JOURNAL OF APPLIED PHYSICS
VOLUME 86, NUMBER 3
1 AUGUST 1999
Design of a InP/In1ⴚ x Gax Asy P1ⴚ y /In0.53Ga0.47As emitter-base junction in a Pnp heterojunction bipolar transistor for increased hole injection efficiency S. Ekbote, M. Cahay,a) and K. Roenker Department of Electrical Engineering, University of Cincinnati, Cincinnati, Ohio 45221
共Received 17 November 1998; accepted for publication 17 April 1999兲 Starting with Burt’s envelope function theory, we calculate the transmission coefficients of holes across an InP/In1⫺x Gax Asy P1⫺y /In0.53Ga0.47As heterointerface while varying the width and gallium and arsenic fractions of the InP lattice-matched quaternary compound (x⫽0.47y). While comparing our results to the case of an abrupt InP/In0.53Ga0.47As interface, we find that the transmission coefficients of both heavy- and light-holes can be enhanced significantly for a 60-Å-wide quaternary layer with an arsenic fraction y⫽0.4 (x⫽0.188). This should lead to an enhanced hole injection efficiency of Pnp heterojunction bipolar transistors using the heterointerface analyzed here as an improved design of the emitter-base junction. © 1999 American Institute of Physics. 关S0021-8979共99兲02615-8兴 I. INTRODUCTION
In1⫺x Gax Asy P1⫺y compound lattice matched to InP (x ⫽0.47y). From Fig. 1, it can be seen that the transmission coefficients of holes from emitter to base should be identical for the cases y⫽0 and 1 since the heterostructure then reduces to an abrupt InP/In0.53Ga0.47As 共Table I兲. When a quaternary compound is present at the interface, we expect the transmission coefficients of holes to be affected due to the interference effects at the two interfaces 共which vary with the y composition兲. An appropriate choice of the y composition and the width of the quaternary layer is therefore expected to lead to enhanced hole injection efficiency 共i.e., hole tunneling from emitter to base兲. The main goal of this article is to determine the optimal values of these parameters through numerical simulations of the transmission coefficients of holes across the heterointerface in Fig. 1 using Burt’s formalism. In Sec. II, we briefly describe Burt’s formalism and Foreman’s boundary conditions and the procedure to determine the transmission coefficients through the structure shown in Fig. 1. Section III contains our numerical examples illustrating the energy and transverse wave vector depen-
Recently, Burt has extended the effective-mass theory to heterostructures using a rigorous envelope-function formalism.1 Soon after, Foreman2 has shown that Burt’s multiband effective-mass theory leads to an unambiguous formulation of the boundary conditions to be satisfied by the envelope function components at an interface. In Ref. 2, Foreman derived the explicit form of the valence-band effective mass Hamiltonian and corresponding boundary conditions at an interface while taking into account the coupling between the heavy, light, and spin–orbit split-off 共SO兲 bands. He showed that the boundary conditions derived from Burt’s Hamiltonian are quite different from those currently found in the literature.3–6 Recently,6 we have shown that the transmission coefficients of heavy and light holes across an abrupt InP/In0.53Ga0.47As interface calculated using Burt’s formalism are quite different from the transmission coefficients calculated using the commonly used ‘‘symmetrization’’ procedure of the 6⫻6 Luttinger–Kohn Hamiltonian.4,6–9 Furthermore, we showed that the transmission coefficients of holes are quite sensitive to the orientation of the hole wave vector component parallel to the heterointerface. The commonly used axial approximation to calculate the hole tunneling current through nanoscale structures starting with the Luttinger–Kohn effective mass Hamiltonian is therefore not accurate.10 In this article, we use Burt’s formalism to analyze the transmission coefficients of holes through the valence band energy profile shown in Fig. 1. Except for the addition of a quaternary layer, this valence band energy profile is typical of the one existing across the emitter-base junction of a Pn p HBT under large forward bias.11 The band diagram has been drawn such that energy is measured positive moving into the valence band. At the interface between the InP emitter and In0.53Ga0.47As base, there is a quaternary
FIG. 1. Schematic representation of the valence band energy profile across the emitter-base junction of a forward biased PnP HBT. The energy is measured positive moving into the valence band. The zero of energy is the bottom of the valence band in the emitter 共InP兲 region. The In1⫺x Gax Asy P1⫺y quaternary layer is assumed to be lattice matched to InP (x⫽0.47y).
a兲
Electronic mail:
[email protected]
0021-8979/99/86(3)/1670/6/$15.00
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© 1999 American Institute of Physics
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J. Appl. Phys., Vol. 86, No. 3, 1 August 1999
Ekbote, Cahay, and Roenker
II. APPROACH
dence of the hole transmission coefficients. This includes a variation of the aluminum composition of the quaternary material to determine the optimal condition for tunneling of both heavy and light holes. Finally, Sec. IV contains our conclusions.
H⫽
冤
⫺S ⫺
R
共 1/& 兲 S ⫺
&R
0
P⫹Q
⫺R
⫺S ⫹
⫺&R
共 1/& 兲 S ⫹
† ⫺S ⫺
⫺R
P⫺Q
C
&Q
R†
† ⫺S ⫹
C†
P⫺Q
† 共 1/& 兲 S ⫺
&R †
⫺&R † 共 1/& 兲 S ⫹
†
&Q
⫺
冑
3 † ⌺ 2 ⫺
冑
⫺
3 † ⌺ 2 ⫹
&Q
where, following Foreman, we have used atomic units (ប ⫽m 0 ⫽1). In Eq. 共1兲, the dagger is used for Hermitian conjugate and the following notations are introduced2 P⫽V h 共 z 兲 ⫹
1 2
共 ␥ 1 k 2 ⫹k z ␥ 1 k z 兲 ,
共2兲
Q⫽ 21 共 ␥ 2 k 2 ⫺2k z ␥ 2 k z 兲 ,
共3兲
冉 冊 冉 冊
) ) 2 2 ¯␥ k ⫺ ⫹ k⫹ , 2 2
共4兲
S ⫾ ⫽)k ⫾ 关共 ⫺ ␦ 兲 k z ⫹k z 兴 ,
共5兲
R⫽⫺
⌺ ⫾ ⫽)k ⫾ 兵 关 31 共 ⫺ ␦ 兲 ⫹ 32 兴 k z ⫹k z ⫻ 关 32 共 ⫺ ␦ 兲 ⫹ 31 兴 其 ,
共6兲
C⫽k ⫺ 关 k z 共 ⫺ ␦ ⫺ 兲 ⫺ 共 ⫺ ␦ ⫺ 兲 k z 兴 ,
共7兲
where k ⫾ ⫽k x ⫾ik y ,
k z ⫽⫺i
d , dz
k 2 ⫽k 2x ⫹k 2y ,
共8兲
and (k x ,k y ) are the components of the transverse wave vector. Furthermore, ˜␥ ⫽ 21 共 ␥ 3 ⫹ ␥ 2 兲 ,
共9兲
TABLE I. Luttinger–Kohn parameters and spin-orbit split-off energy used in the simulations. The values for the In0.53Ga0.47As material are found by linear interpolation.
␥1 ␥2 ␥3 ⌬ 共eV兲
In the exact envelope-function theory of Burt,1 the Hamiltonian describing the interaction of the heavy, light, and SO bands is given by2
0
P⫹Q
GaAs
InAs
InP
6.85 2.1 2.9 0.34
20.4 8.3 9.1 0.38
4.95 1.65 2.35 0.1
1671
冑
†
冑
3 ⌺ 2 ⫺
3 ⌺ 2 ⫹
&Q
冥
,
共1兲
⫽ 21 共 ␥ 3 ⫺ ␥ 2 兲 ,
共10兲
P⫹⌬
⫺C
⫺C †
P⫹⌬
⫽⫺
1 18
⫺
1 18
␥ 1 ⫹ 94 ␥ 2 ⫹ 32 ␥ 3 ,
共11兲
⫽ 61 ⫹ 16 ␥ 1 ⫺ 31 ␥ 2 ,
共12兲
␦ ⫽ 19 共 1⫹ ␥ 1 ⫹ ␥ 2 ⫺3 ␥ 3 兲 .
共13兲
In Eq. 共1兲, ⌬ is the spin-orbit split-off energy and the ␥ i are the Luttinger parameters. Hereafter, hole energies are measured as positive moving into the valence band. In Eq. 共2兲, V h (z) is the valence band energy profile, z being the axis of growth of the heterostructure. The Hamiltonian in Eq. 共1兲 is a 6⫻6 matrix in the basis composed of the 共兩 23,⫾ 23典兲 heavy hole, the 共兩 23,⫾ 21典兲 light hole, and the 共兩 21,⫾ 12典兲 split-off Bloch wave functions at the center of the Brillouin zone. These functions are eigenfunctions of the total angular momentum and diagonalize the spin-orbit interaction. In Burt’s theory, these functions are required to be the same throughout the heterostructure.1 As shown by Foreman,2 the 6⫻6 effective-mass Hamiltonian in Eq. 共1兲 can be block diagonalized into a pair of 3 ⫻3 blocks:2,8
H ⫾⫽
冋
P⫹Q
R⫿iS
R⫾iS †
P⫺Q⫿iC
&R⫿
i &
&R⫾
i &
S
册
&Q⫿i 冑3/2⌺ ,
S † &Q⫾i 冑3/2⌺ † P⫹⌬⫾iC
共14兲
where the upper and lower blocks correspond to the upper and lower signs.
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Ekbote, Cahay, and Roenker
In the Hamiltonians H ⫾ , the P and Q terms are as defined earlier. The new expression for C is obtained from Eq. 共7兲 after replacing k ⫺ by k . The rest of the terms are defined as follows:2 ) R⫽⫺ ␥ k 2 , 共15兲 2
␥ ⫽ 冑˜␥ 2 ⫹ 2 ⫺2 ˜␥ cos共 4 兲 ,
共16兲
S⫽)k 关共 ⫺ ␦ 兲 k z ⫹k z 兴 ,
共17兲
⌺⫽)k 兵 关 31 共 ⫺ ␦ 兲 ⫹ 32 兴 k z ⫹k z 关 32 ( ⫺ ␦ )⫹ 13 兴 其 ,
共18兲
冋
共 ␥ 1 ⫺2 ␥ 2 兲
d dz
⫾ 冑6k 共 ⫺ ␦ 兲
共 ␥ 1 ⫹2 ␥ 2 兲
⫺2& ␥ 2
d ⫾2k 共 ⫺ ␦ ⫺ 兲 dz
d ⫿&k 共 ⫺ ␦ ⫹2 兲 dz
Starting with the upper or lower Hamiltonians, the eigenenergies and corresponding envelope functions of the valence subbands can be obtained by solving the effectivemass equation ⌺ j 关 H i j ⫹V h 共 z 兲 ␦ i j 兴 F j 共 k,r兲 ⫽E 共 k 兲 F i 共 k,r兲 ,
T HH⫽
inc j z,H
,
trans 兩 SOH兩 2 j z,SO , T SOH⫽ inc j z,H
T LH⫽
inc j z,H
⫺2& ␥ 2
␥1
d ⫾&k 共 2 ⫺2 ␦ ⫹ 兲 . dz
d ⫿2k 共 ⫺ ␦ ⫺ 兲 dz
共19兲
and the reflection coefficients as a heavy, light, or SO hole, are given, respectively, by R HH⫽⫺
inc 兩 ⌫ HH兩 2 j ⫺z,H inc j z,H
共20兲
where (i, j)⫽(1,2,3) for the heavy, light, and SO bands. The hole wave functions can then be written in the three regions of Fig. 1 as a linear combination of the eigenfunctions describing left and right moving holes in the heavy, light, and SO bands. Using the boundary conditions for a hole incident from the left, the transmission and reflection coefficients can then be determined after matching the hole wave function at the two interfaces using the boundary conditions given above. The algebra is quite tedious but straightforward and similar to the derivation given in Ref. 7 where we analyze the tunneling of holes above a quantum well. As pointed out by Foreman,2 the unitary transformation which diagonalizes the Hamiltonian in Eq. 共1兲 in the form given by Eq. 共14兲 is strictly valid when the hole transverse wave vector k ⫽(k x ,k y ) is along the 具100典 or 具110典 directions. This requires in Eq. 共16兲 to be equal to 0° or 45°. In order to calculate the transmission and reflection coefficients of holes incident from the InP region, the probability current density must be calculated along the propagation direction for the incident, reflected, and transmitted waves. The details are given in Ref. 6. For instance, the transmission coefficients for an incident heavy hole to be transmitted as a heavy, light, or SO hole, are given, respectively, by trans 兩 LH兩 2 j z,L
册
⫿ 冑6k
⫾2) k
B⫽ ⫿2) 共 ⫺ ␦ 兲 k
trans 兩 HH兩 2 j z,H
where is the azimuthal angle 关 ⫽arctan(ky /kx)兴. Across an interface, we must require the three components of the envelope function to be continuous. If we replace the wave vector k⫽(k x ,k y ,k z ) by the differential operator ⫺i“, the boundary conditions can be obtained by a simple integration of the effective-mass equation with the Hamiltonian in Eq. 共1兲 across the interface. This procedure requires that the three component vectors BF must be continuous, where F are the hole envelope functions solutions of the effective mass equation with Hamiltonian 共14兲 and B is given by2
R SOH⫽⫺
inc 兩 ⌫ SOH兩 2 j ⫺z,SO inc j z,H
R LH⫽⫺
,
inc 兩 ⌫ LH兩 2 j ⫺z,L inc j z,H
, 共22兲
.
In Eqs. 共21兲 and 共22兲, the coefficients IJ and ⌫ IJ are the transmission and reflection amplitudes for a hole incident from the InP region in the Jth band to be transmitted or reflected in the Ith band, respectively. Furthermore, the labels inc and trans mean that the probability current density must be evaluated in the incident and transmitted regions, respectively. As shown in Refs. 6 and 9, the relationship j ⫺z, ␣ ⫽⫺ j z, ␣ holds between the probability current densities corresponding to left ( j ⫺z, ␣ ) and right ( j z, ␣ ) propagating states ( ␣ ⫽H, L, or SO兲. Explicit expressions for the probability current densities are given in Refs. 6 and 9. Current conservation further requires that T HH⫹T LH⫹T SOH⫹R HH ⫹R LH⫹R SOH⫽1, which is helpful to check the accuracy of the numerical simulations. Once the transmission coefficients are found, the tunneling current across the interface can be calculated using a generalization of the expression derived by Chao and Chuang3 J HH⫽
e 43
冕
⍀ h1 ⫹⍀ h2 ⫹⍀ h3
d 3 k关 f L 共 E 兲 ⫺ f R 共 E⫹eV 兲兴
⫻关 T HH共 k兲 ⫹T LH共 k兲 ⫹T SOH共 k兲兴v zHH共 k兲
, 共21兲
⫹
e 43
冕
⍀ h4
d 3 k关 f L 共 E 兲 ⫺ f R 共 E⫹eV 兲兴关 T HL共 k兲
⫹T LL共 k兲 ⫹T SOL共 k兲兴v zHH共 k兲
共23兲
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J. Appl. Phys., Vol. 86, No. 3, 1 August 1999
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and J LH⫽
e 43
冕
⍀l
d 3 k关 f L 共 E 兲 ⫺ f R 共 E⫹eV 兲兴关 T HL共 k兲
⫹T LL共 k兲 ⫹T SOL共 k兲兴v zLH共 k兲 .
共24兲
In Eqs. 共23兲 and 共24兲, f L (E) and f R (E) are the Fermi–Dirac factors in the left 共InP兲 and right (In0.53Ga0.47As) sides of the heterointerface, respectively; v zHH(k) and v zLH(k) are the group velocities of the incident heavy and light holes, respectively.9 The different domains of integration defined in Eqs. 共23兲 and 共24兲 are illustrated in Fig. 8 of Ref. 3. Since the total heavy- and light-hole current densities must be computed by summing the results obtained with the upper and lower Hamiltonians, we compute in the next section the average transmission coefficients obtained with the upper and lower Hamiltonians. At large forward emitter-base bias, only the contribution from the terms proportional to f L (E) will be dominant in Eqs. 共23兲 and 共24兲. In order to increase the hole injection efficiency 共i.e., the hole tunneling current兲 at large forward bias, the contributions of the integrands in Eqs. 共23兲 and 共24兲 must be enhanced. We will show in the numerical examples below that it is indeed the case for the structure shown in Fig. 1 in the presence of a quaternary layer with the appropriate aluminum fraction.
III. RESULTS
In a typical Pnp heterojunction bipolar transistor, the emitter 共InP region in Fig. 1兲 will be doped at a level of several times 1017 cm⫺3. We first consider heavy- and lightholes incident with a transverse wave vector k equal to 0.04⫻(2 /a), where a⫽5.83 Å is the lattice contact of InP. Since the majority of holes in the emitter will be in the heavy-hole band, we first maximize the transmission coefficient of heavy hole through the structure in Fig. 1 by varying the width of the quaternary layer. The results of these simulations are not shown here but lead to the optimum choice of 60 Å for the width of the quaternary layer. First, we calculate the energy thresholds for free propagation in the heavy, light, and SO bands in the InP region as a function of the magnitude of the transverse wave vector. These threshold energies are determined by solving the eigenvalue problem associated to Hamiltonian 共14兲 into a standard eigenvalue problem for k z as described in Ref. 7. The threshold energies are then found numerically by identifying the lowest energy above which k z is purely real in the different bands. Repeating this procedure for different k values, we obtain the energy threshold diagram shown in Fig. 2. For k ⫽0.04⫻(2 /a) and ⫽45°, the energy thresholds for free propagation in the heavy (E ⫺ h ), light (E 1 ) hole bands are found to be equal to 2.5 and 28 meV, respectively. Also shown in Fig. 2 are the ‘‘transverse energies’’ ប 2 k 2 /2m 共HH and LH兲, where m(HH)⫽m 0 /( ␥ 1 ⫺2 ␥ 2 ) and and m(LH)⫽m 0 /( ␥ 1 ⫹2 ␥ 2 ) are the heavy- and light-hole masses, respectively. These transverse energies are above the energy thresholds determined numerically indicating the importance of the SO band in lowering the energy threshold in the various hole bands.12–16
FIG. 2. Dependence on the magnitude of the transverse wave vector (k ) of the threshold energies for free propagation in the heavy-hole (E ⫺ h ), lighthole (E 1 ), and SO band (E SO). The results are shown for an azimuthal angle ⫽45°.
Shown in Figs. 3 and 4 is the variation of the transmission coefficients of heavy holes (T HH) for a 60-Å-wide quaternary layer as a function of the aluminum fraction y. Figures 3 and 4 correspond to a heavy hole whose transverse wave vector k has an azimuthal angle equal to 0° and 45°, respectively. The case y⫽0 corresponds to an abrupt InP/In0.53Ga0.47As interface. In Figs. 3 and 4, the T IJ coefficients are the probabilities of hole conversion from the Jth hole band into the Ith hole band. Figures 3 and 4 indicate that the transmission coefficient T HH can be enhanced substantially, especially at low incident
FIG. 3. Energy dependence of the heavy-hole tunneling coefficients (T HH) as a function of the y composition of the In1⫺x Gax Asy P1⫺y quaternary layer. The y⫽0 and 1 cases corresponds to an abrupt InP/In0.53Ga0.47As interface. Also shown are the tunneling probabilities for heavy-to-light-hole band (T LH) and heavy-to-SO-band (T SOH) conversion. The curve labeled FD is the Fermi–Dirac factor normalized to be equal to 1 for zero incident energy. The latter corresponds to the bottom of the valence band in the InP region. The width of the quaternary layer is set to 60 Å. The azimuthal angle of the incident hole is set equal to 0°. The magnitude of the transverse wave vector k ⫽0.04(2 /a), where a⫽5.83 Å, the lattice constant of InP.
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J. Appl. Phys., Vol. 86, No. 3, 1 August 1999
FIG. 4. Same as Fig. 3 for an incident heavy hole with an azimuthal angle equal to 45°.
energy, for a quaternary layer with an aluminum fraction y ⫽0.4. T HH increases by a factor varying between two and three in the energy range below 20 meV. This factor also depends on the azimuthal angle of the transverse wave vector. The energy region where the enhancement of the transmission coefficient is quite drastic also corresponds to the energy range where the contribution to the integrals for the current density in Eqs. 共23兲 and 共24兲 is most important. This is illustrated by displaying the Fermi–Dirac factor in Fig. 3 normalized to unity at an energy corresponding to the bottom of the valence band in InP. The plot of the Fermi– Dirac factor shows that, in the presence of a quaternary layer, the contribution to the overall tunneling current comes from an energy range where the transmission coefficients of heavy holes are much larger than for the case of a simple interface InP/In0.53Ga0.47As. We also note in Figs. 3 and 4 that the change in the probability of heavy-to-light-hole conversion T LH is not too much affected by the variation in the aluminum fraction.
FIG. 5. Same as Fig. 3 for an incident light hole. T HL and T SOL are the probabilities of light-to-heavy and light-to-SO-hole conversion, respectively. The azimuthal angle of the incident hole is set equal to 0°, k ⫽0.04⫻(2 /a) and a⫽5.83 Å.
Ekbote, Cahay, and Roenker
FIG. 6. Same as Fig. 5 for an incident light hole with an azimuthal angle ⫽45°.
Even though the contribution of T LH to the integral in Eq. 共23兲 is slightly reduced for y⫽0.4 compared to y⫽0, this reduction is largely compensated by the drastic increase in the coefficient T HH going from y⫽0 to y⫽0.4. Furthermore, the probability of conversion from heavy to SO band is very small and occurs over an energy range where the Fermi– Dirac factor is close to zero, making the contribution of the third term in the first integral in Eq. 共23兲 insignificant. A similar enhancement of the transmission coefficient of light hole is also found as indicated in Figs. 5 and 6. Even though Figs. 5 and 6 indicate that an aluminum fraction of y⫽0.6 or 0.7 would also enhance the transmission coefficient T LL we must keep in mind that the enhancement of the transmission coefficients of the predominant heavy-hole population is not as important for these aluminum fractions. As a result of the rapid decay of the Fermi–Dirac factor at large incident energy, the contributions to the current density in Eqs. 共23兲 and 共24兲 from holes with larger transverse wave vector 关 ⬎0.04⫻(2 /a) 兴 is quite small because the
FIG. 7. Same as Fig. 4 for an incident heavy hole with a transverse wave vector k ⫽0.02⫻(2 /a) and a⫽5.83Å.
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J. Appl. Phys., Vol. 86, No. 3, 1 August 1999
FIG. 8. Same as Fig. 6 for an incident light hole with a transverse wave vector k ⫽0.02⫻(2 /a) and a⫽5.83 Å.
threshold energies for free propagation in the heavy- and light-hole bands increase rapidly with the transverse wave vector as shown in Refs. 7 and 11. Finally, Figs. 7 and 8 show that the enhancement of the transmission coefficients T HH and T LL is also observed at lower value of the transverse wave vector 关 k ⫽0.02 ⫻(2 /a) 兴 . For this transverse wave vector and ⫽45°, the energy thresholds for free propagation in the heavy- and light-hole bands in the InP region are found to be equal to 1 and 8 meV, respectively. As before, the change in the contribution of the terms proportional to T IJ coefficients to the integrals in Eqs. 共23兲 and 共24兲 as the aluminum fraction is varied are largely dominated by the leading contributions from the terms proportional to the T II coefficients. Even though not shown here, we have found that trend to be valid for an even smaller value of the transverse wave vector. This leads to an enhanced heavy- and light-hole tunneling current for an emitter-base junction with a quaternary layer compared to the case of simple InP/In0.53Ga0.47As interface. This should lead to an enhanced hole injection efficiency for Pnp heterojunction bipolar transistors making use of an intermediate quaternary layer between emitter and base. IV. CONCLUSIONS
Starting with Burt’s envelope function theory1 and Foreman’s boundary conditions,2 we have calculated the transmission coefficients of holes across a InP/In1⫺x Gax Asy P1⫺y /In0.53Ga0.47As heterointerface while varying the width and gallium and arsenic fractions of the lattice-matched quaternary compound (x⫽0.47y). The valence band profile across that heterostructure is depicted in Fig. 1 as a simple two-step potential. As shown in Ref. 11,
Ekbote, Cahay, and Roenker
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this two-step potential approximates fairly well the valence profile across the emitter-base junction of Pnp heterojunction bipolar transistor with an intermediate layer between the emitter and base for large forward bias approaching the built-in potential of the emitter-base junction. A more rigorous treatment would need to include the effects of spacecharge and band bending in the depletion region at the emitter-base junction but we believe our conclusion will still be valid under large forward bias conditions. While comparing our results to the case of an abrupt InP/In0.53Ga0.47As interface, we find that the transmission coefficients of both heavy and light holes can be enhanced by using a 60-Å-wide intermediate quaternary layer with an arsenic fraction y⫽0.4 (x⫽0.188). 17 This should lead to an enhanced hole injection efficiency of Pnp heterojunction bipolar transistors across the two step heterointerface analyzed here compared to the case of a single abrupt InP/In0.53Ga0.47As heterointerface. We note that Pnp heterojunction bipolar transistors making use of two quaternary layers between an InP emitter and In0.53Ga0.47As base have been reported recently with improved current gain18 compared to devices using single interface emitter-base heterojunction. ACKNOWLEDGMENTS
This work is supported by the National Science Foundation 共ECS-9525942兲. We also acknowledge the Ohio-Cray supercomputing center for the use of their facilities. M. G. Burt, J. Phys.: Condens. Matter 4, 6651 共1992兲. B. A. Foreman, Phys. Rev. B 48, 4964 共1993兲. 3 C. Y. Chao and S. L. Chuang Phys. Rev. B 43, 7027 共1991兲. 4 C. Y. Chao and S. L. Chuang, Phys. Rev. B 46, 4110 共1992兲. 5 A. D. Sanchez and C. R. Proetto, Phys. Rev. B 51, 17199 共1995兲. 6 S. Ekbote, M. Cahay, and K. Roenker 共unpublished兲. 7 S. Ekbote, M. Cahay, and K. Roenker, J. Appl. Phys. 85, 924 共1999兲. 8 D. A. Broido and L. J. Sham, Phys. Rev. B 31, 888 共1985兲. 9 S. L. Chuang, Phys. Rev. B 40, 10379 共1989兲. 10 See for instance A. C. R. Bittencourt, A. M. Cohen, and G. E. Marques, Phys. Rev. B 57, 4525 共1998兲 and references therein. 11 T. Kumar, M. Cahay, and K. Roenker, Phys. Rev. B 56, 4836 共1997兲. 12 S. Ekbote, M. Cahay, and K. Roenker, Phys. Rev. B 58, 16315 共1998兲. 13 C. D. Lee and S. R. Forrest, Appl. Phys. Lett. 57, 469 共1990兲. 14 For the InP/In0.53Ga0.47As interface, we use 1.344 eV and 0.324⫹0.7x ⫹0.4x 2 for the energy gap of InP and In1⫺x Gax As, respectively. For the conduction- and valence-band offsets, we follow Lee and Forrest and use ⌬E c ⫽0.36⌬E g and ⌬E v ⫽0.64⌬E g . 15 P. Dodd and M. S. Lundstrom, Appl. Phys. Lett. 61, 465 共1992兲. 16 S. L. Chuang, Phys. Rev. B 43, 9649 共1991兲. 17 The optimum width of the quaternary layer is expected to be slightly different from the 60 Å found here when the effects of band bending in the valence band energy profile are included in the simulations. 18 L. M. Lunardi, S. Chandrasekhar, and R. A. Hamm, IEEE Electron Device Lett. 14, 19 共1993兲. 1 2
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