APPLIED PHYSICS LETTERS
VOLUME 76, NUMBER 9
28 FEBRUARY 2000
Optical gain and collector current characteristics of resonant-cavity
phototransistors
Chien-chung Lin,a) Wayne Martin, and James S. Harris, Jr.
Solid State Photonics Laboratory, Stanford University, Stanford, California 94305
Fred Sugihwo
Microwave Technology Center, Agilent Technologies, Santa Rosa, California 95403
共Received 8 November 1999; accepted for publication 23 December 1999兲
An analytic model for current gain as a function of optical input power in heterojunction bipolar
phototransistors is developed. The model provides excellent agreement with the dc measurements of
a resonant-cavity-enhanced heterojunction bipolar phototransistor. The model is extended to explain
small-signal results. © 2000 American Institute of Physics. 关S0003-6951共00兲00409-5兴
Heterojunction bipolar phototransistors 共HPT兲 have been
widely investigated over the past 20 years for varying photodetection applications.1 Two-terminal HPTs have demonstrated satisfactory performance in amplification and sensitivity, however, compared to avalanche photodiodes 共APDs兲,
HPTs are not as good in terms of sensitivity.2 Also, the performance of two-terminal HPTs varies according to the input
optical power intensity, which makes it less attractive to optical communication. However, in the fields of the
optoelectronic-integrated circuit 共OEIC兲,3–5 biochip,6
sensing,7 and spectroscopy,8 HPTs still find a wide range of
application. With the incorporation of the resonant cavity,
the functionality of HPTs can be enhanced due to added
wavelength selectivity.9 In order to make HPTs more practical for general detection use, the nonconstant current–gain
characteristic must be fully understood and modeled. In this
letter, we will formulate the optical gain and current characteristics, which can then be adapted for various levels of
illumination. Previously, Chand10 did a similar analysis,
which covers the higher illumination range very well. On the
other hand, the constant gain region observed under low illumination, which is an important operating region for optical receivers, is not addressed. There are numerous experimental results under low illumination, but without
quantitative formulation.11–13 We have developed a model
based on the floating-base configuration, and excellent agreement is obtained with both our devices and other published
measurements.
The current components of a N pn HPT are shown in
Fig. 1. Light is incident from the collector side and is absorbed in the reverse-biased base-collector region. The photogenerated electron–hole pairs will be separated by the external electrical field inside the base–collector region.
Because of the floating base, the external I B component is
zero. We can then write the following equations:
where I c is the collector current, I e is the emitter current, and
I gc is the photogenerated current. I re is the recombination
current in the emitter–base depletion region, and I br is the
recombination current in the neutral base region. I ne and I pe
are the component electron and hole currents in the emitter
and base junction, respectively. For a heterojunction device,
we can write the ratio of I ne to I pe :
冉
* m *pp
D n L p N d m np
I ne
⫽A⫽
* m *pN
I pe
D p L n N a m nN
⫻exp
冉
E gN ⫺E gp
kT
冊
冊
3/2
共 AⰇ1 兲 .
共2兲
Since the base is floating, the collector and emitter currents
must be equal. I br is negligible in modern narrow-base HPTs.
Hence, the generated photocurrent I gc will be equal to (I c
⫹A⫻I re), so that the optical gain of the HPT can be defined
as the following:
I c I c ⫻ 共 A⫹1 兲 I e ⫻ 共 A⫹1 兲
⫽
⫽
I gc I c ⫹A⫻I re
I e ⫹A⫻I re
⫽
共 I ne ⫹I pe ⫹I re兲 ⫻ 共 A⫹1 兲
.
I ne ⫹I pe ⫹I re⫻ 共 A⫹1 兲
共3兲
In the resonant-cavity devices, the generated photocur-
I c ⫽I nc ⫹I gc ,
I e ⫽I ne ⫹I pe ⫹I re ,
共1兲
I gc⫽I pe ⫹I re⫹I br ,
a兲
Electronic mail: cclin@snowmass.stanford.edu
0003-6951/2000/76(9)/1188/3/$17.00
FIG. 1. Current components in the Npn phototransistor.
1188
© 2000 American Institute of Physics
Lin et al.
Appl. Phys. Lett., Vol. 76, No. 9, 28 February 2000
1189
rent I gc can be related to the incident power through the
following 共assuming every absorbed photon generates one
electron–hole pair兲:9
P absorbed
P incident
⫽␩⫽
共 1⫹R 2 e ⫺ ␣ d 兲
1⫺2 冑R 1 R 2 e ⫺ ␣ d cos共 2 ␤ L⫹ ␸ 1 ⫹ ␸ 2 兲 ⫹R 1 R 2 e ⫺2 ␣ d
⫻ 共 1⫺R 1 兲共 1⫺e ⫺ ␣ d 兲 ,
共4兲
where R 1 , R 2 are the reflectivity for top and bottom mirrors
in the devices, ␣ is the absorption coefficient of the quantum
well, d is the thickness of the quantum well, L is the cavity
length, ␸ 1 and ␸ 2 are the phase added by top and bottom
mirrors, respectively, and ␤ is the propagation constant. Using the quantum efficiency of the device ␩ and the incident
power intensity, we can derive the generated current from the
Eq. 共5兲:
I gc⫽Resp⫻ ␩ ⫻ P inicident ,
共5兲
where the Resp is the responsivity at that wavelength.
Considering the thermionic emission and generation–
recombination equations in a standard p – n junction, one can
postulate that I np ⫹I ne ⫽I f ⫽I f 0 (e qV BE /kT ⫺1) and I re
⫽I r0 (e qV BE /nkT ⫺1), where n is the diode ideality factor in
the emitter–base junction. The factor n is nominally between
1 and 2 for homojunctions but can be larger than 2 in a
heterojunction.14 I r0 will be slightly dependent on V BE via
the depletion width modulation, which can be taken into account if a term 冑V bi⫺V BE is included and V bi is the built-in
potential. Using this formula, one can immediately identify
three regions of operation:
共a兲 When the illumination is low, V BE is small, qV BE
ⰆkT, thus expanding the exponential gives:
I f ⬇I f 0 ⫻qV BE /kT, and I re⬇I r0 ⫻qV BE /nkT. Substituting
these two expressions into Eq. 共3兲:
I c 共 I f ⫹I re兲 ⫻ 共 A⫹1 兲
⫽
I gc
I f ⫹I re⫻ 共 A⫹1 兲
冉
冊
FIG. 2. Typical I – V characteristics under different illumination intensities
for the resonant-cavity-enhanced 共RCE兲 HPT.
共c兲 When illumination is high, such that I f ⰇA⫻I re , then
I c 共 A⫹1 兲 I f
⬵
⫽ 共 A⫹1 兲 ⫽h FE⫹1.
I gc
If
共8兲
Usually, HPTs seldom operate in this region. Because I pe
will dominate the current, the ratio of I ne and I pe is actually
the maximum current gain of the device. These three operating regimes have been described qualitatively
elsewhere.1,13 With the above formulation, we can fit the
actual data and extract the diode ideality factor and I f 0 ,I r0 .
A small-signal model can be deduced directly from the
ac components in Eq. 共1兲. If the intensity of the incident light
is modulated with a small variation, the potential between
the emitter–base junction will change correspondingly. If the
small-signal components are denoted as lower-case variables, we can write the following expressions 共the variation
of I r0 can be neglected to first order兲:
qV BE
qV BE
⫹I r0
共 A⫹1 兲
kT
nkT
⬵
qV BE
qV BE
⫹ 共 A⫹1 兲 ⫻I r0
If0
kT
nkT
冉
If0
冊
I r0
共 A⫹1 兲
n
.
⫽
共 A⫹1 兲
I r0
I f 0⫹
n
I f 0⫹
共6兲
Equation 共4兲 will have a constant value, independent of how
much current I c flows through the device in this lowillumination regime.
共b兲 When illumination is in the intermediate regime, I f
⬎I re , but I f ⭐A⫻I re , then
I c 共 A⫹1 兲 I f I f 0
⬵
⬵
⫻ 共 e 共 qV BE /kT 兲共 1⫺1/n 兲 兲 ⬀I 共c1⫺1/n 兲 . 共7兲
I gc
AI re
I r0
This is the general relationship seen in most of the
literature.10
FIG. 3. Experimental results and calculated curves for dc and ac gain dependence on the collector current density of the RCE HPT.
1190
Lin et al.
Appl. Phys. Lett., Vol. 76, No. 9, 28 February 2000
FIG. 4. Theoretical calculation of the gain–current characteristics for past
experimental results.
is likely caused by a larger measurement error for the weak
signal since our laser source is not well power stabilized.
Also, the second reason is because the dark current is at the
same order of the photogenerated current. The values of I f 0
and I r0 determined from this model are 3.0628⫻10⫺9 and
7.812⫻10⫺9 A, respectively. The diode ideality factor n is 2.
The lowest incident optical power the device can detect is 15
nW. Figure 4 shows the fit using our model to data extracted
from other papers.11,13 The excellent agreement in these plots
to devices fabricated by other groups and of different dimensions suggests that our model is a widely applicable way to
describe the current gain as a function of optical power in
HPTs.
In conclusion, we have developed a simple, yet powerful
model for HPT devices. This model can be used to evaluate
the parameters, such as the ideality factors I f 0 and I r0 in
phototransistors. The excellent agreement between our
model and experimental results of several groups demonstrates the accuracy and generality of our theory.
i c ⫽I C,total⫺I C,dc
1
⫽I f 0 共 e q 共 V BE⫹ v be 兲 /kT ⫺e qV BE /kT 兲
⫹I r0 共 e q 共 V BE⫹ v be 兲 /nkT ⫺e qV BE /nkT 兲
⫽I f 0 e qV BE /kT 共 e q v be /kT ⫺1 兲 ⫹I r0 e qV BE /nkT 共 e q v be /nkT ⫺1 兲
⬵ 共 I f ⫹I f 0 兲 ⫻
q ␯ be
q ␯ be
⫹ 共 I re⫹I r0 兲 ⫻
.
kT
nkT
共9兲
Similarly,
i gc⫽ 共 I f ⫹I f 0 兲 ⫻
q v be
q v be
⫹ 共 I re ⫹I r0 兲 ⫻
⫻ 共 A⫹1 兲 ,
kT
nkT
共10兲
therefore,
h f e⫽
i c 关 I f ⫹I f 0 ⫹ 共 I re⫹I r0 兲 /n 兴共 A⫹1 兲
⫽
.
i gc I f ⫹I f 0 ⫹ 关共 I re⫹I r0 兲 /n 兴共 A⫹1 兲
共11兲
Figure 2 shows typical I – V characteristics of our
resonant-cavity HPT devices with various incident optical
powers. The resonant-cavity HPT was originally designed
for tunable devices. The detailed structure and fabrication
can be found elsewhere.15 The solid line in Fig. 3 shows the
dc gain and the dashed line gives the ac gain fitted by this
model. The scattering of the data at lower collector currents
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