Modeling of tunnel junctions for current injection in Vertical
Cavity Surface Emitting Lasers (VCSELs)
Grigore Adrian IORDACHESCU, Joël JACQUET1
SUPELEC, Laboratoire Matériaux Optiques Photonique et Systèmes, Unité Mixe de
Recherche - 7132,
2 rue Edouard Belin, 57070 METZ France
Unité Mixte internationale Georgia Institute of Technology – Centre National de la
Recherche Scientifique (CNRS), UMI 2958
ABSTRACT
Tunnel junctions have been widely used in the fabrication of Vertical Cavity Surface Emitting Lasers
since it allows fabrication of low electrical resistance as well as low optical absorption Bragg mirrors. The
basic idea is to inject holes through a highly doped reverse biased n+/p+ tunnel junction. We present in this
paper a review of the different materials that can be used for various wavelength applications ranging from
UV (GaN) to IR (GaSb). We have elaborated a new modelling tool that has been validated for homo-junctions.
The results show that the injection efficiency is directly linked to the energy gap of the material and to the
effective mass of the electrons and light holes. The first important discussion is related to the condition on
doping levels to get the material degenerate. Low band gap materials such as InAs or GaSb semiconductors
are well appropriate to realise tunnel junctions with moderate doping levels. At the opposite large band-gap
materials as GaN or AlN require very high doping levels to reach the tunnelling condition. GaSb based
VCSELs emitting in the infrared region (2-3µm) can use very efficiently such electrical injection scheme. On
the other side, it will be much less beneficial to use it for surface emitting laser emitting in the ultra violet
wavelength range. Comparison with published papers will be discussed as well as preliminary work done in
the case of hetero junctions.
Keywords : Tunnel junction, Esaki Junction, semiconductor material, semiconductor lasers, Vertical Cavity
Surface Emitting Lasers (VCSELs),
1. INTRODUCTION
A better understanding of the tunneling conduction and its mathematical model has become
increasingly important with the advent of VCSELs in the laser industry. This is due to the tunneling junctions’
capacity to enhance the VCSEL’s performances by significantly reducing its series resistance and, as a
consequence, its Joule heating. The specific sheet resistence of the tunnel junction can become as small as
3⋅10-6 Ωcm2 for the case of InP-based semiconductor materials[1]. The exact employment of such a method
will not be discussed here, but it could be found in References[1]. The purpose of this article is to present a
review of two theoretical methods used to describe the tunneling conduction and to show some new
interesting computer simulations which rely on them. One such method for tunneling current estimation is the
Esaki integral, which sadly needs for its solution the value of the tunnel effect probability for all the energies in
its domain of integration. The benefit for choosing it is that it works for both heterojunctions and
homojunctions alike. Due to the ever-increasing importance of heterojunctions in the VCSEL industry, we’ll be
more and more constrained to using this method in future simulations. As a result, we’ll give an account of it
towards the end of this article. On the other hand, the method on which many of the simulations from this
paper were based upon is the one by Demassa and Knott[2]. According to this method, the I-V characteristic of
a tunnel diode is obtained by using a very good algebraic approximation of the experimental results (eq 1), in
which the two unknown parameters of the tunneling current can be computed theoretically.
1
Joel.Jacquet@Supelec.fr
Physics and Simulation of Optoelectronic Devices XVII, edited by Marek Osinski,
Bernd Witzigmann, Fritz Henneberger, Yasuhiko Arakawa, Proc. of SPIE Vol. 7211,
721113 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.808468
Proc. of SPIE Vol. 7211 721113-1
2. SIMULATION TOOL
For better grasping the signification of the two parameters, please refer to figure 1.
(1)
In order to widen the applicability domain of this method we have removed all the calculus
appoximations from the original paper when computing Vp. Another thing one must pay attention to when
using this method is some printing errors found in [2], and later on spread in [3, 4] regarding the use of Kane
method for calculating Jp[5] (eq 2).
J*
Tuiiiiel
''
4
6/
/&Ex
-'ICu
Figure 1 : The J-V characteristic of a tunnel
diode and the representation of Jp and Vp
V
p
(2)
For the signification of physical quantities in (2) please turn to the original paper by Demassa
and Knott[2], where you’ll also notice the erroneous lack of the minus sign in the exponential.
Sadly, the big limitation of this method when comparing to the Esaki integral is that its use is
restricted to homojunctions. Nevertheless, its embodiment as a Matlab application yielded very interesting
qualitative and even exact quantitative results (where the verification allowed) for such materials as
GaSbyAs1-y, AlxGa1-xAs, GaxIn1-xAsyP1-y, Ge, InP, InSb, InN, InAs, GaSb, GaAs, AlN, GaN. We’ll make a run
through all of them in the next section. Besides the universal constants, the specific physical quantities
needed for each of those materials in order to apply this method are presented in Table 1. Their values have
been taken or calculated from data made available by Ioffe Institute[6]. For ternary and quaternary
compounds, the respective physical quantities have been written as functions of relative concentrations x and
y.
Proc. of SPIE Vol. 7211 721113-2
Table 1 : definition of parameters
ε
dielectric permittivity
Eg
band gap
Mn
electron effective mass for density-of-states calculations
Mn,c
electron effective mass for conductivity calculations
Mp
hole effective mass for density-of-states calculations
Mlp
light hole effective mass
The validity of this method has been quantitatively verified in the case of Ge by Demassa and Knott[2]
for all the positive values of the bias applied on the junction. A simpler way to check the results returned by
this method is to do it for only the peak value of the direct tunneling current (Jp). This does not assure us that
our plot is quantitatively correct for all bias voltages, but predicting the correct value for Jp is of most
importance in realizing the strength of the tunneling effect. As an example of such practical confirmation, we’ll
take the case of a GaAs tunnel diode[7] with a high density of deep-states of 1020cm-3. In this experiment a
current density of 16kA/cm2 and a negative-conductance of (226Ωμm2)-1 are obtained[7]. If we look at the
simulation results in figure 2, we’ll notice a peak tunnel current density of 15kA/cm2 and a negativeconductance of (330Ωμm2)-1. The orders of magnitude are similar but in general we would expect the
experimental values be less performant than the theoretical ones because of the damage we inflict to the
crystalline structure when we use such high levels of impurities. We don’t see such a phenomenon here
though, partly because at high doping levels, even a slight deviation from set values may produce dramatic
changes in the measured tunnel effect. So what should be followed next should be the order of magnitude
and not the exact values of the simulation results.
GaAs
GaAs
Na=1 .00e+020 cm3
Nd=1 .00e+020 cm3
Na=1.00e+020 cm3
Nd=1.00e+020 cm3
1
N
E
0
0
500
1000
1500
V(mV)
Proc. of SPIE Vol. 7211 721113-3
-40
-20
V(mV)
-30
-10
0
Figure 2 – Matlab simulation of a GaAs tunnel diode with a high density of deep-states of 1020cm-3
3. DISCUSSION
We have investigated various semiconductors materials commonly used in for laser fabrication. Table 2
summarizes the data we have used for these materials. The last two columns give the limit of the degeneracy
(minimum doping level to get the Fermi level in the valence or conduction band for p and n doping
respectively).
Eg
Mn
Mlh
(eV)
Nd_min
Na_min
cm-3
cm-3
GaSb
0,726
0,04
0,05
3E+17
1E+19
GaN
3,45
0,2
0,26
3 E+18
1E+20
AlN
6,2
0,4
0,24
1E+19
1E+21
InAs
0,354
0,023
0,026
1E+17
1E+19
InP
1,344
0,073
0,089
1E+18
1E+19
InN
2
0,11
0,27
1E+18
1E+20
InSb
0,17
0,013
0,015
1E+17
1E+19
GaAlAs
2
0,1
0,11
1E+18
1E+19
GaAsSb
0,78
0,05
0,066
3E+17
1E+19
Table 2 : Energy gap and effective masses of electrons and light holes for the materials we have used.
3.1 Influence of the Energy gap on Tunnel condition
We can observe that the degeneracy is reached with lower doping levels for n as compared to p material.
The second observation is that the larger the energy bandgap, the higher the doping level needs to be to for
the degeneracy to be reached. Doping levels as high as 1019 cm-3 are necessary for AlN n material
(Eg=6.2eV) to have an efficient tunneling capacity. On the contrary, InSb can be used at a doping
concentration around 2 orders of magnitude lower. For p doping, the concentration limit rises to 1019 cm-3 for
AlN ; this means that this material will absolutely not be a good candidate to be used in optical devices. The
minimum doping concentrations for the other materials are at least 1019 cm-3. All these observations are
illustrated in figure 3.
Proc. of SPIE Vol. 7211 721113-4
1,E+21
Nd or Na (cm-3)
1,E+20
1,E+19
1,E+18
Nd
Na
1,E+17
0
1
2
3
4
5
6
7
Eg (eV)
Figure 3 : Minimum doping level to reach degeneracy for p and n material with various bang gap energies
3.2 Influence of doping level on Tunnel characteristics.
We have then investigated the performance of tunnel homojunctions made with these materials. We
first calculated the values of Vp as a function of p doping concentration (figure 4) for a constant n doping of
1020 cm-3. Vp increases with this doping concentration and is lower for the higher band gap materials. We
have after that taken the case of an InSb junction separately and calculated Vp as a function of p doping
concentration for different n doping levels. The results can be seen in figure 5.
250
200
Vp (mV)
150
100
GaAlAs
GaInAsP
GaN
50
GaSb
InP
0
0,0E+00
InSbSb
5,0E+19
1,0E+20
Na
1,5E+20
2,0E+20
(cm-3)
Figure 4 : Calculated Vp as a function of p doping concentration for tunnel
homojunctions made with the materials investigated in this work.
Proc. of SPIE Vol. 7211 721113-5
300
250
Vp(mV)
200
150
100
Nd=1e+18
Nd=5e+18
Nd=1e+19
Nd=5e+19
Nd=1e+20
50
0
1E+19
1E+20
1E+21
Na(cm-3)
Figure 5 : Calculated Vp as a function of p doping concentration
for InSb tunnel homojunction at different n doping concentrations.
3.3 Influence of doping level on Jp
Very often in the literature one often encounters a physical measure which comprises both the effects
of n and p doping. This quantity is named the reduced concentration of impurities and it is written as n*. It is
defined as the product of donor and acceptor impurity concentrations divided by their sum. In figure 6 we
have reported exactly the influence of this measure on the maximum current density Jp, considering all
impurities active (n=Nd and p=Na). Because this type of graphic is usually hard to follow, we have also
calculated Jp as a function of p doping level (figure 7), the n doping level beeing kept at a constant value of
1020 cm-3, for the same materials as in figure 4.
Proc. of SPIE Vol. 7211 721113-6
Jp (A/cm²)
1E+06
1E+05
1E+04
1E+03
1E+02
1E+01
1E+00
1E-01
1E-02
1E-03
1E-04
1E-05
1E-06
1E-07
1E-08
1E-09
1E-10
1,0E-10
AlGaAs
GaInAsP
GaN
GaSb
InP
InSb
1,5E-10
2,0E-10
2,5E-10
3,0E-10
n*-1/2 (cm-3/2)
Figure 6 : Calculated Jp as a function of n*
Nd=1e+20cm-3
1E+06
1E+04
Jp(A/cm2)
1E+02
1E+00
1E-02
AlGaAs
GaInAsP
GaN
GaSb
InP
InSb
1E-04
1E-06
1E-08
0E+00
1E+20
Na(cm-3)
2E+20
Figure 7: Calculated Jp as a function of p doping concentration for tunnel
homojunctions made with the materials investigated in this work
Proc. of SPIE Vol. 7211 721113-7
3.4 Influence of Eg on Jp and Vp
We have plotted Vp and Jp for Ga(1-x) AlxAs n+p+ homojunction as a function of the relative
concentration x. The impurities concentrations of the junction in this example are Na=Nd=5e+19cm-3. When x
varies from 0 to 1, the energy gap covers the 1.4 to 3 eV range – figure 8. We observe that Vp is not strongly
affected over the whole range. Opposing it, Jp decreases from 500 mV for GaAs material down to nearly 0 for
AlAs composition (the exact value is in fact 0,25A/cm2, but because we are using a linear scale, we can’t
differentiate it from 0).
600
Jp
500
Vp
Jp (A/cm²) or Vp (mV)
400
300
200
100
0
1,4
1,6
1,8
2
2,2
2,4
2,6
2,8
3
Eg (eV)
Figure 8 : Calculated Vp and Jp as a function of the relative concentration x
for a homojunction of Ga1-xAlxAs doped with n=p=5e+19cm-3
3.5 Injection current in reverse biased junctions
Finally, we arrived to the most important feature of this work: the tunnel current that can be injected in
reverse biased tunnel junctions realised out of the different materials investigated. In this section, we calculate
the I(V) characteristics. An example of such calculation is shown in figure 9. The result is as expected for a
tunnel junction. When watching the next three figures, we have to be aware of the fact that we are neglecting
the resistive effects given by the lengths of the two sides of the junction or by the resistive contacts we are
Proc. of SPIE Vol. 7211 721113-8
using to mount the junction into a circuit. Those effects tend to become increasingly powerful as we increase
the injected current into the junction.
Ga(O.47)In(0.53)As(0.46)P(0.54)
Na=5.00e+019 cm3
Nd=5.00e+019 cm3
-0.5
-00
0
200
400
V(mV)
600
800
Figure 9 : J(V) characteristic for forward and reverse biased InGaAsP tunnel homojunction.
The region of interest is on the reverse bias side. We record the voltage value which is necessary to
reach 10kA/cm² in the junction, V10. The best materials are the ones with the lowest V10 values; in that case
the electrical consumption will be the lowest and the thermal heating reduced as well. The V10 values are
reported in table 3 for six different homojunctions with doping level of 5⋅1019 cm-3 for both p and n type
materials. V10 varies from 2 mV for InSb to 1.34 Volts for GaN based materials respectively. A clear
correlation between V10 and the Energy gap of the material is observed. This is illustrated in figure 10.
V @10kA/cm²
(mV)
Eg (eV)
GaN
InP
InSb
GaAlAs
GaAsSb
GaInAsP
1344
3,45
218,9
1,344
2
0,17
631,8
2
19,44
0,78
76,17
0,9
Table 3 : Necessary voltage values to reach 10 kA/cm² in the reverse biased
19
-3
tunnel homojunction of different materials doped with n=p=5⋅10 cm .
In figure 11, we have plotted the values of the current injected in reverse biased homojunctions
polarized at -1V for different materials with varying composition. Considering the high value of the calculated
current density, it’s obvious that we are dealing with an ideal junction and that in a real experiment the
thickness of the junction induces resistive effects that should be accounted for.
Proc. of SPIE Vol. 7211 721113-9
1400
'4
1200
V (mV) @ 10kA/cm²
1000
800
600
400
200
0
0
1
2
3
4
Eg (eV)
Figure 10 : Graphic showing the correlation between the necessary reverse voltage needed to reach
a current density of 10kA/cm2 and the bangap of the base material on which the homojunction is realised.
1E+09
1E+08
J (A/cm²) @ -1 Volt
1E+07
1E+06
Al(x)Ga(1-x)As
1E+05
Ga(0.47)In(0.53)As(x)P(1-x)
GaAs(1-x)Sb(x)
1E+04
0
0,2
0,4
0,6
0,8
1
X or 1-X
Figure 11 : Value of the current injected in reverse biased homojunctions polarized at -1V
for different materials with varying composition.
Proc. of SPIE Vol. 7211 721113-10
4. PROSPECTS
As we have seen in the previous section, a large number of possible simulations may be realised
when we have a theoretical model that can be easily numerically implemented. The increasing role the
heterojunctions are playing in this industry compel us to realize in their case the same kind of work we have
done for the homojunctions. In this respect, we need a broader theory that can account for the effective mass
and bandgap discontinuities between the two sides of those junctions. As we have already stated in the
Introduction, the starting point of this endeavor is the original Esaki integral. We will use an improved form of
it[8]:
(3)
where we prefer using β=(kT)-1 instead of the temperature for avoiding any confusion between the usual
temperature notation and T(E), which represents the transmission probability (the probability for the electric carrier of
energy E to surmount the potential barrier). This probability doesn’t depend on only the energy of the carrier, but also on
the geometry of the potential barrier. Thus, in order to find T(E) we’ll need the solution of the Schrödinger equation (eq
4) in which V(x) coincides with the potential energy profile across the junction.
(4)
The method for solving eq. 4 by expanding it in a series of algebraic equations is to be found in [8]. At this
stage the problem of not having a coherent theory for calculating the discontinuities in the energy levels of conduction
and valence bands forces us to introduce a junction specific parameter (ΔEc – the discontinuity of the conduction band at
the interface between the two materials). Thus, for all types of heterojunctions for which this parameter is measured,
we’ll be soon able to obtain the same kind of graphics as for the homojunctions in the previous section.
5. ACKNOWLEDGMENTS
This work has been realized in the frame of the ANR MIREV project. The authors would like to thank the Conseil
Regional de la Lorraine (France) for financial support of this project.
6. REFERENCES:
[1] Markus ORTSIEFER - Fabricating buried tunnel junctions for InP-based VCSELs. Solid State Technology, February
2006
[2] Thomass DEMASSA and David KNOTT – The Prediction of Tunnel Diode Voltage-Current Characteristics. Vol. 13
pp 131-138 Pergamon Press 1970
[3] Manish MEHTA, Danny FEEZELL, David BUELL, Andrew JACKSON, Larry COLDREN and John BOWERS –
Electrical Design Optimization of Single-Mode Tunnel-Junction-Based Long-Wavelength VCSELs. IEEE Journal of
Quantum Electronics Vol. 42, No. 7, July 2006
[4] Manish MEHTA – High-Power, High-Bandwidth, High-Temperature Long-Wavelength Vertical-Cavity SurfaceEmitting Lasers. PhD Dissertation – University of California Santa Barbara, June 2006
[5] Evan O. KANE – Zener Tunneling in Semiconductors. Vol 12 pp 181-188 Pergamon Press 1959
[6] Institut IOFFE. New Semiconductor Materials. Characteristics and Properties. 1998-2001
[7] Janet L. PAN, J.E. McMANIS, L. GROBER and J.M. WOODALL – Gallium-arsenide deep-level tunnel diode with
record negative conductance and record peak current density. Solid-State Electronics, Volume 48, Issues 10-11,
October-November 2004
Proc. of SPIE Vol. 7211 721113-11
[8] William R. FRENSLEY. Heterostructure and Quantum Well Physics. http://www.utdallas.edu/~frensley/technical/
1995
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