Design of an electrically pumped SiGeSn/GeSn/SiGeSn double-heterostructure

midinfrared laser
G. Sun, R. A. Soref, and H. H. Cheng

Citation: Journal of Applied Physics 108, 033107 (2010); doi: 10.1063/1.3467766

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 JOURNAL OF APPLIED PHYSICS 108, 033107 共2010兲

Design of an electrically pumped SiGeSn/GeSn/SiGeSn double-

heterostructure midinfrared laser
G. Sun,1 R. A. Soref,2 and H. H. Cheng3,a兲
1
Department of Physics, University of Massachusetts Boston, Massachusetts 02125, USA
2
Sensors Directorate, Air Force Research Laboratory, Hanscom Air Force Base, Massachusetts 01731, USA
3
Center for Condensed Matter Science, National Taiwan University, Taipei 106, Taiwan
共Received 11 April 2010; accepted 28 June 2010; published online 5 August 2010兲
This paper presents the conception, modeling, and simulation of a silicon-based group-IV
semiconductor injection laser diode in which the GeSn-alloy active region has a direct band gap
wavelength in the 1.8 to 3.0 ␮m midwave infrared for 6%–12% ␣-Sn. The strain-free monolithic
P-type semiconductor/Intrinsic semiconductor/N-type semiconductor 共PIN兲 bulk heterostructure,
grown lattice matched upon a relaxed GeSn-buffer on silicon-on-insulator, is believed to be
manufacturable in a complementary metal-oxide semiconductor fab. Detailed modeling is given for
the type-I band offsets, carrier lifetimes, infrared gain profile and laser threshold current density Jth
in a Fabry–Perot cavity having 20– 100 cm−1 loss. The laser’s temperature of operation is
determined by a combination of the radiative lifetime and the nonradiative lifetime due to unwanted
Auger electron-hole recombination. If we keep Jth below 10 kA/ cm2, then we find that this laser
requires cooling in the 100–200 K range, whereas Jth at 300 K appears to be too high for a practical
device. However, the GeSn quantum-well laser diode does offer a pathway to room-temperature
operation. © 2010 American Institute of Physics. 关doi:10.1063/1.3467766兴

I. INTRODUCTION pler because the exact layer thickness of the DHS active
region is not as critical as it is in a multiple QW device and
Silicon-germanium-tin technology is an excellent new should therefore be easier to implement. Our simulation in-
group-IV technology for the monolithic integration of active cludes the effects of nonradiative Auger recombination be-
and passive waveguided components on silicon or silicon- tween electrons and holes because that physical mechanism
on-insulator 共SOI兲.1,2 Within group-IV photonics, the is the key determinant of the diode’s temperature of opera-
group-IV electrically injected laser diode is a key component tion. Our result indicates that this DHS laser would operate
that is missing thus far, and that laser is the focus of this below room temperature. Nevertheless, with its simplicity
article. GeSn is probably the best-known Sn-containing alloy for device growth, it could very well lead to the first electri-
and for this material the infrared region is most natural for cally injected group-IV laser that can be monolithically inte-
radiative band-to-band emission because the direct band gap grated on Si or SOI. In addition, once the material growth
of this crystalline alloy is narrower than that of elemental Ge, becomes mature, such a GeSn/SiGeSn lattice-matched struc-
the semiconductor in which optically pumped Ge-on-Si las- ture can be extended to room-temperature QW structures that
ing at 1.6 ␮m has already been observed.3 This article pre- require lower carrier densities for adequate optical gain be-
sents design-and-simulation results on an electrically in- cause of the reduced density of states and the simultaneous
jected P-type semiconductor/Intrinsic semiconductor/N-type Auger suppression.
semiconductor 共PIN兲 double heterostructure 共DHS兲 diode in
which a layer of GeSn is the active region and a SiGeSn
ternary layer forms each of the two cladding regions. The II. BAND STRUCTURE OF GESN/GESISN DHS
compositions of GeSn and SiGeSn are chosen in such a way Energy-band theory,8 plus a few FTIR absorption
that not only they form type-I band alignment but also are experiments,9 indicate that the band gap of unstrained crys-
lattice matched. The laser would be situated on a relaxed talline GeSn changes from indirect to direct as the percent of
buffer layer of GeSn-upon-silicon or SOI 共Ref. 4兲 whose ␣-Sn is increased. Since band offsets between ternary Sn-
lattice parameter is the same as that of the GeSn/SiGeSn containing alloys and Si or Ge are not known experimentally,
laser, hence the entire laser structure is unstrained. The re- we follow the assumptions made in Ref. 1 and calculate the
sults reported here compliment the previous simulations of a conduction band minima for the lattice-matched heterostruc-
1.55 ␮m GeSn-quantum-well 共QW兲 laser,5 a strained GeSn/ ture consisting of Ge1−zSnz and a ternary Ge1−x−ySixSny based
GeSiSn QW laser,6 and a terahertz Ge/SiGeSn quantum cas- on Jaros’ band offset theory,10 which is in good agreement
cade laser.7 with experiment for many heterojunction systems. For ex-
The band-to-band laser diode simulated in this paper has ample, this theory predicts an average valence band offset,
an emission wavelength in the 1.7 to 1.8 ␮m midwave in- ⌬E␯,a␯ = 0.48 eV for a Ge/Si heterointerface 共higher energy
frared. Compared to the prior art5–7 this device is much sim- on the Ge side兲, close to the accepted value of ⌬E␯,a␯
= 0.5 eV. The basic ingredients of our band alignment cal-
a兲
Electronic mail: hhcheng@ntu.edu.tw. culation are the average 共between heavy, light, and split-off

0021-8979/2010/108共3兲/033107/6/$30.00 108, 033107-1 © 2010 American Institute of Physics

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TABLE I. Band parameters at various valleys used in the band alignment calculation 共Refs. 9–11兲.

EGe ESi ESn bGeSi bGeSn bSiSn

Valley 共eV兲 共eV兲 共eV兲 共eV兲 共eV兲 共eV兲

L 0.66 2.0 0.14 0.0 ⫺0.11 0.0

⌫ 0.795 4.06 ⫺0.413 0.21 1.94 13.2

hole bands兲 valence band offset between the two materials EX共Ge1−xSix兲 = 0.931 + 0.018x + 0.206x2 共5兲
and the compositional dependence of the band structure of
the ternary alloy. For the Ge/ ␣-Sn interface Jaros’ theory 共in electron volt兲 for Ge1−xSix alloys.14 On the other hand, the
predicts ⌬E␯,a␯ = 0.69 eV 共higher energy on the ␣-Sn side兲. empirical pseudopotential calculations of Chelikovsky and
Thus, relative to the average valence band of Ge, the average Cohen place this minimum at 0.90 eV in ␣-Sn, virtually the
valence band position for Ge1−x−ySixSny is simply a linear same as its value in pure Ge.15 We thus assume, as in Ref. 1,
interpolation that the position of this minimum in ternary Ge1−x−ySixSny
alloys is independent of the Sn concentration y, and thus is
⌬E␯,a␯共Ge1−x−ySixSny兲 = − 0.48x + 0.69y. 共1兲 also given by Eq. 共5兲. Obviously, the calculation of band
structures outlined above is an approximation that is subject
Similarly, with these spin-orbit splitting values ⌬so共Ge兲
to experimental corrections as more measurements become
= 0.295 eV, ⌬so共Si兲 = 0.043 eV, ⌬so共Sn兲 = 0.800 eV,11 the
available. This implies that the compositions of Ge1−zSnz and
spin-orbit splitting for Ge1−x−ySixSny is
Ge1−x−ySixSny are necessarily adjusted in the DHS to arrive
⌬so共Ge1−x−ySixSny兲 = 0.295共1 − x − y兲 + 0.043x + 0.800y. at the band structure that is being proposed here. But it
should be pointed out that the laser behavior depends only on
共2兲
the band structure, and the results obtained in this design
The top of the valence band for Ge1−x−ySixSny can then be should be valid albeit at slightly different binary and ternary
determined as compositions.
The ␣-Sn composition dependence of the conduction
E␯共Ge1−x−ySixSny兲 = E␯,a␯共Ge1−x−ySixSny兲 band gaps for Ge1−zSnz at the three valleys L, ⌫, and X is first
⌬so共Ge1−x−ySixSny兲 calculated using Eqs. 共4兲 and 共5兲 to establish the crossing
+ . 共3兲 point where the ⌫-point band gap drops below that of the
3
L-point. Figure 1共a兲 shows that for ␣-Sn composition greater
The minima of the conduction band at points L and ⌫ can than z ⱖ 6%, Ge1−zSnz becomes direct band gap which can
then be calculated by evaluating the compositional depen- serve as the active layer in the DHS. Fixing at this compo-
dence of the band gaps of the ternary alloy as sition Ge0.94Sn0.06, we then looked for a lattice matched
Ge1−x−ySixSny that can be used as cladding layers for
E共Ge1−x−ySixSny兲 = EGe共1 − x − y兲 + ESix + ESny Ge0.94Sn0.06. The requirement for Ge1−x−ySixSny being the
− bGeSi共1 − x − y兲x − bGeSn共1 − x cladding is that it must form type-I band alignment with
Ge0.94Sn0.06. Such a simultaneous requirement can be satis-
− y兲y − bSiSnxy, 共4兲 fied by the additional degree of freedom in the Ge1−x−ySixSny
where both x and y can be tuned. Using the Vagard’s law for
where EGe, ESi, and ESn are the band gap of Ge, Si, and ␣-Sn,
the lattice constant of Ge1−x−ySixSny
respectively, at those points, and the bowing parameters
bGeSi, bGeSn, bSiSn have been discussed in Refs. 12 and 13. a共Ge1−x−ySixSny兲 = aGe共1 − x − y兲 + aSix + aSny, 共6兲
These values at L and ⌫ points have been given in Table I.
Finally, for the indirect conduction band minimum near where the lattice constants aGe = 5.64613 Å, aSi
the X-point, Weber and Alonso find = 5.43095 Å, and aSn = 6.48920 Å for Ge, Si, and ␣-Sn,11

FIG. 1. 共Color online兲 共a兲 Band gaps

of Ge1−zSnz at L, ⌫, and X vs Sn com-
position z, and 共b兲 ⌫-point band align-
ment between lattice matched
Ge1−x−ySixSny and Ge0.94Sn0.06 vs the
Sn composition y.

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 033107-3 Sun, Soref, and Cheng J. Appl. Phys. 108, 033107 共2010兲

follows. For a given carrier concentration, we can derive

quasi Fermi levels at a specific temperature 共T兲 for electrons
in the conduction band 共E fc兲, and for light and heavy holes in
the valence band 共E f ␯兲. The carrier lifetime in this
Ge0.94Sn0.06 region with a direct band gap of
Eg,⌫共Ge0.94Sn0.06兲 = 0.619 eV should be determined by the
radiative and Auger recombination processes. At lasing
threshold, the radiative process is spontaneous consisting of
electron-heavy hole 共e-hh兲 as well as electron-light hole
共e-lh兲 recombination. The spontaneous emission rate per unit
volume in the energy interval E → E + dE due to the e-hh
process can be calculated as16
FIG. 2. 共Color online兲 Illustration of the GeSn/GeSiSn DHS on a lattice
matched GeSn relaxed buffer on Si or SOI substrate along with its band
alignment.
rsp,e-hh共E兲dE =
n̄e2E兩M b兩2
2␲ 3
m20␧0ប2c3
冉 冊
2mr
ប2
3/2

⫻共E − Eg兲1/2 f c共Ee兲f v共Ehh兲dE, 共7兲

respectively, we can vary the Si and ␣-Sn compositions si-
multaneously to yield exactly the lattice constant of where e is the electron charge, m0 the free electron mass, ␧0
Ge0.94Sn0.06. Adding the band gaps to the top of the valence the permittivity of vacuum, ប the Planck constant, c the
band Eq. 共3兲, we obtain the band alignment between them at speed of light in vacuum, n̄ the index of refraction, Eg the
the ⌫-point as shown in Fig. 1共b兲. band gap, and 兩M b兩2 the average matrix element for the Bloch
The ⌫-valley of the Ge0.94Sn0.06 active layer as shown in states.16,17 The occupation probabilities of an electron and a
Fig. 1共a兲 lies below L-valley. For the lattice matched heavy hole at the states that are separated by a photon energy
Ge1−x−ySixSny cladding layer, it can be shown by Eq. 共4兲 with E with the same k in the reciprocal space are
parameters in Table I that the position of its L-valley in-
creases asymptotically from 0.785 to 1.257 eV as the Sn
composition y increases from 0.06 to 0.20, consistently
冋 冉 冊册
f c共Ee兲 = 1 + exp
Ee − E fc
k BT
−1
,

above its ⌫-valley.

It is clear that there is a wide range of Sn composition
0.06⬍ y ⬍ 0.2 in which the ternary Ge1−x−ySixSny forms
冋 冉 冊册
f ␯共Ehh兲 = 1 + exp
E f ␯ − Ehh
k BT
−1
, 共8兲

type-I confinement with Ge0.94Sn0.06, i.e., the band alignment respectively, where kB is the Boltzmann constant, and the
provides carrier confinement for both elections and holes electron and hole energies in the conduction and valence
when Ge0.94Sn0.06 is sandwiched by the Ge1−x−ySixSny band
cladding layers. In the material growth front, there are
currently challenges in incorporating high Sn compositions mr
Ee = 共E − Eg兲,
into the alloy, we have therefore chosen to design lasers me
based on relatively small Sn compositions. In particular,
we propose a laser made of the lattice matched mr
Ehh = 共E − Eg兲, 共9兲
Ge0.94Sn0.06 / Ge0.75Si0.15Sn0.1 DHS 共Fig. 2兲. It can be esti- mhh
mated from Fig. 1 that the band gap of the active region
are computed through the reduced effective mass
Ge0.94Sn0.06 is Eg,⌫共Ge0.94Sn0.06兲 = 0.619 eV, while the con-
duction band offset at ⌫-point ⌬Ec,⌫ = 149 meV, and that of memhh
valence band ⌬E␯ = 54 meV. The laser device shall be grown mr = , 共10兲
me + mhh
on a relaxed Ge0.94Sn0.06 buffer on Si substrates to ensure
that the entire structure is strain free. in which me and mhh are the electron and heavy hole effec-
tive mass, respectively. Obviously, the spontaneous emission
rate rsp,e-lh共E兲dE for the radiative e-lh process can be ob-
III. CARRIER LIFETIME tained similarly. The total spontaneous emission rate per unit
volume can then be evaluated by integrating over the photon
Since the active region of Ge0.94Sn0.06 in this DHS has
energy E above the band gap Eg as
small amount of Sn, we shall use ⌫-point Ge parameters in
the following calculations by considering that the only effect
of incorporating the 6% Sn is to lower the ⌫-point of the Rrad = 冕 Eg
⬁
关rsp,e-hh共E兲 + rsp,e-lh共E兲兴dE. 共11兲
band gap to yield a direct band gap. The laser device under
consideration has a PIN DHS where the active Ge0.94Sn0.06 The radiative lifetime can then be obtained by ␶rad = n / Rrad
region is undoped. The band-to-band lasing transitions occur where n is the carrier density. The result for a carrier density
as stimulated emissions triggered by recombination of of n = 1018 / cm3, is shown in Fig. 3 for a range of tempera-
electron-hole pairs that are injected into this region. The ac- ture. It can be seen that the radiative lifetime for a fixed
tive region in the DHS is always wide enough for the carrier carrier density is rather insensitive to temperature change,
dynamics to be treated as bulk-like. The analysis proceeds as showing a slight increase with the temperature.
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 033107-4 Sun, Soref, and Cheng J. Appl. Phys. 108, 033107 共2010兲

me mhh + me
␮= , ␣c = , 共14兲
mhh mhh + 2me
the threshold energy required for the second electron to par-
ticipate in the Auger process
mhh + 2me
ET = Eg , 共15兲
mhh + me
兩M a兩2 is the matrix element of coulomb interaction between
two electrons evaluated at ET,16 and introducing kT
= 冑共2me / ប2兲ET

FIG. 3. 共Color online兲 Radiative and Auger recombination lifetime as a

function of temperature for the carrier concentration of 1018 cm−3 in the
I0 = 冕0
⬁
k2共k + kT兲2共k + 2kT兲2

冋 册
active region of the Ge0.94Sn0.06 / Ge0.75Si0.15Sn0.1 DHS.
ប2
⫻exp − k共k + 2kT兲 dk. 共16兲
2mekBT
The other important recombination mechanism is the
nonradiative Auger process where the recombination of an Other Auger processes involving different particles can be
electron-hole pair takes place by transferring energy and mo- evaluated following a similar procedure. Among them, the
mentum to a third particle which could be either an electron above illustrated e-hh Auger process involving a second
or a hole. We shall use the example of the Auger process electron dominates, and the result for the Auger recombina-
involving an electron as the third particle to illustrate the tion lifetime obtained by ␶aug = n / Raug is shown in Fig. 3 for
calculation procedure whose recombination rate per unit vol- n = p = 1018 / cm3. Clearly, the Auger lifetime decreases rap-
ume can be approximated as Raug = Cn2 p with an Auger co- idly with the increase in temperature as suggested by Eq.
efficient C, and for nondegenerate semiconductor, the elec- 共13兲. At higher temperature, it surpasses radiative rate to be-
tron and hole concentrations come the dominant recombination mechanism in the active
region.

n = Nc exp − 冉 Ec − E fc
k BT
冊
, IV. OPTICAL GAIN AND THRESHOLD CURRENT
For a given injected carrier concentration n = p, we can

冉 冊
calculate the optical gain at a photon energy E due to e-hh
E f ␯ − E␯ recombination as16
p = N␯ exp − , 共12兲

冉 冊
k BT
e2ប2兩M b兩2 2mr 3/2
␥e-hh = 共E − Eg兲1/2关f c共Ee兲
where Nc and N␯ are the effective density of states for con- ␧0m20cn̄E ប2
duction and valence band, respectively. According to Ref. 16, + f v共Ehh兲 − 1兴. 共17兲
the Auger rate per unit volume due to the recombination of
an e-hh pair involving a second electron as the third particle A similar expression can be obtained for the e-lh process
can be calculated as ␥e-lh. Thus, the total optical gain is then ␥ = ␥e-hh + ␥e-lh. The
profiles of optical gain are shown in Fig. 4. At a fixed tem-

冉 冊
perature, the optical gain increases with the carrier concen-
me␣2c 兩M a兩2 n2 p ET − Eg
Raug = I0 exp − , 共13兲 tration as the population inversion increases and the peak
4␲ ប 共1 + 2␮兲 Nc N␯
4 3 3/2 2
k BT shifts to higher photon energy as higher energy states in con-
duction and valence bands are occupied 关Fig. 4共a兲兴. For the
where under the assumption of parabolic bands same carrier concentration, greater optical gain is always ob-

FIG. 4. 共Color online兲 Optical gain

profile 共a兲 for a range of carrier con-
centration from 1018 to 5 ⫻ 1018 cm−3
at 300 K and 共b兲 for a fixed carrier
concentration of 2 ⫻ 1018 cm−3 in the
temperature range from 100 to 300 K.

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 033107-5 Sun, Soref, and Cheng J. Appl. Phys. 108, 033107 共2010兲

FIG. 5. 共Color online兲 Maximum optical gain as a function of the nominal FIG. 6. 共Color online兲 Nominal current density at threshold as a function of
current density for the temperature range from 100 to 300 K. temperature for different losses.

tained at lower temperatures as shown in Fig. 4共b兲. require a lower density of carriers to achieve optical gain
The carriers that are responsible for the optical gain are because of their reduced density of states and their simulta-
excited by external pumping. A preferred pumping method is neous suppression of the Auger process. The recent result by
current injection capable of sustaining the carriers character- expitaxial technique exhibiting very fine control of GeSiSn
ized by their lifetime. It is convenient to use the nominal alloy layers has indeed opened the pathway to developing
current density Jn to describe the pumping current in the GeSn QW lasers. There have already been proposals for la-
DHS, which is defined as current per unit volume given in sers based on strained GeSn/GeSiSn QW structures.5,6 By
the unit of ampere per centimeter square per micron, and reducing the active layer thickness to ⬃100 Å in our DHS,
when multiplied by the active region thickness, yields a we plan to investigate lattice-matched strain free GeSn/
pumping current density. At lasing threshold where the GeSnSi multiple QW lasers. These encouraging efforts will
stimulated emission is just about to take place, the nominal potentially lead to room temperature electrically injected la-
current density is related to carrier recombination rates when ser diodes that can be monolithically integrated on Si or SOI.
radiative and Auger processes are dominant The laser simulated here emits at a photon energy that is
Jn = e关Rrad共n兲 + Raug共n兲兴. 共18兲 higher than the ⌫ band gap energy of the active bulk layer.
Thus this DHS laser is really capable of operating anywhere
Since both the optical gain and the recombination rates de- within the 1.7 to 3.0 ␮m mid IR band because the emission
pend on carrier density n, we can now relate the optical gain wavelength desired is chosen simply by adjusting the Sn
with the nominal current density. Figure 5 shows the maxi- content of the active layer to a value in the 6%–12% range as
mum optical gain obtained from the gain profile in Fig. 4 as seen in Fig. 1共a兲. Then the longer-wave laser is designed by:
a function of the nominal current density for a temperature 共1兲 taking the GeSn buffer lattice to match the new active
range from 100 to 300 K. GeSn lattice, 共2兲 choosing the ternary cladding composition
In order to achieve lasing, the optical gain must be high to match the buffer lattice and to provide type-I offsets to the
enough to compensate for the loss. There are various loss active region, and 共3兲 calculating as before the carrier life-
mechanisms such as free-carrier absorption, waveguide loss, time, gain and threshold properties versus temperature. It is
and mirror loss that contribute to the required infrared gain. clear that the longer wave DHS would, once again, require
We have examined the nominal current density necessary to cooling in the 100 to 200 K regime.
generate sufficient optical gain to overcome a range of losses Looking to the future when a multiple-QW structure is
共20– 100 cm−1兲 under different temperatures as shown in substituted for the present three layer structure, we envision
Fig. 6. that room-temperature CW operation will be feasible for the
At T = 300 K, for an active region of 0.1 ␮m, the cur- new GeSn QW laser diode at shorter-than-DHS wavelengths
rent density that is required to compensate for loss of of operation—an emission that could be shifted towards the
20 cm−1 exceeds 10 KA/ cm2, and needs to be much 1.55 ␮m telecom range if desired because of the blueshift in
higher if more loss exists. While it is always desirable to subband-to-subband energy induced by the quantum confine-
have lasers operate at or above room temperature, such a ment.
large pumping current makes it rather unlikely with the
Ge0.94Sn0.06 / Ge0.75Si0.15Sn0.1 DHS. Looking at the estimates
V. CONCLUSION
in Fig. 6, if we pick a reasonable “ceiling” for current den-
sity, we conclude that the GeSn DHS laser requires cryo- We propose a simple group-IV laser made of
genic cooling in the 100–200 K range. Ge0.94Sn0.06 / Ge0.75Si0.15Sn0.1 DHS where the direct
We have found that the strong temperature dependence Ge0.94Sn0.06 active region is confined by two Ge0.75Si0.15Sn0.1
of threshold current in bulk heterostructure long wave lasers cladding layers. The compositions of both active and clad-
is due to the high density of carriers and the Auger recom- ding layers are determined to yield lattice matching and
bination process. One possible solution to the room- type-I band alignment between them. The entire DHS would
temperature problem is to work with GeSn QW lasers that be grown on a relaxed Ge0.94Sn0.06 buffer on either Si or SOI,
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