Anomalous capacitance–voltage profiles in quantum wells explained by a quantummechanical modelSudakshina Kundu, Dipankar Biswas, and Reshmi Datta Citation: Journal of Applied Physics 81, 2030 (1997); doi: 10.1063/1.364061 View online: http://dx.doi.org/10.1063/1.364061 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/81/4?ver=pdfcov Published by the AIP Publishing

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

93.180.53.211 On: Tue, 18 Feb 2014 12:28:51

[This a

Anomalous capacitance–voltage profiles in quantum wells explainedby a quantum mechanical model

Sudakshina Kundu, Dipankar Biswas,a) and Reshmi DattaDepartment of Electronic Science, Calcutta University, 92 Acharya Prafulla Chandra Road,Calcutta-700009, India

~Received 22 July 1996; accepted for publication 8 November 1996!

We have developed a quantum mechanical model for understanding and explaining thecapacitance–voltage~C–V! carrier profiles observed in quantum wells~QW!. The external fieldimposed on the QW during C–V profiling changes the carrier distribution of the system. This modelconsiders the effects of field and quantum confinement of the carriers in the well. The resultsobtained by iterative solutions of Schrodinger’s and Poisson’s equations give a better understandingof the experiments than the previous models where quantum confinement is ignored. ©1997American Institute of Physics.@S0021-8979~97!03304-5#

yata-econ

umheieto

zeeisone-

elwiifm

r’

osbui

liutive

ll,reub

ub-

en-

rs

e-

-

the

heofierry

Capacitance–voltage~C–V! measurements are widelused for carrier profiling of semiconductors and to estiminterface states.1,2 We have recently shown that C–V mesurements with time in the form of deep level transient sptroscopy~DLTS! can be used to determine the conductiand valence band offsets of quantum wells accurately.3,4

When an external field is imposed on a single quantwell for C–V profiling, a number of changes take place. Tbands bend~Fig. 1!, as carriers get depleted from the barron the Schottky side, and collect in the quantum wellincrease the two-dimensional~2D! carrier concentration. In-side the quantum well the motion of the carriers is quantialong the well width. The carriers are free to move in planparallel to the interface, making their motion 2D. Due to thquantum confinement, the entire carrier distributichanges5,6 so that the estimation of the carrier profile bcomes quite involved.

Our model accounts for this quantum effect and the fidependent phenomena inside the quantum well alongthe band offset. The carrier distribution and the peak shwith temperature are computed here when this quantumchanical model is applied to ap-type Si/SiGe/Si quantumwell. Self-consistent, iterative solutions of the Schrodingeand Poisson’s equations are adopted.

The cosine wave function representing the twdimensional hole gas~2DHG! present in the quantum well ichanged in the presence of an external field as givenBastard. In our model, the wave function is modified to sthe requirements of a 2DHG.

The quantum confinement of the carriers leads to spting of the degenerate heavy and light hole bands of the bsemiconductor. Due to valence band anisotropy, the effecmasses of the 2D holes are modified from their bulk valuFrom a knowledge of the measured effective massesheavy and light holes in the subbands of the quantum we7,8

it is evident that for low doping and moderate temperatualmost 90% of the holes populate the first heavy-hole sband.

a!Institute of Radio Physics and Electronics, Calcutta University, 92 AchaPrafulla Chandra Road, Calcutta 700 009, India

2030 J. Appl. Phys. 81 (4), 15 February 1997 0021-8979/9rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

93.180.53.211 On: Tue,

e

-

r

ds

dthtse-

s

-

yt

t-lkes.of

s-

The wave function representing the first heavy-hole sband is given by

wv5N~b!ucos~pz/L1d!exp @b~Z/xL21/2x!# ~1a!

wb5N~b!ucos~p/21d!expF2q0S ZL21

2D G ~1b!

in the well and barrier, respectively.N~b! is a normalization constant andb is a variational

parameter obtained by a process of minimization of theergy term

E~b!5E0@11b2/4p21f$21/b22b/~4p21b2!

2 12 coth~2b/2!%# ~2!

d5tan21~q01b/x!/p2p/2 ~3!

is a phase factor

q0252m* L

2~E02DEv!/\. ~4!

Herex is a fraction andL is the total well width so thatxLrepresents the fractional well width over which the carrieare piled up due to the imposed fieldF. E0 is the groundstate,DEv is the difference in the valence-band offset btween the quantum well and the barrier layer

and f5qFxL/E0 . ~5!

Without any field, Eq.~1a! reduces to the cosine function.

The carrier concentration is obtained by integratingproduct of the 2D density-of-states~2D-DOS! function,Fermi function, and the probability density obtained from tcarrier distribution in the quantum well. The concentrationholes within the quantum well and those split into the barraway from the Schottky side are given respectively bya

7/81(4)/2030/3/$10.00 © 1997 American Institute of Physicsject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

18 Feb 2014 12:28:51

n-e

enreb

arreenort

ann

r-n

ionethnerein

riseheseakally.thesider-era-iththeyewde-Wu

er

re-

ith

[This a

pw5EL~122x!/2

L/2

p2DN2~b! cos2S pz

L1d D

3exp 2bS z

xL2

1

2xD lnU 11exp~E02EF!

KT

11exp~DEv2EF!

KT

Udz~6a!

pB5EL/2

a

p2D8 N2~b!cos2S p

21d Dexp22q0S zL2

1

2D3 lnU11exp

~DEv2EF!

KT Udz. ~6b!

Starting with a trial valueF, the fractional well-widthxL anda trial hole concentration within the well, the 2D-hole cocentration is obtained by iterating. Repeated iterations givunique set of values forF andx.

Our main objective is to explain the anomaly betwethe carrier profile obtained experimentally by C–V measument and the theoretical interpretations offeredTittelbach–Helmrich.9 Our model is applied to ap-type Si/SiGe/Si quantum well shown in Fig. 1. The calculationsdone at different temperatures and repeated for diffevalence-band offsets in the expected ranges of 25–40 m

Figure 2 shows the carrier distribution inside the quatum well with the reverse field. As the field increases, mcarriers are depleted from the barrier and are pushed intoquantum well where they pile up, raise the peak height,cause the peak position to shift away from the Schottky eThe peak carrier concentration changes from.4.031015

cm23 at equilibrium to nearly 8.031016 cm23 at a field re-quired to bring the depletion edge into the well. It is inteesting to note that without any field the peak concentratiovery close to the theoretical result of T-H.9

The variations of the peak height and the peak positwith temperature are shown in Fig. 3, for a valence baoffset of 25 meV and at a field required to completely dplete the cap layer. The peak position shifts towardsSchottky end with a rise in temperature as in that obtaiexperimentally. The shift in peak position with temperatuin the 3D model is in the opposite direction and rema

FIG. 1. Bending of the valence band of the Si–SiGe–Si QW under revbias.

J. Appl. Phys., Vol. 81, No. 4, 15 February 1997rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

93.180.53.211 On: Tue,

a

-y

entV.-ehedd.

is

nd-ed

s

unexplained. The carrier peak height also decreases within temperature because the carrier confinement diminiswith an increase in temperature. Such decrease in pheights is more pronounced when measured experimentThis seems to arise from the fact that the error betweenactual peak and the measured peak increases at a conable rate due to an increase in Debye length with tempture. Figure 4 plots the variation of the peak heights wtemperature for both the theoretical model as well asexperimental data of T-H. The nature of variation of Deblength is also shown in Fig. 4. It is apparent that at lotemperatures the measurement error diminishes with thecreasing Debye length. In fact, it has been shown byet al.10 and Johnsonet al.11 that the effect of Debye length

se

FIG. 2. Hole concentration inside the quantum well on application ofverse bias. Field increases in steps of 12 kV/m.

FIG. 3. Variation of peak height and peak position inside the QW wtemperature. Left hand half of the well is shown.

2031Kundu, Biswas, and Dattaject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

18 Feb 2014 12:28:51

anathtiotdanen4–dio

toubhanthnee

ndexrrivsis

theanll asomrva-gect

ortheer-n-p-

tron.

s.

ated

. A.

tt.

-

and

[This a

becomes very pronounced if the carrier distribution risesfalls very rapidly within a couple of Debye lengths as inquantum well. So the higher the temperature and largerDebye length the lower the measured carrier concentracompared to its actual value. The spatial resolution cannoless than a couple of Debye lengths. The experimentalof Wu et al.10 gives a clear picture of this underestimatioFor a very sharp ion implanted profile, C–V measuremgives an experimental carrier concentration that is abouttimes less than the actual value. This effect is clearly incated in our calculated value of the peak height whichroughly four times higher than the experimental dataT-H.9 But the theoretical peak height of the model of T-H9 isnearly 4 times less than the 3D measured value. Althoughcomputation presented were done for the first heavy-hsubband, they were checked by including the light-hole sband as well. The difference in the two results was less t3%. Inclusion of the first light-hole subband lengthens acomplicates computational procedures without alteringresults significantly. It seems logical not to include such unecessary complexities, particularly, when the error betwtheory and experiment is so high.

There is still some ambiguity about the ratio of the baoffsets. Results for different valence-band offsets in thepected range are shown in Fig. 5. It is seen that the caconfinement and peaking are strongly dependent on thelence band offset. When the valence band offset is conered to be 40 meV, the peak carrier concentration rise131017 cm23.

FIG. 4. The experimental (E) and theoretical (T) peak heights vs temperature. The Debye length variation with temperature is also shown.

2032 J. Appl. Phys., Vol. 81, No. 4, 15 February 1997rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

93.180.53.211 On: Tue,

d

enbeta.t5i-sf

hele-nde-n

-era-d-to

We therefore conclude that our model outlines howcarrier distribution inside the quantum well changes withimposed field and variations of temperature. These as wethe shift in the peak position with temperature obtained frour model support the nature of the experimental obsetion. But it should be noted that without precise knowledof the valence band offset it is difficult to compute the exavalue of the carrier peak concentration.

Acknowledgments:The authors are grateful to ProfessB. R. Nag., Professor P. K. Basu, and Dr. J. B. Roy ofInstitute of Radio Physics and Electronics Calcutta Univsity for their useful discussions. R.D. is thankful to the Coucil of Scientific & Industrial Research for the financial suport.

1D. Biswas P. R. Berger, U. Das, J. E. Oh, and P. Bhattacharya, J. ElecMater.18, 137 ~1989!.

2H. Kromer, W.-Y. Chien, J. S. Harris, Jr., and D. D. Edwell, Appl. PhyLett. 36, 295 ~1980!.

3N. Debber, D. Biswas, and P. Bhattacharya, Phys. Rev.40, 1058~1989!.4P. Bhattacharya, N. Debber, and D. Biswas, Int. Symp. GaAs and RelCompounds~1989!.

5Doyeol Ahn and S. L. Chuang, Appl. Phys. Lett.49, 1450~1986!.6G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B28,3241 ~1983!.

7J. P. Cheng, V. P. Kesan, D. A. Grutzmacher, T. O. Sedwick, and JOtt, Appl. Phys. Lett.62, 1522~1993!.

8D. X. Xu, C. D. Shen, M. Willander, and G. V. Hansson, Appl. Phys. Le58, 2500~1991!.

9K. Tittelbach-Helmrich, Semicond. Sci. Technol.8, 1372~1993!.10C. P. Wu, E. D. Douglas, and C. W. Mueller, IEEEED-22, 319 ~1975!.11W. C. Johnson and P. T. Panousis, IEEEED-18, 955 ~1971!.

FIG. 5. The carrier distribution and peak heights for different valence-boffsets at different temperatures.

Kundu, Biswas, and Dattaject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

18 Feb 2014 12:28:51