Vol. 34, No. 9
Journal of Semiconductors
September 2013
Comparison of band-to-band tunneling models in Si and Si–Ge junctions
Jiao Yipeng(矫亦朋)1 , Wei Kangliang(魏康亮)2 , Wang Taihuan(王泰寰)1 , Du Gang(杜刚)2 ,
and Liu Xiaoyan(刘晓彦)2; Ž
1 Shenzhen
2 Institute
Graduate School, Peking University, Guangdong 518055, China
of Microelectronics, Peking University, Beijing 100871, China
Abstract: We compared several different band-to-band tunneling (BTBT) models with both Sentaurus and the
two-dimensional full-band Monte Carlo simulator in Si homo-junctions and Si–Ge hetero-junctions. It was shown
that in Si homo-junctions, different models could achieve similar results. However, in the Si–Ge hetero-junctions,
there were significant differences among these models at high reverse biases (over 2 V). Compared to the nonlocal
model, the local models in Sentaurus underrated the BTBT rate distinctly, and the Monte Carlo method was shown
to give a better approximation. Additionally, it was found that in the Si region near the interface of the Si–Ge
hetero-junctions, the direct tunneling rates increased largely due to the interaction of the band structures of Si and
Ge.
Key words: hetero-structure; Monte Carlo device simulation; carrier transport; band-to-band tunneling
DOI: 10.1088/1674-4926/34/9/092002
PACC: 7340G; 7340L; 7115Q
1. Introduction
Much work has been done on the mechanism of band-toband tunneling (BTBT), which is the basic principle for tunneling field effect transistors (TFET), especially in indirect semiconductorsŒ1 4 , such as Si and Ge. However, because the L
and valleys of the conduction band have small energy differences (0.14 eV) in Ge, direct tunneling dominates the total tunneling processŒ5; 6 . While in Si, the situation is just the opposite
due to a larger difference, 2.3 eV, between the main valley and
sub-valley. What’s more, the BTBT mechanism becomes more
complex in Si–Ge hetero-structures, which are regarded as one
of the most important methods to optimize TFETŒ7 . Hence, the
accuracy of BTBT models in Si–Ge hetero-junctions is critical
to investigate the performance of these devices. However, few
works so far have discussed this issueŒ8; 9 .
In this paper, we will mainly focus on the comparison
among three widely-used BTBT models when they are employed in Si homo-junction and Si–Ge hetero-junction simulation at reverse biases. At the same time, the performance of
our recently developed two-dimensional (2D) full-band Monte
Carlo (FBMC) simulator with BTBT models is also evaluated
(by comparison with Sentaurus). This FBMC simulator is then
used to study the BTBT mechanism in Si–Ge hetero-junctions
with diverse doping concentrations.
2. Physical models
DD
np
n2i; eff
.n C ni; eff /.p C ni; eff /
:
(2)
The effect of D will be discussed in Section 4.
In the device with a steep p–n junction and high field, exceeding (approximately) 8 105 V/cm, Schenk’s modelŒ12 is
used to compute the phonon-assisted BTBT rate,
2
RBTBT
3=2
exp
6 .FC /
6
6
D AS D" 6
h!
4
exp
kt
.FC / 3=2
C
1
FC
"
!
1
!3
FC
exp
7
" 7
7;
7
h!
5
exp
kt
(3)
with
There are three BTBT tunneling models that can be chosen
in Sentaurus: Hurkx’s model, Schenk’s model and the nonlocal
path model.
The most widely used, Hurkx’s model, givesŒ10; 11 ,
RBTBT D AD"p e
This model is based on a two-band model. RBTBT is the
BTBT rate and A and B are parameters, which will be discussed in detail below. " is the local electric field. For the direct
tunneling process, P D 2, and for the phonon-assisted tunneling process, P D 2.5. D, called the net generation factor, can
be calculated from,
B="
:
(1)
FC˙ D exp BS .Eg ˙ h!/3=2 :
(4)
The upper sign refers to the generation process (np < n2i; eff )
and the lower sign refers to the recombination process (np
> n2i; eff ). h! is the energy of the transverse acoustic phonon,
about 19 meV in Si ( !X) and 8.6 meV in Ge (both !X
† Corresponding author. Email: xyliu@ime.pku.edu.cn
Received 21 February 2013, revised manuscript received 2 April 2013
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© 2013 Chinese Institute of Electronics
J. Semicond. 2013, 34(9)
Jiao Yipeng et al.
Table 1. The values of parameters A, B and P with tunneling directions along [100] in Eq. (1). The “Direct BTBT” and “Indirect BTBT”
columns list the results calculated in Ref. [14], and the “Sentaurus” columns list the default values in SentaurusŒ16 .
Material
Si
Ge
Direct BTBT
A (cm 3 s
Indirect
BTBT
1/
1.35 1020
1.46 1020
3.29 1015
1.67 1015
Sentaurus
Direct
BTBT
4 1014
9.1 1016
101
6.04
Fig. 1. Simulated BTBT current density of an Si linearly graded junction compared with the experimental data.
and !L)Œ13 , and " is the local electric field. As and Bs are
parameters, which are 8.977 1020 cm 1 s 1 V 2 and 2.14667
107 Vcm 1 eV 3=2 , respectivelyŒ12 .
For a more accurate calculation, the nonlocal BTBT models in the Sentaurus TCAD should be switched on in the device simulation. This model allows the simulator to dynamically search for the tunneling path that has a direction which is
opposite to the gradient of the valence band. To compute the
BTBT rate of such a method is very complex and the method
has been discussed in Ref. [14], but in the uniform electric-field
limit and one-dimensional case, these equations can be reduced
to a form the same as Eq. (1)Œ15 .
For both Si and Ge, the values of A; B and P for Eq. (1)
have been listed in Table 1.
For Si, where the indirect transitions are dominant in the
BTBT process, the theoretical values of A and B are 3.29 1015 cm 3 s 1 and 2.38 107 V/cm from Table 1. While from
the experimental dataŒ17 , parameter A varies between 2 1015 and 3 1015 cm 3 s 1 , and B is approximately 2.0 107 V/cm. Therefore, in this paper, the values of parameters A
and B for Si are finally chosen as 3.0 1015 cm 3 s 1 and 2.0
107 V/cm, respectively, reproducing the experimental data
in Ref. [10] very well (Fig. 1).
3. Simulation method
The 2D FBMC simulator has been discussed in detail in
Refs. [18, 19]. The band structures, including four conductions bands and three valence bands, were calculated from
the local empirical pseudo-potential method. Scattering mechanisms, including acoustic and optical phonon scattering, and
impact ionization scattering for both Ge and Si were consid-
B (MV/cm)
Indirect
Sentaurus
BTBT
23.8
6.55
19
4.9
Direct
BTBT
P
Indirect
BTBT
Sentaurus
2
2
2.5
2.5
2.5
2.5
Fig. 2. Simulation results for the Si step homo-junction by Sentaurus
and the MC simulators at forward bias. The valence and conduction
band at 0.5 V is shown in the inset.
ered.
The non-self-consistent mode was used in our Monte
Carlo (MC) simulator as follows. Firstly, we obtained the electrostatic potential for the whole device using the drift–diffusion
method, and then this potential was kept frozen during the
simulation. For each step, the program calculated the BTBT
generation or recombination rate over the whole device using
Eq. (1) or Eq. (3), and the generated particles with charge Pc ,
by
Pc D RBTBT St;
(5)
where RBTBT is the rate of BTBT generation and recombination, and S is the area of the meshŒ9 .
As a comparison with the MC method, in Sentaurus
we used the Fermi-Dirac statistics model (Fermi), the driftdiffusion carrier transport model, the doping–dependent mobility model (DopingDep), and the high-field velocity saturation
model (HighField saturation). The doping-dependent bandgap-narrowing model (BandGap narrowing) was also switched
on.
4. Results and discussions
In order to verify the validity of the MC simulator, the results of an Si diode at a series of forward biases (0.15–1 V)
were obtained (Fig. 2). At a forward bias, BTBT can hardly
occur since the conduction band edge is located above the valence band edge (see the inset of Fig. 2). It is found that the
forward currents computed by the two simulators agree well
with each other when the reverse bias is above 0.5 V. That is to
say, there is no difference between the results calculated from
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J. Semicond. 2013, 34(9)
Jiao Yipeng et al.
Fig. 3. Simulated BTBT current density of an Si homo-junction with
three BTBT models in Sentaurus and the MC simulator. The solid line
plots the results in Ref. [10].
them if BTBT does not occur. These results prove the reliability of our MC simulator, as well as the reasonability of the
comparisons in this paper.
Figure 3 plots the BTBT currents of an Si homo-junction
at various reverse biases, from 1 to 4 V. It is worth noting that
the current density obtained from Schenk’s model is one order of magnitude less than that from Hurkx’s model in both
Sentaurus and the MC simulator in the range 1.5–4 V (reverse
bias). This can also be verified from the difference of RBTBT
between the two models (Fig. 4). In addition, although the dynamic nonlocal path BTBT model demonstrates a wider spatial range of generation and recombination rate in the device,
the currents are almost the same as the values calculated from
Hurkx’s model in an Si homo-junction.
On the other hand, at very low reverse biases (0–1.5 V),
the current obtained from the MC simulator is much smaller
than that from Sentaurus. In Fig. 5, it is found that at the peak
of the BTBT rates, D drops to very low values in the MC simulator, while it is still maintained at about 1 (which means that
there is no attenuation for Rbtbt / at the same position in Sentaurus. Because D has a close relationship with the carrier density
(Eq. (2)), it is obvious that the carrier density in the MC method
does not reach the expected value at a low reverse bias due to
this method’s drawback in addressing a retarding field, resulting in a relatively low BTBT current.
However, in the Si–Ge hetero-structure, the situation is
very different. The I –V characteristics of two step diodes (Si
and Si–Ge) are shown in Fig. 6, and they have the same doping concentrations. With both models (Hurkx and Schenk), the
currents in the Si–Ge hetero-junctions are larger than the ones
in the Si homo-junctions. Figure 7 gives a detailed comparison of the currents calculated with different BTBT models in
an Si–Ge diode. It should be noted that in order to have a full
understanding of these models, we also introduced the direct
and indirect BTBT parameters listed above to our MC simulator, but they are still treated as local models. Obviously, in a
local BTBT model of Sentaurus (Hurkx’s model), the current
increases rapidly at a low voltage and then tends to be saturated. The nonlocal model gives a larger current value than the
local one at the same bias, especially when the voltage comes
Fig. 4. Simulated BTBT rate of an Si homo-junction with two models
at different biases.
Fig. 5. The D factor (right vertical axis) and BTBT (left vertical axis)
rates calculated by Hurkx’s model at 1 V (reverse bias). The curves
with square symbols are the results from Sentaurus, and the curves
with triangular symbols are from the MC simulator. The inset shows
the electrostatic fields from the two simulators.
to more than 1 V (reverse bias), in agreement with the BTBT
rates in Fig. 8. However, the results of the MC method are more
sophisticated. At low reverse voltages, from 1 to 2 V, the current calculated from the MC method is very close to that from
the local Hurkx’s model, and this is because the model applied
in our MC simulator is also a local one. However, when the
reverse bias exceeds 2 V, the current increases rapidly in ex-
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J. Semicond. 2013, 34(9)
Jiao Yipeng et al.
Fig. 6. Comparison of the current density of Si and Si–Ge diodes with
two different BTBT models.
Fig. 8. The BTBT rates of the Si–Ge hetero-junction calculated under
reverse biases of 1 V and 4 V.
Fig. 7. Simulation results using different methods and BTBT models.
ponential form. It is noteworthy in Fig. 8(a) that from 0.10 m
to 0.13 m, the BTBT rates using the MC method are larger
than the corresponding values by Sentaurus at a reverse bias of
4 V. Hence, this implies that at a high reverse voltage (more
than 3 V from Fig. 7), the local model in Sentaurus underestimates the spatial range and values of BTBT in the Si–Ge
hetero-junctions, and the MC method can give current values
very close to the ones with the nonlocal models, although this is
very time-consuming. Additionally, for Sentaurus, in a heterojunction it is sometimes very difficult to obtain convergence
results if the nonlocal models are switched on. Hence, in order
to get a suitable result, one must weigh the pros and cons of the
different methods.
For Ge devices, it has been reported that the direct BTBT
process dominates over the indirect BTBT processŒ6; 14 . Similarly, in the Si–Ge hetero-junction used in Fig. 7, the direct
tunneling current is about four orders of magnitude larger than
the indirect one at the same bias, which is attributed to the band
structures of Ge. Figure 9 is a comparison of RBTBT in three
different Si–Ge diodes (A, B and C) using both Hurkx’s and
Schenk’s models by the MC method at a reverse bias of 3 V,
where this method is regarded to make a more accurate prediction for the BTBT current than the local models in Sentaurus
and have a better convergence than the nonlocal models, ac-
Fig. 9. Simulated BTBT rates of three different hetero-structure
diodes. The donor concentration of the n-region for each diode is
1020 cm 3 . The inset shows the I –V curves from 1 to 4 V (reverse
bias).
cording to the discussions above. The parameters in our MC
simulator are the same as the default values in Sentaurus, and
the doping of n-regions for all three devices is 1020 cm 3 . The
BTBT rate calculated from Hurkx’s model can be regarded as
the total BTBT rateŒ14 . From diodes A and C, we find that the
doping concentration has little effect on the BTBT rate differences of the two models, which seems to be an intrinsic property of the materials. This can also be verified by the compar-
092002-4
J. Semicond. 2013, 34(9)
Jiao Yipeng et al.
isons between diodes A and B, which have the same doping
profiles but different material composition. In diode A, where
the depletion region mainly lies in the Ge region, the indirect
BTBT rate has a smaller percentage than the direct one, while
in diode B, the situation is just the opposite—indirect transition takes a larger proportion. The BTBT rate curves of diode
B also demonstrate that direct tunneling increases significantly
at the Si region near the interface of the hetero-junction. This
phenomenon is due to the electron in the valance band edge of
Si being able to tunnel into the valley in the conduction band
of Ge near the interface, which is obviously direct tunneling.
[4]
[5]
[6]
[7]
5. Summary
In this paper, we compared three different BTBT models
with both a Sentaurus TCAD and 2D FBMC hetero-junction
simulator in diverse Si homo-junctions and Si–Ge heterojunctions. It was shown that in Si homo-junctions, these models achieved the same results in predicting the BTBT rates
and currents at reverse biases. However, for the Si–Ge heterojunctions, there were some contradictions in these models, especially at high reverse biases, and an underestimation of the
tunneling rates and spatial range by local models could be seen
from our comparisons.
It was also found that our improved MC simulator with
BTBT models has good computation accuracy compared
with Sentaurus, as well as good convergence for the heterojunctions. In addition, we utilized our MC simulator to invest
the BTBT phenomenon in three different abrupt Si–Ge heterojunctions, and interactions of the direct and indirect transitions
near the interface of the Si and Ge regions were observed for
the first time.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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