Например, Бобцов

Theoretical investigation and analysis of time response in heterostructure Geiger-APD Теоретическое исследование и анализ временного отклика в гетероструктуре APD-Гейгер

УДК 621.383

© 2012 г. Mehdi Dehghan
Department of Electrical Engineering, Firoozabad Branch, Meymand Center, Islamic Azad University, Meymand, Iran
E-mail: m_dehghan592@yahoo.com

In this paper the mean current impulse response and standard deviation in Geiger mode for heterostructure APD are determined. The model is based on recurrence equations. These equations are solved numerically to calculate the mean current impulse response and standard deviation as a function of time. In this structure we illustrate the multiplication region with different ionization threshold energies that the impact ionization of the injected carrier type is localized and the feedback carrier type is suppressed. In fact for this structure, better control of spatial distribution of impact ionization for both injected and feedback carriers can be achieved. By enhancing the control of impact-ionization position, the structure achieved to high gain and very low noise.
Keywords: Avalanche Photodiode, Geiger mode, I2E structure, Dark Count.

Сodes OCIS: 040.1345.

Received 29.03.2012.

1. Introduction
Avalanche photodiodes (APDs) are famously known as detectors in long-haul fiber optic systems and Geiger mode applications due to their advantage of high internal gain generated by avalanche multiplication [1]. According to the local-field avalanche theory, both the multiplication noise and the gain-bandwidth product of APDs are determined by the ratio of the electron and hole ionization coefficients of the semiconductor in the multiplication region. Since this ratio is a material property, for a given electric field, efforts to improve the APD performance have focused on optimizing the electric field profile and characterizing new materials. A great deal of research has been devoted to reducing the multiplication noise of APDs by suppressing the feedback process and making the impact-ionization process itself more deterministic [2].
Recently, lower multiplication noise and higher gain-bandwidth products have been achieved by submicrometer scaling of the

thickness of the multiplication region. The heterostructure APD (HAPD) was first proposed by Chin et al [3] to minimize the excess noise factor. An electron or hole may gain extra energy when cross over a band-edge discontinuity which would enhance the impact ionization. Proper selection of the heterojunction material layer will determine the degree of enhancement in ionization coefficient. However, Chia et al [4] found that no enhancement of ionization coefficients was observed in AlxGa1–xAs/GaAs HAPDs. Their measurements suggested that the excess energy gained by the carriers crossing the heterojunction interface is negligible. Later, it has been proved by Kwon et al [5] that low noise could be achieved in HAPDs when the finite initial energy of carriers entering the multiplication region.
Above breakdown the APD operates in the Geiger mode, so that a single absorbed photon can generate a measurable current APDs used as single photon detectors are referred to as single photon avalanche diodes (SPADs). Groves et al [6] showed how to calculate the mean current

62 “Оптический журнал”, 79, 12, 2012

Energy [eV]

Undoped multiplication region with AlxCa1–xAs of different Al%
Al% = 60%
30% 15%
P 0%

N Ec


40 70 80


Distance from P–I interface [nm]


Fig. 1. Band structure of multiplication region in the I2E structure.

impulse response and standard deviation in PIN APD structure for Geiger mode. In this letter we generalize the technique of Groves to show mean current and standard deviation of impact ionization engineered (I2E) structure in Geiger mode. Band structure is shown in Fig. 1. The multiplication region consisted of a 40 nm thick Al0.6Ga0.4As followed by 30 nm Al0.3Ga0.7As, 10 nm Al0.15Ga0.85As and 20 nm GaAs.
The paper is organized as follows. In Section 2, the modified model to calculate the mean current impulse response and standard deviation by solving the recurrence equations is presented. In section 3, we demonstrate the results for I2E structure. Finally we close this paper by conclusion, in section 4.
2. Theory of Model
We consider an APD with a multiplication region of width w. A parent photo-electron is injected into the multiplication region at x = 0

with a fixed velocity ve under the effect of an electric field. After traveling a fixed dead space de, in the x-direction, the electron becomes capable of impact ionizing with an ionization coefficient . Upon ionization, an electron-hole
pair is generated, so that the parent electron is
replaced by two electrons and a hole. The hole travels in the (–x)-direction and becomes capable
of impact ionizing with an impact ionization coefficient  only after traveling a dead space dh. This avalanche of ionization events continues until all carriers exit the multiplication region.
In the case of multiplication with a fixed dead space de, the probability density function (pdf) of carriers vs time  and distance  is given by

he (, ) = ïïìïíïî0 exp éë-( -de )ûù (- /ve )

 

£ >

de de



hh (, ) = ïïíîïïì0exp éë-( -dh )ùû (- /vh )

 

£ >

dh dh



where de and dh are the electron and hole dead spaces, respectively; ve and vh are the velocity of the electrons and holes, respectively;  and 
are the ionization rates for electrons and holes,
respectively, that often modeled by standard
equation [7, 8] –






Ec E




Here A, Ec and m are the parameters taken from [9]. With integration of this distribution
function over the total time, the position
dependent ionization pdf is given as

òhe(h) ()=

¥ 0





The recurrence equation for electron and hole mean current impulse response are given by [10]

Ie (z, t) = Pe (z, t) Ie (z, t) +

ò ( )min(w-z,vet) 2 0

Ie (z + , t - /ve )


Ih (z + , t - /ve )

he ()d,

Ih (z, t) = Ph (z, t) Ih (z, t) +

ò ( )min(w-z,vht) 2 0

Ih (z + , t - /vh )


Ie (z + , t - /vh )

hh ()d,

(5) (6)

where the first terms on the right-hand side of these equations represent the contributions from the injected, primary currents Ie(h)0(z, t).

The probabilities that the injected carriers avoid ionizing before exiting the multiplication region before time t is given by

“Оптический журнал”, 79, 12, 2012


òPe (z, t) =1-




òPh(z, t) =1-





(7) (8)

The initial current from electrons and holes

can be calculated as

Ie0 (z, t) = ìîïíïïïq0ve /w

t t

> £

(w (w

- z) ve - z)/ve



Ih0 (z, t) = îìïïíïïq0vh /w

t t

> £

(w (w


z)/ve z)/ve



The standard deviation of the impulse response
can be determined by developing the recurrent
expressions for the second order statistics of
Ie (z, t), Ih (z, t) using the same technique used for the mean currents. The second moment of the impulse response i2 (z, t) = I2 (z, t) can be computed by

ò òIe2(z, t) = Pe (z, t) Ie20(z, t) +

w-z d

t 0


Ie2(z + , t - )


Ie(z + , t - )


Ih2 (z + , t - ) + 4 Ih (z + , t - ) ´ Ie (z + , t - ) ûúù´he (, )d,

ò òIh2(z, t) = Ph (z, t) Ih20(z, t) +

z d

t 0


Ih2(z + , t - )


Ih(z + , t - )


Ie2 (z + , t - ) + 4 Ih (z + , t - ) ´ Ie (z + , t - ) ùûú´hh (, )d.

(11) (12)

And the standard deviation of I (z, t) can then be obtained using [11]

(z, t) = i2 (z, t)-i2(z, t).

The dark current is defined by [12] as


( )Id




XVBias wm





2wm XVBias +




XVBias Rd


(13) (14)

1 =

2mc /Eg h2




2mc0.5 Eg1.5 qh


X = XX+1,


Rd is the parasitic leakage resistance, m is the effective mass of electron,  is the constant
depending on the shape of the tunneling barrier,
h is the Planck’s constant, and Eg is the energy gap.

3. Results and Discussion
Figure 2 shows the mean current impulse response to pure electron injection for I2E avalanche photodiode which normalized to the injected primary current qve/w, as a function

Dimensionless Mean Current Impulse Response

of normalized time tve/wt, where wt is the total thickness of multiplication region (wt = 100 nm). We assume equal ionization parameters for electrons and holes, take de = dh = 0.5 nm, ve = vh = 105 m/s.
By contrast, in Geiger mode the standard deviation grows with an exponential rate twice that of the mean current impulse response, so that the standard deviation grows with the same exponential growth rate as its mean current impulse response. With having of standard deviation we can found the behavior of excess
1000 0,4 0,8 1,2 1,6 2 tve/wt
Fig. 2. Mean current impulse response for I2E structure.

64 “Оптический журнал”, 79, 12, 2012

Dimensionless Mean Current Impulse Response

108 Impact Ionization Engineering (I2E) APD
107 PIN APD, Wm = 100 nm 106 105 104 103 102 101 1000 0,4 0,8 1,2 1,6 2
tve/wt Fig. 3. Comparison of dimensionless standard deviation in PIN and I2E structure.
10–6 Simulation Experiment

Dark Current (A)





7 8 9 10 Bias Voitage (V)


Fig. 5. Comparison of dark current in PIN-APD and I2E structure at T = 300 K.


Pulse Width

Breakdown Voltage

Over Voltage

Bias Voltage

Hold-off Time

Voltage Pulse Period


Fig. 6. Periodic voltage pulse used in Geiger mode.

Dark Current (A)


78 9 Bias Voitage (V)



Fig. 4. Experimental and calculated dark current as a function of bias voltage at T = = 300 K.

noise in this structure. In Fig. 3 we compare the standard deviation in I2E structure and
Al0.6Ga0.4As homojunction with w = 100 nm, de = dh= 0.5 nm, ve = vh = 105 m/s, k = 1. According to this figure we can found that the
Al0.6Ga0.4As homojunction exhibits the highest standard deviation and ultimately highest excess noise compared to I2E structure.
Experimental and calculated dark current for I2E structure as a function of applied bias
at T = 300 K are shown in Fig. 4 [13]. From
the calculations, we can see that the breakdown
voltage for this structure is 11.5 V. For low
bias, the dark current is dominated by diffusion

current and parasitic leakage current, and for higher bias, the tunneling current plays an important role. As seen from this figure, the calculated results are in agreement with the measured results.
Figure 5 shows comparison of dark current in PIN-APD and I2E structure at T = 300 K. According to this figure, low dark current was achieved in I2E structure.
We use of the 4V periodic voltage pulse as shown in Fig. 6, for bias APD above the breakdown in Geiger mode.
Breakdown voltage for this APD is 11.5 V. Figure 7 shows dark count rate as a function of over voltage for I2E structure at T = 300 K. From this figure we can see that with increase of over voltage, the dark count rate is increased.
A separate absorption and multiplication (SAM) Al0.6Ga0.4As/GaAs avalanche photodiode with I2E structure in the multiplication region was shown in Fig. 8. The low noise characteristics associated with the initial-energy effect can be

“Оптический журнал”, 79, 12, 2012


Log (Dark Count per Second) Dimensionless
Standard Deviation

12,5 Vbr = 11,5 V
12 T = 300 K
9,05,5 1 1,5 2 2,5 3 3,5 Over Voitage (V)
Fig. 7. Dark count as a function of over voltage at T = 300 K for I2E structure.

1014 W1 = 20 nm, W2 = 80 nm
1012 W1 = 30 nm, W2 = 70 nm W1 = 40 nm, W2 = 60 nm
1010 W1 = 50 nm, W2 = 50 nm
1000 0,4 0,8 1,2 1,6 2 tve/wt
Fig. 9. Dependence of the standard deviation in Al0.6Ga0.4As/GaAs heterostructure APDs on the width of the Al0.6Ga0.4As energy-build up layer.

Absorption Multiplication

P+ h+



w1 w2

incurring significant multiplication events, while a second GaAs layer (w2) with lower bandgap energy is used as the primary carrier
multiplication layer.
In the calculations, the standard deviation is
computed for different widths of the Al0.6Ga0.4As layer while the total multiplication layer width
was fixed at 100 nm. The results are shown in
Fig. 9. The figure shows that with increase of energy-buildup layer thickness (w1) the standard deviation and ultimately the excess noise factor
are increased.

Al0,6 Ga0,4 As


4. Conclusions

100 nm
Fig. 8. Separate Absorption and Multiplication APD with I2E structure in the multiplication region.
achieved by utilizing a two-layer multiplication region. A high bandgap Al0.6Ga0.4As material, termed the energy-buildup layer (w1), is used to elevate the energy of injected carriers without

In this paper, we used recurrence equation to investigation of mean current impulse response and standard deviation for I2E structure with four and two layer heterostructure multiplication regions in the Geiger mode. By enhancing the control of the impact ionization position, we achieved to low noise for these structures. We anticipate that this approach can be incorporated into separate absorption and multiplication structures and extended to long-wavelength materials for optical communication applications.


1. Tan C.H., David J.P.R., Plimmer S.A., Rees G.J., Tozer R.C., Grey R. // IEEE Trans. Electron Devices. 2001. V. 48. P. 1310–1317.
66 “Оптический журнал”, 79, 12, 2012

2. Groves C., Chia C.K., Tozer R.C., David J.P.R., Rees G.J. // IEEE J. Quantum Electron. 2005. V. 41. P. 70–75. 3. Chin R., Holonyak N., Stillman G.E., Tang J.Y., Hess K. // Electron. Lett. 1980. V. 16. P. 467–469. 4. Chia C.K., Ng B.K., David J.P.R., Rees G.J., Tozer R.C., Hopkinson M., Airey R.J., Robson P.N. // J. Appl. Phys.
2003. V. 94. P. 2631–2637. 5. Kown O. // IEEE J. Quantum Electron. 2003. V. 39. P. 1287–1296. 6. Groves C., Tan C.H., David J.P.R., Ress G.J., Hayat M.M. // IEEE Trans. Electron Devices. 2005. V. 52.
P. 1527–1534. 7. Moll J.L., Meyer N. Solid State Electron. 1961. V. 3. P. 155–161. 8. Saleh M.A., Hayat M.M., Saleh B.E.A., Teich M.C. IEEE Trans. Electron Devices. 2000. V. 47. P. 625–633. 9. Plimmer S.A., David J.P.R., Grey R., Rees G.J. IEEE Trans. Electron Devices. 2000. V. 47. P. 1089–1097. 10. Tan C.H., Hambleton P.J., David J.P.R., Tozer R.C., Rees G.J., Lightwave J. Technol. 2003. V. 21. P. 155–159. 11. Hayat M.M., Saleh B.E.A., Lightwave J. Technol. 1992. V. 10. P. 1415–1425. 12. Chen W., Liu Sh. IEEE Journal of Quantum Electronics. 1996. V. 32. 2105–2111. 13. Wang S., Sidhu R., Zheng X.G., Li X., Sun X., Holmes A.L., Jr., Campbell J.C. IEEE Photon. Technol. Lett.
2001. V. 13. P. 1341–1348.

“Оптический журнал”, 79, 12, 2012