RAPID COMMUNICATION

Generation and recombination in two-dimensionalbipolar transistors

Behnaz Gharekhanlou • Sina Khorasani

Received: 4 February 2014 / Accepted: 21 March 2014 / Published online: 8 April 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract We study the effects of recombination and

generation process on the operation of bipolar junction

transistor based on two-dimensional materials, and in

particular, graphone. Here, we use Shockley–Read–Hall

model to study these process. First, we investigate the

current–voltage characteristics of a graphone p–n junction

considering generation and recombination process. Then,

we calculate the estimated changes in current gain, cutoff

frequency, and output characteristics of a graphone bipolar

junction transistor designed in a recent study.

1 Introduction

The recent growth in the science and technology of planar

two-dimensional (2D) materials, initiated with the discov-

ery of graphene [1], has brought unprecedented possibili-

ties to the world of nanoelectronics. Since the theoretical

predication [2] and subsequent experimental observation

[3] of the first stable 2D hydrocarbon, numerous applica-

tions are introduced for this 2D material. This material is

obtainable through hydrogenation of graphene, which may

be either double-sided or single-sided, respectively named

as graphane or graphone [2–4].

Besides having many unique properties of graphene, the

desirable energy gap of graphone makes it possible to

chemically introduce n and p regions, thus allowing fab-

rication of p–n rectifying junctions and bipolar junction

transistors (BJTs). Chemical doping of 2D materials is

generally not trivial and it has been so far successfully

demonstrated for few 2D materials, most recently including

MoS2 [5].

As we have shown in our previous publications [6, 7],

the theory of ideal 2D p–n junctions is completely different

from what is known and widely used for 3D bulk junctions.

We have developed an extensive theory of 2D diodes [6]

and BJTs [7]. In 2D junctions, we deal with 2D surface

charges in depletion region, which may be appropriately

modeled with infinitely many thin line charges [6]. Unlike

the bulk model, the electric field in the neutral region of 2D

junction is non-zero and extends well to the outside of the

depletion layer. This property has been illustrated in Fig. 1.

Generation and recombination are the processes that

usually considered in the non-ideal operation of p–n

junctions and bipolar transistors. In particular, since the

most of the gapped 2D materials including graphone,

MoS2, and WSe2 allow direct transitions in the infrared or

red spectrum, novel applications readily follow the

recombination and generation in the 2D depletion layer.

These, for instance, include the spontaneous light emission

in 2D LEDs [8–10] and 2D solar cells [11].

Here, we study these non-ideal processes in the opera-

tion of designed graphone p–n junction and BJTs [6, 7],

using the standard Shockley–Read–Hall model. Needless

to emphasize, the same modification occurs in the theory of

2D junctions based on other 2D gapped materials.

2 Shockley–Read–Hall (SRH) model

Any perturbation in the thermodynamic equilibrium of a

semiconductor activates processes to restore the system

back to the equilibrium. These are the recombination when

pn [ n2i and thermal generation when pn\n2

i . One of the

best models that can be appropriately describe these

B. Gharekhanlou � S. Khorasani (&)

School of Electrical Engineering, Sharif University of

Technology, P. O. Box 11365-9363, Tehran, Iran

e-mail: khorasani@sharif.edu

123

Appl. Phys. A (2014) 115:737–740

DOI 10.1007/s00339-014-8402-7

phenomena is the Shockley–Read–Hall (SRH) statistics

[12, 13], given by the relationships.

U ¼ U0 pn� n2i

� �

U0 ¼rnrpvthNt

rn nþ ni exp Et�Ei

kBT

� �h iþ rp pþ ni exp Ei�Et

kBT

� �h i

ð1Þ

where rnðpÞ is the electron (hole) capture cross section and

Nt is the trap density at the trap energy Et. Normally, U0

attains a maximum value when Et � Ei, so U is reduced to.

U ¼ rnrpvthNt

rn nþ nið Þ þ rp pþ nið Þ pn� n2i

� �ð2Þ

We consider first the generation current under the

reverse bias condition. When carriers are below their

thermal equilibrium density (pn\n2i ), generation of carri-

ers rather than recombination of excess carriers will occur.

The net generation rate can be found by.

U ¼ � rnrpvthNtni

rp 1þ p=nið Þ½ � þ rn 1þ n=nið Þ½ � � �ni

sg

ð3Þ

where the generation carrier lifetime sg is given by.

sg ¼1þ n=nið ÞrpvthNt

þ 1þ p=nið ÞrnvthNt

¼ 1þ n

ni

� �sp þ 1þ p

ni

� �sn

ð4Þ

in which, sn pð Þ is electron (hole) life time. The electric

surface current density due to the generation in the deple-

tion region is hence given by.

Jgen: ¼Z WD

0

Uj jdx � q Uj jWD �qniWD

sg

ð5Þ

where WD is width of the depletion layer.

The total reverse current can be approximated by sum of

the drift and diffusion components in the neutral regions

and generation current in the depletion region as.

Jtotal ¼ JdiffþJdriftþJgr ð6Þ

Under forward bias pn [ n2i

� �, recombination process

dominates over the generation, and therefore the pn product

takes the form.

pn ¼ n2i exp

EFn � EFp

kBT

� �

¼ n2i exp

qV

kBT

� � ð7Þ

Under the assumption that rn ¼ rp ¼ r, substitution of

(7) in (2) yields.

U ¼rvthNtn

2i exp qV

kBT

� �� 1

h i

nþ pþ 2ni

¼rvthNtni exp qV

kBT

� �� 1

h i

exp EFn�Ei

kBT

� �þ exp

Ei�EFp

kBT

� �þ 2

ð8Þ

The maximum value of U in the depletion region, is located

at where Ei is halfway between EFn and EFp. So, we obtain.

U ¼rvthNtni exp qV

kBT

� �� 1

h i

2 exp qVkBT

� �þ 1

h i

� 1

2rvthNtni exp

qV

2kBT

� �ð9Þ

The recombination current is therefore finally deter-

mined as.

Jrecom: ¼Z WD

0

q Uj jdx � qniWD

2sexp

qV

2kBT

� �ð10Þ

As mentioned before, this current will be added to drift

and diffusion current components under forward bias in (6)

using the relationship Jgr ¼ Jrecom: � Jgen:.

3 Generation–recombination process in p–n junctions

In one of our previous works, we have predicted expo-

nential current–voltage characteristics for a graphone p–n

junction [6]. Here, we investigate effects of generation and

recombination process on current–voltage characteristics of

that p–n junction. Such exponential behavior in 2D recti-

fying junctions has been now observed [11].

Fig. 1 Electrostatic model for 2D surface charges

738 B. Gharekhanlou, S. Khorasani

123

We follow our design for a 2D p–n junction with surface

dopant concentrations ND ¼ 1� 1012cm�2 and

NA ¼ 2� 1011cm�2. Using our derived formula [7], the

depletion widths are found to be xp0 ¼ 140:4 nm and

xn0 ¼ 28 nm. It is important to note that the depletion

region width (W ¼ xp0 þ xn0) of a 2D p–n junction is

dependent on the applied voltage with relation [7].

Vbi � Va ¼q NA þ NDð Þ

2pe� xp0 ln

W

xp0

� �þ xn0 ln

W

xn0

� ��

ð11Þ

The ideal 2D surface current density is.

J ¼ Jn driftþdiffð Þ þ Jp driftþdiffð Þ

¼ qln E þ VT

d

dx

� �np

x¼�xp0

þqlp E � VT

d

dx

� �pn

x¼xn0

¼ Jn0 þ Jp0

� �exp

V

VT

� �� 1

�

ð12Þ

Table 1 enumerates the numerical values of the hole and

electron current densities expected for this p–n junction.

What remains now is to simply add the generation and

recombination components using (5, 6, 10, 11). The result of

calculations for this graphone p–n junction is illustrated in Fig. 2.

4 Generation–recombination processes in BJTs

As it was considered for a p–n junction in previous section,

generation and recombination current components can be

similarly added to currents due to both of the reverse and

forward biased junctions of a BJT.

In our previous work, we have designed an n–p–n BJT

based on graphane [7] with surface dopant concentrations

NE ¼ 1� 1012cm�2, NB ¼ 1� 1011cm�2, and NC ¼ 1�1010cm�2. The current gain of this transistor obtained to be

b ¼ 138. Using (5) and (10), one can find the new current

gain of this transistor to be slightly lower, equal to 127. The

cutoff frequency of this transistor as a function of collector

current density is shown in Fig. 3. The maximum value

of this frequency is found to be about 93 GHz at

0.48 mA lm-1. For this transistor without considering

generation and recombination process, the maximum cutoff

0 5 10 15 2010

-5

100

105

1010

1015

V/VT

|J/J

0|

Ideal reversrIdeal foward

ForwardReverse

Fig. 2 Current–voltage characteristics of a graphone p–n junction

10-5

100

20

40

60

80

100

Collector current density JC(A/cm)

Cut

off F

requ

ency

(GH

z)

Fig. 3 Cutoff frequency versus collector current density

Fig. 4 Output characteristics of graphone n–p–n transistor in com-

mon emitter configuration

Table 1 Numerical values of

hole and electron surface

current densities

Jp0ðpA cm�1Þ Jn0ðpA cm�1Þ

0.332 0.310

Generation and recombination 739

123

frequency was 77 GHz at 0.38 mA lm-1, hence the

maximum speed is improved due to non-ideal effects at the

cost of some excessive power consumption. Another

specification that shifts due to non-ideal processes is the

output characteristics shown in Fig. 4. Here, the inverted

current gain is br ¼ 0:02.

5 Conclusions

In this letter, we first studied the generation and recombi-

nation non-ideal effects on the operation of a rectifying 2D

p–n junction. Then, we extended this theory to study the

non-ideal effects in a graphone n–p–n transistor. The cur-

rent gain of the designed transistor under normal and

inverted bias configuration was noticed to decrease to 127

and 0.02, respectively. However, the maximum cutoff

frequency of this transistor was significantly enhanced

(about 20 %) to reach the value of 93 GHz at a collector

current density of 0.48 mA lm-1.

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