness. Com- parison between the model predictions (squares, lines, and triangles) with the MEDICI simulation (circles) shows good agreement.

Fig. 5 shows the collector and base currents against the baseeemitter voltage for a linearly graded hetero- junction bipolar transistor. The AI,Ga, ~I As/GaAs het- erojunction bipolar transistor has a mole fraction 0.3, linearly graded emitter-base heterojunction of 300 A, emitter doping 5 x 10" cm-', base doping 5 x 10" ~ m - ~ , collector doping 5 x 10" ~ r n - ~ , emitter thickness 0.1 pm. base thickness 0.1 pm, collector thick- ness 0.3 pm. and emitter area 80 pm'. The physical and empirical parameters used in our calculation include E , , = 1.773 eV, n i l = 2368.6 ~ m - ~ , nt2 = 2.05 x lo6 ~ m - ~ , JHs0 = 1 x I O - ' ' A. and nL = 1.4. In Fig. 5 the lines represent the

model predictions and the symbols represent the experi- mental data [16]. The agreement between the model pre- dictions and experiment results over a wide range of VHE is fair.

I € € Proc.-Circuits Deuices Syrr . b'ol 142. MO. 4, August 199.5

E, , = 1.423 eV, T~ = 1 x I O - ' O s ,

7 028 O 317

o . i , , , , , " 0 50 100 150 200 250 300 350 400

setback layer thickness A

U

I 0 v o l t

.........h ..... 99.99.9.1 .............. "L 0 25

- 0 1 3

0 1 0 50 100 150 200 250 300 350 LOO

graded layer thickness, A

b

Fig. 4 0 MEDIC1 sirnulation LI Setback layer h Graded layer

Junction capacitance against layer thickness at different biases

10-19

06 0 7 08 09 1 1 1 1 2 1 3 1 4 1 5

base-emitter voltage, V

Fig. 5 linearly graded heterojuncfion bipolar transistor

Collector and base currents against base-emitter voltage /or

W I , expenrnenl ~ I,rnodel A 1.experirnenl

0-0 1,rnodel

259

The agreement can be improved by increasing the base and collector currents at moderate VBE (by adjusting J,,, . nL , T ~ , T ” ~ , z P 9 , 7”) which also reduces the collector current, and by adding base and emitter resistance effect ( I , R , + I , RE drop) in our model at high V,, . Neverthe- less, the agreement between the model prediction and experiment around the turn-on voltage of the HBT (GEon = 1.3 V) is fairly good. We thus use this bias point to evaluate the sensitivity of setback and graded-layer thicknesses on the current gain and transconductance of the HBT.

Fig. 6a shows the current gain against graded-layer thickness at V,, = 1.25, 1.3, and 1.35 V, respectively. The current gain increases with graded-layer thickness at all

2001

0 50 100 150 200 250 3 0 0

g raded layer thickness, A

0

Fig. 6A emitter rolrages

Currem gain against graded-layer thickness at diferent base-

X , = O A Y , = 5 x I O l - c m - ’

250 I graded h e t e r o p n c t l o n X g 1200 A

200

xg: 100 a

c a 2 , c c : /

I ’ abrupt hetero ‘ u n c t ion

5 0 , /// ~-

LO 80 120 ’60 ZOO

setback layer thmckness, h

b

Fig. 66 graded injer rhicknessec VBL = I 3 v V, = 5 x IO’ -cm- ’

260

Current gain against setback-laver thicknes, /or diflerent

biases owing to reduction of electron potential barrier at the emitter-base heterojunction. The increase of current gain, however, saturates around 130 A. Further increase of graded-layer thickness even decreases the current gain slightly. This is because of the increase of space-charge region recombination in the graded layer at larger X , . The sensitivity of current gain against setback-layer thickness at different graded-layer thicknesses is shown in Fig. 6b. The current gain is evaluated at V,, = 1.3 V. For the abrupt heterojunction bipolar transistor, the use of a setback layer certainly increases the current gain until it reaches the saturation point. For the linearly graded het- erojunction bipolar transistor with X , = 50 A, the use of a setback layer still contributes to the increase of current gain. For the linearly graded heterojunction bipolar tran- sistor with X , = 100 and 200 8, in Fig. 6b, however, the current gain decreases with increasing setback-layer thickness. This phenomenon is explained as follows. When the heterojunction bipolar transistor has enough junction grading, the emitter-base heterojunction is similar to a homojunction. The injection efficiency cannot be further improved by the use of a setback layer. The use of a setback layer in this case introduces the increment of space-charge recombination current and decreases the current gain of the HBT.

The transconductance against graded-layer thickness as a function of setback-layer thickness is shown in Fig. 7a. The transconductance increases with graded-layer thickness owing to an increase of sensitivity of collector current with respect to the change of baseeemitter voltage. When the setback-layer thickness is increased, the transconductance becomes less sensitive to graded- layer thickness. Similar behaviour can be seen in Fig. 76 which shows the transconductance against setback-layer thickness as a function of graded layer thickness. The transconductance increases with setback-layer thickness and becomes insensitive to X , at larger graded layer thickness.

The cutoff frequency against the collector current density is depicted in Fig. 8. The AI,,Ga,,,As/GaAs het- erojunction bipolar transistor has the emitter doping 5 x 1 0 ” ~ m - ~ , base doping 1 x l O ” ~ m - ~ , collector doping 5 x loL6 cm-,, emitter thickness 0.17 pm, base thickness 0.1 pm, collector thickness 0.3 pm, emitter area 10pm2, and collector area 100pm2. In Fig. 8 the solid line with empty squares represents the isothermal model, the filled squares represents the numerical results published in the literature [19], and the solid and dotted lines represent the present model with self-heating effect at V,, = 2.5 and 5.0 V, respectively. The increase of cutoff frequency at low collector current density is due to the decrease of emitting charging time (7, cc l/Jc). When the collector current density is sufficiently high, the cutoff fre- quency decreases with J , owing to an increase of junc- tion temperature resulting from the self-heating effect. Self heating decreases the saturation velocity in the collector-base depletion layer and elecron mobility in the base which increases the collector-base depletion-layer transit time and base transit time, respectively. The iso- thermal model obviously cannot account for the self- heating effect. The present model with self-heating effect is compared with the numerical model. Good agreement between the present model prediction and numerical results is obtained. Self-heating effects on the common- emitter current gain and transconductance of the HBT are presented. Fig. 9 plots the current gain against graded-layer thickness for the isothermal model and the present model at V,, = 1.4 volts and V,, = 3 , 6, and 9

I E E Proc.-Circuits Devices Syst., Vol. 142, N o . 4, August 1995

volts. The current gain increases with graded-layer thick- ness and then decreases with X , when self heating becomes significant. Self heating causes the rise of junc- tion temperature which increases the base and collector

50 - N I 0 2. 4 0 - !i

E

z 20-

30-

- 2

1

x, : 233 A

150-

100-

U)

E aJ

C 0 U

3 D C

" VI

L

2

U- E

a. v ' 0

3 V C 0 v vi C

-

? L

I

0 4c 80 120 160 200

graded layer th ickness A

Q

' o o o l g r a d e d heterojunct8on X g . 2 C O A

1 I C ? A

'0,

abrupt heterojuPctlon

C ' I

1 3 8C 12c 160 200

setback !ayer tmckness, A

b

Fig. 7 i , , = I ? \ U Graded la?er h Setback laber

7 ratirconducrnrice aqainst /ar.er thichnecs for different X,

currents of the HBT. The increase of base current, however. is larger than that of collector current. This results in a decrease of current gain at larger X , . Fig. 10 shows the transconductance against the graded-layer thickness for the isothermal model and the present model under the same bias conditions in Fig. 8. The trans- conductance increases with graded layer thickness at smaller X , and decreases with graded-layer thickness at larger X , . The increase of y, at smaller X , is due to improvement of injection efficiency by increasing graded- layer thickness. The decrease of 9, at larger X , is due to the self-heating effect which increases the collector current as well as junction temperature.

I E E Proc.-Crrcuits Decrccs Sysr., I ' d 142. No. 4. August 1995

5 Conclusions

Common-emitter current gain, transconductance, and cutoff frequency of the abrupt and linearly graded hetero- junction bipolar transistors with or without a setback

col lector current density. Alcrn'

Fig. 8 0-0 isothermal model W-m -~

~~

Cutofffrequency against collector current density

numerical model V,, = 2.5 V madel V,, = 2 5 V model V,, = 5.0 V

200 I

50v 04

100 150 200 i 0 50

graded layer th ickness, A Fig. 9 thermal models V,, = 1 4 V , X, = 0 A

Current gain against graded-layer thickness for isothermal and

~ ~~ isothermal model model V,, = 3 V model V,, = 6 V model VCt = 9 V

~~~~

~~

. . . . . . .

layer have been evaluated. Comparisons between the iso- thermal model, the numerical model, and the present model including the self-heating effect are demonstrated. Under the isothermal condition, the current gain increases with graded-layer and setback-layer thicknesses, but decreases slightly with setback-layer thickness at larger X , owing to an increase of space-charge recomb- ination. The transconductance increases with graded- layer and setback-layer thicknesses because of an increase of sensitivity of collector current with respect to the change of baseeemitter voltage. When self heating is accounted for, the current gain and transconductance

26 1

increase a.ith graded-layer thickness at smaller X , and decreases with graded-layer thickness at larger X,. The degradation of g, results from higher junction tem-

3 1 0 50 103 150 200 250

graded layer thickness , 8

Transconductance uyainsr graded-layer thickness /or isother- Fig. 10 mal and thermal models

mthermal model model I, = 3 V model I,, = 6 V madel C,, = Y V

r;, = I 4 v. x, = n A

. . . . . . .

perature at large power dissipation. The cutoff frequency increases with collector-current density owing to a reduction of emitter charging time J l/J,) and decreases with J , when self heating is significant. Self heating increases the junction temperature and decreases the saturation velocity in the collector-base depletion layer and electron mobility in the base which increases the collector-base depletion-layer transit time and base transit time.

6 References

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262 IEE Proc.-Circuits Deuices Syst., Vol . 142, No. 4 , August 1995