Microelectronics Journal 37 (2006) 919–929 An analytical model for discretized doped InAlAs/InGaAs heterojunction HEMT for higher cut-off frequency and reliability Ritesh Gupta, Sandeep k. Aggarwal, Mridula Gupta, R.S. Gupta Semiconductor Device Research Laboratory, Department of Electronic Science, University of Delhi South Campus, New Delhi - 110 021, India Received 26 November 2005; accepted 29 January 2006 Available online 31 March 2006 Abstract In the proposed work the model has been formulated for discretized doped HEMT, where the conventional uniformly doped, pulsed doped and delta doped structure are the special cases. An expression for sheet carrier density has been formulated considering the effect of doping-thickness product and has been extended to calculate drain current, transconductance, capacitance and cut-off frequency of the device. The model also takes into account the non-linear relationship between sheet carrier density and quasi Fermi energy level to validate it from subthreshold region to high conduction region. The results so obtained have been compared with pulsed doped structure to validate the model. The analysis concentrates on the distance of doping from the heterojunction and gate electrode. Different design criteria have been given to dope the carriers (amount and distance) in different regions to optimize the performance for higher sheet carrier density/parallel conduction voltage/effective parallel conduction voltage (V c V off ) to increase the transconductance, cut-off frequency and reliability of the device. r 2006 Elsevier Ltd. All rights reserved. Keywords: InAlAs/InGaAs heterostructure; Discretized doped; Delta doped; Uniformly doped; Pulsed doped; Parallel conduction; Breakdown voltage 1. Introduction High electron mobility transistors (HEMTs) consisting of InAlAs/InGaAs heterostructure, play a vital role in optical fiber communication and millimeter wave power applications subject to higher transport properties of InGaAs and larger sheet carrier density in the two- dimensional quantum well. However, some analog applica- tions of HEMTs are still limited by the reduced breakdown voltage of these devices, which limits the power applica- tions of HEMTs. In general, this problem is related to the properties of InAlAs/InGaAs material systems, in parti- cular due to enhance impact ionization effects in the narrow band gap (0.73 eV) of In 0.53 Ga 0.47 As or tunneling due to low Schottky barrier height (0.66 eV) of In 0.52 Al 0.48 As [1–8]. The effect of low breakdown voltage due to tunneling can be lowered by the enhancement of the Schottky layer and has been done by using undoped InAlAs layer directly beneath the gate [9] or by increasing the Al-mol fraction in the insulator or by using ternary semiconductor layer adjacent to the semiconductor contact (In 0.2 Ga 0.8 As/GaAs and In 0.48 Ga 0.52 P/GaAs heterostructure) [10–13] or by moving a portion of the dopants from the top InAlAs layer to the buffer layer [14,15]. Doping the carrier nearer to the heterointerface enhances the device performance by increasing 2-DEG electron density [16–19]. On the contrary, higher doping layer at the surface directly adjacent to the Schottky contact is needed to increase the Schottky barrier height [20]. Higher 2-DEG sheet carrier density leads to PBE effect resulting in early breakdown of the device [8]. In addition, the maximum sheet carrier density is also limited by the doping thickness product [21]. Increasing the Schottky layer thickness increases the drop across the Schottky layer thickness that can lead to early breakdown of the device [22]. Uniformly doped or pulsed doped structure may not fulfill all these requirements at the same time and a discretized doping profile is clearly needed to achieve desired value of sheet carrier concentration with ARTICLE IN PRESS www.elsevier.com/locate/mejo 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.01.012 Corresponding author. Tel.: +91 11 4115580; fax: +91 11 4110606. E-mail address: [email protected] (R.S. Gupta).
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ARTICLE IN PRESS
0026-2692/$ - se
doi:10.1016/j.m
�CorrespondE-mail addr
Microelectronics Journal 37 (2006) 919–929
www.elsevier.com/locate/mejo
An analytical model for discretized doped InAlAs/InGaAsheterojunction HEMT for higher cut-off frequency and reliability
Ritesh Gupta, Sandeep k. Aggarwal, Mridula Gupta, R.S. Gupta�
Semiconductor Device Research Laboratory, Department of Electronic Science, University of Delhi South Campus, New Delhi - 110 021, India
Received 26 November 2005; accepted 29 January 2006
Available online 31 March 2006
Abstract
In the proposed work the model has been formulated for discretized doped HEMT, where the conventional uniformly doped, pulsed
doped and delta doped structure are the special cases. An expression for sheet carrier density has been formulated considering the effect
of doping-thickness product and has been extended to calculate drain current, transconductance, capacitance and cut-off frequency of
the device. The model also takes into account the non-linear relationship between sheet carrier density and quasi Fermi energy level to
validate it from subthreshold region to high conduction region. The results so obtained have been compared with pulsed doped structure
to validate the model. The analysis concentrates on the distance of doping from the heterojunction and gate electrode. Different design
criteria have been given to dope the carriers (amount and distance) in different regions to optimize the performance for higher sheet
carrier density/parallel conduction voltage/effective parallel conduction voltage (Vc�Voff) to increase the transconductance, cut-off
High electron mobility transistors (HEMTs) consistingof InAlAs/InGaAs heterostructure, play a vital role inoptical fiber communication and millimeter wave powerapplications subject to higher transport properties ofInGaAs and larger sheet carrier density in the two-dimensional quantum well. However, some analog applica-tions of HEMTs are still limited by the reduced breakdownvoltage of these devices, which limits the power applica-tions of HEMTs. In general, this problem is related to theproperties of InAlAs/InGaAs material systems, in parti-cular due to enhance impact ionization effects in thenarrow band gap (0.73 eV) of In0.53Ga0.47As or tunnelingdue to low Schottky barrier height (0.66 eV) of In0.52Al0.48As [1–8].
The effect of low breakdown voltage due to tunnelingcan be lowered by the enhancement of the Schottky layer
e front matter r 2006 Elsevier Ltd. All rights reserved.
ejo.2006.01.012
ing author. Tel.: +9111 4115580; fax: +91 11 4110606.
and has been done by using undoped InAlAs layer directlybeneath the gate [9] or by increasing the Al-mol fraction inthe insulator or by using ternary semiconductor layeradjacent to the semiconductor contact (In0.2Ga0.8As/GaAsand In0.48Ga0.52P/GaAs heterostructure) [10–13] or bymoving a portion of the dopants from the top InAlAslayer to the buffer layer [14,15]. Doping the carrier nearerto the heterointerface enhances the device performance byincreasing 2-DEG electron density [16–19]. On thecontrary, higher doping layer at the surface directlyadjacent to the Schottky contact is needed to increase theSchottky barrier height [20]. Higher 2-DEG sheet carrierdensity leads to PBE effect resulting in early breakdown ofthe device [8]. In addition, the maximum sheet carrierdensity is also limited by the doping thickness product [21].Increasing the Schottky layer thickness increases the dropacross the Schottky layer thickness that can lead to earlybreakdown of the device [22]. Uniformly doped or pulseddoped structure may not fulfill all these requirements at thesame time and a discretized doping profile is clearly neededto achieve desired value of sheet carrier concentration with
ARTICLE IN PRESSR. Gupta et al. / Microelectronics Journal 37 (2006) 919–929920
high breakdown voltage. The breakdown mechanism andspeed of the device depends on the details of the devicedesign i.e. the thickness and doping concentration of thediscretized doping profile, recess width, channel composi-tion, etc. and an optimization is needed for its requiredapplications.
In the proposed work the model has been formulated fordiscretized doped HEMT, where the conventional uni-formly doped, pulsed doped and delta doped structure arethe special cases. For the analysis, In0.52Al0.48As/In0.53Ga0.47As HEMT with a wide gate recess is considered forobtaining a high breakdown voltage, as wide recessstructure reduces both the transverse electric field in thechannel and the vertical electric field at the edge of the gateelectrode. An expression for sheet carrier density has beenformulated considering the effect of doping–thicknessproduct and has been extended to calculate drain current,transconductance, capacitance and cut-off frequency of thedevice. The results so obtained have been compared withpulsed doped structure to validate our model. The analysisconcentrates on the distance of doping from the hetero-junction and gate electrode. Different design criteria havebeen given to dope the carriers (amount and distance) indifferent regions to optimize the performance for highersheet carrier density/parallel conduction voltage/effectiveparallel conduction voltage (Vc�Voff) to increase thetransconductance and cut-off frequency without compen-sating the reliability of the device.
2. Theoretical considerations
The basic structure of an InAlAs/InGaAs HEMT usedin the analysis is shown in Fig. 1, consists of InP substrate;
Fig. 1. Device structure for discretized doped InAl
InAlAs undoped buffer; undoped InGaAs layer to formthe 2-DEG channel; InAlAs undoped spacer-layer ofthickness ds; and Si-doped InAlAs layer doped with dopingprofile discretized into regions of constant doping densitiesN1;N2;N3; . . . andNn of thickness d1; d2; d3; . . . and dn,respectively. The basic charge control equation for twodimensional electron gas (2-DEG) along the channel usedin the analysis is given by [24]
where b ¼ eq:d , Vg is the gate voltage, d is the channel depth
and k1, k2 and k3 are the fitting parameters used to obtain theequivalent expression for 2-DEG sheet carrier density and thequasi Fermi level for InAlAs/InGaAs system. The thresholdvoltage, Voff for the discretized doped profile is found to be
Voff ¼ fb � DEc þ k1
�q
e
Xn
i¼1
Nid2i
2þXn�1i¼1
Nidi
Xn
j¼iþ1
dj
!, ð2Þ
where fb ¼ Schottky barrier potential (�0.4V for theundoped InAlAs layer), DEc ¼ the conduction band discon-tinuity at the InAlAs/InGaAs interface (�0.52V). The aboveexpression of the sheet carrier density is limited by itsmaximum value or doping thickness product
Xn
i¼1
Nidi
!
which ever is less.
As/InGaAs heterostructure, InP based HEMT.
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The maximum value of 2-DEG sheet carrier densitydepends on the transfer of carriers from different dopingregions to the quantum well, doping density and thethickness of the regions in discretized doped layer. If all thecarriers are assumed to be transferred from region m fromthe heterointerface to the quantum well then the maximumsheet carrier concentration is found to be
and the corresponding depletion width is found to be
W m ¼ ds þXm�1i¼1
di þnsom �
Pm�1i¼1 Nidi
Nm
. (4)
Fig. 2. Contours for maximum 2-DEG sheet carrier density for various valu
Nd1 ¼ 3/2.Nd, (dashes) for Nd1 ¼ Nd/2 and (solid) for conventional pulsed do
In general, maximum sheet carrier concentration is givenby
nso ¼ nsom if ds þXm�1i¼1
dipW mpds þXm
i¼1
di (5)
and the corresponding parallel conduction voltage is foundto be
V c ¼ fb �q
2:e:Xn
i¼mþ1
Ni:d2i �
q
e:Xn�1
i¼mþ1
Ni:di:Xn
j¼iþ1
dj
�q:Nm
2:e:
Xn
i¼m
di �nsoðmÞ �
Pm�1i¼1 Ni:di
Nm
!20@
�Xn
i¼mþ1
di
!21A. ð6Þ
The expression of drain current is given by [23]
Id ¼W qmoðf ðy1Þ � f ðy0ÞÞ
CB2ðL� DLþ ðmoVd=vsatÞÞ(7)
in which
f ðyÞ ¼ A2yþy2
2þ
4Ay3=2
3
es of doping thickness product and Schottky layer thickness. Dotted for
ped structure, i.e. Nd1 ¼ Nd.
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and
y1 ¼ ðb:k2Þ2þ 4:b:ð1þ b:k3Þ:ðV geff � VdÞ,
y0 ¼ ðb:k2Þ2þ 4:b:ð1þ b:k3Þ:ðV geff Þ,
A ¼ �bk2; b ¼ 2ð1þ bk3Þ; C ¼ �4bð1þ bk3Þ,
where Vgeff ( ¼ Vg�Voff) is the effective gate voltage, W
and L are the gate width and length, respectively and DL ischannel length modulation factor. The effect of parasiticresistances can be considered by replacing Vd byVd�Id(Rs+Rd) and Vg by Vg�IdRs. Transconductancecan be obtained by differentiating the drain current withrespect to gate voltage at constant drain voltage andcorresponding cut-off frequency can be obtained from theexpression
f c ¼gm
2pCgs(8)
in which, gm is the transconductance and Cgs is thechannel–gate capacitance.
The channel–gate capacitance can be found easily byfollowing the same approach as suggested by Nawaz et al.[24], in which charge associated with the gate terminal is
Fig. 3. Contours for maximum effective gate voltage for various values of do
2.Nd, (dashes) for Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped stru
defined as
Qg ¼ �Qt þ qW
Z L
0
nsðxÞ dx; (9)
where x is any position along the channel relative to thesource side (x ¼ 0) and for discretized doped structure
Qt ¼ qWLXn
i¼1
Nidi
!.
As ns is related to the channel potential so the integral in (9)is first transformed into channel potential by using theexpression of drain current given by
Ids ¼ qWnsðxÞvs.
Using a simple velocity field relationship
vs ¼
moE
1þE
Ec
; if EpEc;
moEc if EXEc;
8>><>>:
ping thickness product and Schottky layer thickness. Dotted for Nd1 ¼ 3/
cture, i.e. Nd1 ¼ Nd.
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in the linear region
dx ¼qWnsmo
Ids�
1
Ec
� �dV
on solving further, we get
Qgjl ¼ �Qt þ qW
Z Vd
0
nsðV ÞqWnsðV Þmo
Ids�
1
Ec
� �dV
(10)
and the corresponding channel–gate capacitance is givenby
Fig. 4. Contours for maximum drain current for various values of doping th
(dashes) for Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped structure,
f 2ðyÞ ¼qWns
C
qWnsmo
Ids�
1
Ec
� �.
In the saturation region the expression for gate charge isfound to be
Qgjs ¼ �Qt þ qW ðL� DLÞnsjV¼Vdsat, (12)
where DL is the channel length modulation factor. Thecorresponding channel–gate capacitance is given by
Cgsjs ¼ qW ðL� DLÞqns
qV gs
����V¼Vdsat
� nsjV¼Vdsat
qDL
qV gs
!.
(13)
It could be noted from (7) that at constant channeldepth, if we consider the variation of drain current witheffective gate voltage in place of gate voltage, theexpression of drain current becomes independent ofvertical thickness and doping variation in the device. Orin other words devices having constant channel depth showidentical characteristics with effective gate voltage irrespec-tive of the doping variation. Furthermore, the character-istics shows similar trend with effective gate voltages aswith gate voltage, where maximum value of transconductance
ickness product and Schottky layer thickness. Dotted for Nd1 ¼ 3/2.Nd,
i.e. Nd1 ¼ Nd.
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is controlled by its position with respect to parallelconduction voltage. Doping variation can further be usedto achieve the desired value of effective gate voltage atwhich parallel conduction starts.
2.1. Discretized doped structure as a pulsed doped structure
Doping variation in pulsed doped structure is given by
NðyÞ ¼
0 if 0pypds;
Nd if dspypds þ da;
0 if ds þ dapypds þ da þ dsh;
8><>: (14)
where ds is the spacer layer thickness, da is the doping layerthickness and dsh is the Schottky layer thickness. Dopingvariation in discretized doped structure is given by
NðyÞ ¼
0; 0pypds;
N1; dspypds þ d1;
N2; ds þ d1pds þ d1 þ d2;
..
.
Nn; ds þPn�1i¼1
dipypds þPni¼1
di:
8>>>>>>>>><>>>>>>>>>:
(15)
Fig. 5. Contours for Maximum Transconductance for various values of doping
(dashes) for Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped structure,
Comparing these two structures for same number ofcarriers gives
Nn ¼ 0 and dn ¼ dsh,
Xn�1i¼1
di ¼ da,
and
Xn�1i¼1
Nidi ¼ Nd:da. (16)
Assuming the doping to be equally spaced with thicknessd 0, in that case
d 0 ¼da
n� 1,
and
Xn�1i¼1
Ni ¼Nada
d 0(17)
thickness product and Schottky layer thickness. Dotted for Nd1 ¼ 3/2.Nd,
i.e. Nd1 ¼ Nd.
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Substituting the above values in (2), the expression forthreshold voltage for discretized doped structure havingidentical carriers in doping region with pulsed dopedstructure reduces to
Voff ¼ fb � DEc þ k1
�qd 0
2eNdda 1þ
2ðda þ dshÞ
d 0
� �� 2d 0
Xn�1i¼1
iNi
( )
ð18Þ
for identical doping concentration in the discretized region,the above expression reduces to
Voff ¼ fb � DEc þ k1 �q:Nd:d
2a
2:e: 1þ
2:dsh
da
� �, (19)
which is found to be the same expression as reported in [23]for pulsed doped structure, thus showing the validity of theproposed model. Similarly, substituting (17) in (3) andassuming identical doping concentration in the discretizedregion, the expression of sheet carrier concentrationreduces to
Fig. 6. Contours for maximum gate capacitance for various values of doping
(dashes) for Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped structure,
where
A1m ¼ Ndds and A2m ¼2Nde
qðDEc � Ef Þ
which is found to be the same expression as reported in[21,25] for pulsed doped structure, thus showing thevalidity of the proposed model.
3. Result and discussion
As mentioned earlier, the discretized doped region isdivided into n regions, where doping in region-n is assumedto be zero so that region-n can replace Schottky layer topulse doped structure. Furthermore, for simplicity, regionsare assumed to be equispaced and there are only threeregions in which region-2 and region-1 together form thedoping region of the pulsed doped structure and region-3replaces the Schottky layer in the pulsed doped structure.To elaborate the advantage of discretized doped structurethree different cases have been considered for identicalcarriers in the doping region of the pulsed doped structure.
1.
thic
i.e.
For the first case, doping in region-1 is assumed to behalf of the doping in the pulsed doped structure and forequivalent carrier concentration doping in region-2
kness product and Schottky layer thickness. Dotted for Nd1 ¼ 3/2.Nd,
Nd1 ¼ Nd.
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Fig
(da
R. Gupta et al. / Microelectronics Journal 37 (2006) 919–929926
becomes 3/2 times the doping in the pulsed dopedstructure.
2.
For the second case, doping in both the regions isassumed to be the same.
3.
For the third case, doping in region-1 is assumed to be3/2 times the doping in the pulsed doped structure andfor equivalent carrier concentration doping in region-2becomes half of the doping in the pulsed dopedstructure.
The variation of maximum value of sheet carrierconcentration for discretized doped structure is shown inFig. 2 for all the three cases with various values of doping-thickness product and Schottky layer thickness of theequivalent pulsed doped structure. It can be seen from thefigure that higher doping-thickness product and Schottkylayer thickness are the favorable conditions for larger valueof sheet carrier density. Initially for devices having identicalmaximum sheet carrier density, increase in Schottky layerthickness required decrease in doping-thickness product. Insuch devices, doping thickness product is larger then themaximum sheet carrier density and such device may sufferfrom parallel conduction that can lead to decrease intransconductance at higher gate voltages and is notrequired. At higher value of Schottky layer thicknessdoping-thickness product required to be constant and for
. 7. Contours for maximum cut-off frequency for various values of doping
shes) for Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped structure,
theses devices the value of doping thickness product canreplace the value of maximum sheet carrier density. Thesedevices are useful as these are the optimized devices in thesense that these devices will not suffer from parallelconduction. Varying the doping concentration in differentregions shifted the range of optimized value of Schottkylayer thickness, as for the three cases shown in the figure. Inother words splitting the pulsed doped region into tworegions generates different range of Schottky layer thick-ness that can be used to consider other parameters likebreakdown voltages, barrier height, threshold voltages, etc.The maximum value of sheet carrier concentrationobtained from the graph varies from 1� 1016 to2.5� 1016m�2 with the variation of doping thicknessproduct from 0.5� 1016 to 5� 1016m�2 and Schottky layerthickness from 0 A (uniformly doped) to 190 A (deltadoped structure).Transconductance increases to its maximum value and
then decreases with increase in gate voltage. At constantchannel depth (d) the same variation has been observedwith effective gate voltage (Vgeff ¼ Vgs�Voff), in which, Voff
is the threshold voltage of the device independent of thevertical thickness and doping concentration. In HEMT, thedecrease in transconductance is either due to high value ofparasitic resistances or due to parallel conduction. Thedecrease in transconductance due to parasitic resistance
thickness product and Schottky layer thickness. Dotted for Nd1 ¼ 3/2.Nd,
i.e. Nd1 ¼ Nd.
ARTICLE IN PRESSR. Gupta et al. / Microelectronics Journal 37 (2006) 919–929 927
can be controlled by decreasing the value of parasiticresistances while the effect of parallel conduction isuncontrollable but can be pushed towards higher gatevoltage by increasing the parallel conduction voltage. Oneexpects an increase in transconductance and cut-offfrequency by increasing parallel conduction voltage (Vc)and maintaining constant threshold voltage or by increas-ing the threshold voltage and maintaining the maximumgate voltage constant or by increasing Vc�Voff. Thevariation in maximum effective parallel conduction voltage(Vc�Voff) for discretized doped structure for all the threecases with various values of doping-thickness product andSchottky layer thickness of the equivalent pulsed dopedstructure is shown in Fig. 3. From the figure the varia-tion of doping thickness product from 0.5� 1016 to5� 1016m�2 and Schottky layer thickness from 0 A(uniformly doped) to 190 A (delta doped structure) leadsto change in maximum value of effective parallel conduc-tion voltage from 0.8 to 1.4V.
The effect of enhancement in maximum effective gatevoltage can be seen through the variation of drain current,transconductance, capacitance and cut-off frequency inFigs. 4–7 for all the three cases with various values ofdoping-thickness product and Schottky layer thicknessof the equivalent pulsed doped structure. Drain currentvaries from 0.4 to 0.9A/mm, where transconductance
Fig. 8. Contours for threshold voltage for various values of doping thickness pr
Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped structure, i.e. Nd1 ¼ N
changes from 0.78 S/mm to 0.91 S/mm for a gate lengthof 0.25 mm. Capacitance also increases from 710 fF/mm to800 fF/mm, which resulted in cut-off frequency from 173 to182GHz.Threshold voltage controllability can be seen from
Fig. 8. From the figure threshold voltage varies from�0.4 to �1.6V for the variation in doping thicknessproduct from 0.5� 1016 to 5� 1016m�2 and Schottky layerthickness from 0 A (uniformly doped) to 190 A (deltadoped structure).Now larger the sheet carrier density larger will be the
electrons traversing from doped InAlAs to InGaAs. Eachelectron will gain energy equivalent to conduction banddiscontinuity. So a very less energy is needed to break thebond. Such effect is usually referred to as parasitic bipolareffect (PBE). To consider the PBE the approach suggestedby Higuchi et al. [8] is used to calculate the breakdownvoltage of the device and has been plotted in Fig. 9. Fromthe analysis breakdown voltage varies from 14 to 19V.
4. Conclusion
In the proposed work the model has been formulated fordiscretized doped InAlAs/InGaAs heterojunction HEMT.The expression of maximum sheet carrier density has beenformulated considering the effect of doping–thickness
oduct and Schottky layer thickness. Dotted for Nd1 ¼ 3/2.Nd, (dashes) for
d.
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Fig. 9. Contours for breakdown voltage for various values of doping thickness product and Schottky layer thickness. Dotted for Nd1 ¼ 3/2.Nd, (dashes)
for Nd1 ¼ Nd/2 and (solid) for conventional pulsed doped structure, i.e. Nd1 ¼ Nd.
R. Gupta et al. / Microelectronics Journal 37 (2006) 919–929928
product. The comparison has been made between pulseddoped structure and equivalent discretized doped struc-tures to validate the model and is found to be in excellentagreement. The variation of Schottky layer thickness foridentical carriers leads to increase in maximum sheetcarrier concentration/effective parallel conduction voltage/transconductance/capacitance/cut-off frequency. Altera-tion of doping concentration in equivalent discretizeddoped structure can be used to optimize the conventionalpulsed doped structure for better performance. Maximumvalue of sheet carrier density, effective gate voltage, draincurrent, transconductance, capacitance, cut-off frequencyand breakdown voltage obtained from the analysis is2.5� 1016m�2, 1.4V, 0.9A/mm, 0.91 S/mm, 800 fF/mm,182GHz, 14V, respectively.
Acknowledgement
Authors are thankful to Council of Scientific andIndustrial Research (CSIR), Government of India andDefence Research Development Organization (DRDO),Ministry of Defence, Government of India, for providingthe necessary financial assistance.
References
[1] S.R. Bahl, J.A. del Alamo, Physics of breakdown in InAlAs/
