Scattering of Carriers by Charged Dislocations in Semiconductors

0.18 MB·English

by  Bhavtosh Bansal

Scattering of Carriers by Charged Dislocations in Semiconductors
∗
Bhavtosh Bansal , V. Venkataraman
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
(Dated: February 2, 2008)
Thescatteringofcarriersbychargeddislocationsinsemiconductorsisexaminedwithintheframe-
4
workoftheBoltzmann transport theory. Theratiosofquantumandtransport scatteringtimesare
0
evaluatedafteraveragingovertheanisotropyintherelaxationtime. Anewapproximateexpression
0
forthecarriermobilityisproposedandthevalueoftheHallscatteringfactoriscomputed. Change
2
in the resistivity when the dislocations are tilted with respect to the plane of transport is deter-
g mined. Finally an expression for the relaxation time is derived when the dislocations are located
u
within thesample with a uniform angular distribution.
A
PACSnumbers: 72.10.Fk,72.20.Dp
2
1

I. INTRODUCTION II. ISSUES ADDRESSED
]
n
n
- Epitaxial growth of thin semiconductor films on sub- (i)Thescatteringpotential,duetoitscylindricalsymme-
s strates which have a large lattice constant mismatch re- try,ishighlyanisotropicandthevalidityoftheextension
i
d sults in the films being strained. Depending on the ofthe relaxationtimeapproachtothisproblemhasbeen
. growth conditions and the films’ thickness, this strain questioned[2, 7, 13].
t
a can either partially or fully relax through a formation of (ii) Po¨d¨or’s [1] expression for the relaxation time
m
various possible kinds of lattice defects. Among these
- defects edge dislocations are prominent and have a pro- 8ǫ2m∗2 ¯h2
nd nouncedeffectonthemobilityofcarriers. Whilethethe- τ = N e2Q2λ(4m∗2λ2 +v⊥2)3/2 (2)
o oryforchargeddislocationscatteringwasfirstformulated d
c toexplainthelowtemperaturemobilityofplasticallyde-
[ formedsemiconductors[1],interestindislocationscatter- is apparently physically incorrect. v⊥ is the component
of electron velocity perpendicular to the dislocation axis
ing has revivedin contextof GaN [2, 3, 4, 5, 6, 7, 8]. In-
1 and N is the number of dislocations per unit area, all
v deedit isimportantinallepitaxiallygrownmaterials[9] d
assumed to be parallel and independent. Only the per-
6 onmismatchedsubstrates,aswellasbulk crystalswhose
pendicular component of the impinging electron’s veloc-
5 growth techniques have not yet been mastered [10].
ity contributes to scattering and the component parallel
2
8 An edgedislocationis a rowofdanglingbonds formed tothedislocationisunaffected. Equation2isfinitewhen
0 by an abruptly terminated plane somewhere inside the v⊥ →0, whereas in this limit, τ should diverge.
4 crystal[2]. Thislocaldeparturefromtetragonalcoordina- (iii)The methodofenergyaveragingemployedbyPo¨d¨or
0 tion produces acceptor states in the energy gap, forming has been questioned[4].
t/ one dimensional lines of charge. The effective screened (iv) Due to (iii), the tensor nature of resistivity is not
a electrostaticpotentialenergy,U(x⊥)isthuscylindrically evident in the final expression. In particular if the dislo-
m
symmetric if the extent of the edge dislocation is taken cations are tilted at an angle with respect to the direc-
- to be infinite[11, 12]. tionperpendiculartotheplaneoftransport,itisdifficult
d
n to give anything better than a rough estimate[8] in the
o Qe present theory. Most often, the effect of dislocation ori-
c U(x⊥)= 2πǫK0(x⊥/λ) (1) entation is disregarded and µ⊥ is replaced by a scalar
:
v number.
i (v) Quantum and classical scattering times were
X whereQisthechargeperunitlength,K isthemodified
0 calculated[14] without averaging out the anisotropy in
r zeroth order Bessel function of the second kind, ǫ=ǫ0ǫr the problem.
a
is the dielectric constant, λ is the screening length and
(vi) There are correctionsto the measured Hall mobility
x is the distance from the dislocation line in a perpen-
⊥ due to the Hall scattering factor.
dicular plane, r = f(x ,θ,z). These one dimensional
⊥ (vii)In general, dislocations may not be all parallel.
lines of charge have detrimental effects on the transport
Points (i) and (ii) above were what led us to the re-
properties of charge carriers.
examination of this problem. Nevertheless, a first prin-
ciples treatment of the calculations shows that Po¨d¨or’s
40-year old formula is indeed correct. The reasons for
∗Present Address: Department of Condensed Matter Physics the apparent discrepancy are traced to the limits where
the solution of the Boltzmann equation is invalid. This
and Materials Science, Tata Institute of Fundamental Research,
Mumbai-400005, India,ElectronicMail: [email protected] is discussed in the next section.
2
III. THEORETICAL FORMULATION Here θ is the angle between k⊥,k′⊥ which lie on a circle
parallel to the xy-plane since k is independently con-
z
served. The wave vectors in the summation in equation
We start from the Boltzmann equation within the lin-
(7) are three dimensional.
ear response regime[15]. Then upto first order in elec-
Within the Born approximation
tric field, the perturbed distribution function may sym-
bolically be written as fk = f0k +φk, where φk is de- 2π
viation from an equilibrium distribution in presence of Wk,k′ = ¯h [Lzδkz,kz′]2δ(Ek−Ek′)
a perturbing external electric field F. In absence of
2
a thermal gradient and a magnetic field, the linearized × 1 dx⊥U(x⊥)ei(k⊥−k′⊥)·x⊥ (8)
Boltzmann equation for carriers described by spherical (cid:20)LxLyLz Z (cid:21)
parabolic band reduces to
Here L ,L and L are the crystal dimensions over
z y x
which the plane wave electron states are normalized
e¯h ∂f
m∗F·k ∂E0k = Wk′,k[φk′ −φk] (3) and the length of the ‘infinite’ dislocation has been lim-
k′ ited to the size of the crystal along the z direction.
X
The energy conserving delta function, δ(Ek − Ek′) =
Wptelrakin,nkeg′ wipsaovtteehnesttiatarltaegnsi,svietkinonabnydraetkqe′u,abtienitownpere(en1s)e.nincieFtioaorlfsatchnaedttesfircnianatg-l (bt∂ooEtδhk/z∂,tkhkz′e)−inp1eδtr(hpkee−nsudkmi′c)um=laart(ih¯oaknn/.dmTt∗hh)ue−s1,p(aaksr/apkllr⊥ee)lvδic(ookum⊥slp−yocnkle⊥′ani)mtsdeudoef,
fromchargeddislocations,the scatteringrate isgivenby the electron momenta are separately conserved. Since,
Wk,k′ = δ(kz −kz′)δ(k−k′)g(|k′⊥ −k⊥|). g(|k′⊥ −k⊥|) Σk′ → LxLy/(2π)2 dk′⊥, an overall factor of area re-
is the part depending on only a function of in-plane mo- ma⊥insinthedenominatoraftertheprimedmomentahave
menta (shownbelow). Thus (a)collisions areelastic, (b) been integrated overR. This simply means that the scat-
thecomponentsoftheincidentelectron’smomentawhich teringdue to a singlechargeddislocationis ineffective in
are parallel and perpendicular to the dislocation line are a large sample.[16] When there are many charged dislo-
separately conserved, (c) no electric field develops along cations within this area which are all parallel, one can
the dislocation axis, i.e. F·k = F⊥ ·k⊥. This imme- simply replace (L L )−1 by N the dislocation density
x y d
diately implies that no relaxation time can be defined per unit area if the interference terms can be neglected.
along the direction parallel to the dislocations’ axis. In The Fourier transform in equation (8) can be approxi-
other words, for time independent electric field, there is mately written as [4]
nosteadystatesolutiontotheBoltzmannequationifthe
collision term is zero. Nevertheless, one may physically ′ Qeλ2
argue that 1/τz = 0. The argument is clear within the U(|k⊥−k⊥|)≃ ǫ(1+|k′⊥−k⊥|2λ2) (9)
variational formalism[15] where one defines the sample
resistivity in terms of the Joule-heat dissipated due to
a finite current. With constraints (a)-(c) in mind, we IV. RESULTS
shall choosea φk which solvesthe Boltzmann’s equation
exactly. Ansatz: A. Transport Lifetime
e¯h ∂f
φk =−m∗τ(k,k⊥)F⊥·k⊥∂Ek (4) peFrproenmdiecquulaartioton(t7h)e, tdhiselorcealatxioantioanxistimise in the direction
Substituting φk in equation(3) yields ¯h3ǫ2
τ⊥(k,k⊥)= Q2e2λ4m∗N [1+(2k⊥λ)2]3/2 =τ⊥(k⊥)
∂f d
0kF⊥·k⊥ =F⊥· Wk,k′ (10)
∂E
k′ This is exactly what Po¨d¨or had derived (equation (2))
X
× ∂f0k′τ(k′,k⊥′ )k′⊥− ∂f0kτ(k,k⊥)k⊥ (5) atinodn.kW⊥h=ile0inimtphlrieeesdaimfiennitseioτn⊥saenveenlecatftreornowuirthrekd=eri0vais-
∂E ∂E
(cid:20) (cid:21) unphysical (there is no associated phase space), an elec-
Fromenergyandperpendicularmomentumconservation, tron with k⊥ = 0 and kz 6= 0 corresponds to a physical
situation. The inconsistency in the final formula results
∂f0k′ ′ ′ ∂f0k from the breakdown of the validity of the assumed solu-
∂E τ(k ,k⊥)= ∂E τ(k,k⊥). (6) tion, φk = 0 for k⊥ = 0 in equation (4). This condition
isoutsidethescopeofthepresentschemeofthesolution,
ThusthelinearizedBoltzmannequationisexactlysolved which is otherwise consistent.
if The anisotropy in τ necessitates a further angular av-
eraging for a comparison with any physical quantity as-
1
= Wk,k′(1−cosθ) (7) sociatedwithameasurementwhichinvolvesathermody-
τ(k,k⊥) namicdistributionofelectrons. Thistransportscattering
k′
X
3
time is directly connected to mobility, µ = (e/m)hhτii, a subject of lively debate sometime back. Many par-
where hhii denote an energy average, (see below) over allel interpretations for level broadening [18] have been
a distribution function of appropriate degeneracy. In a suggested. Some semiclassical arguments even favour a
fully degeneratesystem,usingequation(13), thissimpli- small angle cutoff. This fact may be particularly impor-
fies to hhτtrii=(3/4) 0πsin3θτ⊥dθ. tant in two dimensions where it could rescue the quan-
tum scattering time from a divergence [14] in a simple
R
and physically meaningful way, the small angle cutoff θ
c
B. Quantum Scattering Time (in radians) being inversely proportional to the Landau
level index n, θ ≃π/2n [19, 20].
c
A quantum scattering time, τ⊥q(k⊥) is, by definition, Despite the preceding remarks,the conceptof a quan-
tum scattering time finds a widespread use in literature
equation(7), but without the (1−cos θ) factor andmay
(for example, references [20, 21, 22]). Therefore we have
be calculated similarly. This was recently done by Jena
plotted the suitably defined ratio h1/τqihhτ ii of the
and Mishra[14]. tr
transport and quantum scattering times for a three di-
τ⊥q(k⊥)= Q2e2¯hλ34ǫm2∗Nd[11++(22k(⊥k⊥λ)λ2)]23/2 (11) mitseerp,nlskoiottn/eqadl da.esgqeanefurianstcetthicoeanrsrimoiefprtlgehaewsadinvimefivegneucsrtieoonr1l.iensTdsehppeaegrnardamepneht-
F TF TF
However, the angular dependence of τq must also to Thomas-Fermi screening function. The largeness of this
⊥
ratio is often regarded as a measure of ‘anisotropy’ of
be averaged out. The meaningful quantity is h1/τqi =
scattering. [23]. The real space anisotropy of the dis-
(2/π) π/2[τq(θ)]−1dθ and is often connected to the fi-
0 location potential is different from the anisotropy in its
nite amplitude and width of the Shubnikov-de Haas or
Fourier transform, which is more a measure of the ef-
R
de Haas-van Alphen oscillations. The quantum scatter-
fective range of the potential. An additional averaging
ingtimemaybelookeduponasaneffective‘Dingle’tem-
causes the transport to quantum scattering times ratio
perature, T ∼(h¯/2πk )h1/τqi.
D B to be larger than what was calculated by in reference
Nevertheless, while comparing Shubnikov amplitudes,
[14].
the scattering rates are better calculated between Lan-
dau wave functions and with a density of states at the
Fermi level modified by the magnetic field, as was done C. Mobility
long back by Vinokur[17] for the essentially the same
problem. Furthermore, literature on the connection be-
In calculating mobility, the averaging procedure em-
ployed by Po¨d¨or has been called ‘unspecified’[4] and
hence it is worked out below. For dislocations along the
(cid:13) z-axis,thecurrentandelectricfielddirectionscoincideas
10(cid:13) long as the measurement is done in the xy-plane. Then,
9(cid:13) After Averaging (this paper)(cid:13) jx =nehhvxii and hhvxii=µ⊥Fx where
Jena and Mishra(cid:13)
8(cid:13) Short Ranged(cid:13) hhv ii= k(f0k+φk)vx = kφkvx (12)
s(cid:13) x
me 7(cid:13) P k(f0+φk) P kf0k
g ti 6(cid:13) Or P P
n
of scatteri 45(cid:13)(cid:13) (cid:13) σxx = em2¯h∗22(2π2)3 Z kx2(cid:18)−∂∂fE0(cid:19)τ⊥(k,k⊥)d3k (13)
o 3(cid:13) Or
ati
R 2(cid:13) ¯h5ǫ2 π
σ = dθsin2θ
1(cid:13) xx 4π2m∗3Q2Ndλ4 Z0
∞
0.0(cid:13) 0.5(cid:13) 1.0(cid:13) 1.5(cid:13) × dkk4 −∂f0 1+(2kλsinθ)2 3/2 (14)
k(cid:13) /q(cid:13) ∂E
F(cid:13) TF(cid:13) Z0 (cid:18) (cid:19)
(cid:2) (cid:3)
The integralsmustnow be evaluatednumerically. Equa-
tion (13) has the unpleasant feature of depending very
FIG. 1: The ratio of dislocation scattering limited transport strongly on screening length and thus at low tempera-
and quantum scattering times for a degenerate electron gas. tures turns out to be dependent on the model used for
the temperature dependent of carrier concentration and
screening. Asimpleanalyticexpressioncanbeestimated
tween scattering times for dislocations’ strain field and by interpolating the two integrals ( dθ and dk) be-
deHaas-vanAlphenoscillationamplitudesinmetalswas tweenthetwoextremescases,whenthefirsttermismuch
R R
4
smallerand whenit is muchlargerthan the secondterm j = σ E + σ E , the Hall scattering factor r is
x xx x xy y H
in square brackets in equation (13). This is significantly defined as
better than Po¨d¨or’shightemperatureapproximation[11] σ
xy
(kλsinθ ≫1). Therelativepercentageerrorsareplotted rH =neBσ2 (17)
in figure 2 as a function of the dimensionless parameter xx
8m∗kBTλ2. It can be seen that this approximationof the wheretheoff-diagonalconductivity,σ ,forcarrierswith
h¯2 xy
integral never deviates from the numerically calculated parabolic energy dispersion which are distributed along
exact answer by more than 5%. isotropic constant energy surfaces is
90(cid:13) (cid:13) σxy = ¯he23mB∗ Z 4dπ3k3τ⊥2∂∂fE0 (cid:18)∂∂kEx(cid:19)2"1+(cid:18)eτm⊥∗B(cid:19)2#−(118)
This work(cid:13)
80(cid:13) 'High temperature' (cid:13) From equations (13), (18) and (17) the Hall scattering
approximation (Podor)(cid:13)
70(cid:13) factor for nondegenerate carriers at high temperatures
(i. e. 8m∗kBTλ2 ≫ 1) approaches a value of 2.17, ob-
60(cid:13) h¯2
r (cid:13) tained by dropping the second term in square brackets
o 50(cid:13)
r in equation (13). At lower temperatures, its value is de-
r
E
% 40(cid:13) (cid:13) pendent on the model of carrier density and screening
but always smaller. The anisotropy in scattering makes
30(cid:13)
thevaluehigherthantheHallfactorforionizedimpurity
20(cid:13)
scattering which is 1.93.
10(cid:13)
0(cid:13)
E. Effect of Dislocation Tilt
-10(cid:13)
0.1(cid:13) 1(cid:13) 10(cid:13) 100(cid:13)
8m(cid:13) (cid:13) *(cid:13)k(cid:13) T (cid:13)2(cid:13) / h(cid:13)2(cid:13) Assume that dislocations are all parallel, but now
B(cid:13) at a longitude φ and latitude θ with respect to the z-
axis while the measurement is being done in the xy-
plane. A unit vector along this dislocation axis is dˆ =
FIG.2: The relative percentage errors (µexact−µapprox ×100) xˆsinθ sinφ+yˆsinθ cosφ+zˆcosθ. Because the electric
µexact
inour formulaand P¨od¨or’sapproximation withrespect tothe field is developed only along the direction perpendicular
teexdacatseaxpfruenscstiioonneovfalduiamteednsniounmleesrsicpaallrya.mTetheerg8rmap∗khh¯B2sTaλr2e.plot- tyoietldhse(dwisiltohcactoisonasn’daxsiins,aFb⊥br=evρiaj⊥ted=tρo[jc−a(njd·dˆs))dˆ]which
1−s2θs2φ −s2θsφcφ −cθsθsφ
Assuming that the electrons are distributed according
ρ′ =ρ −s2θsφcφ 1−s2θc2φ −cθsθsφ (19)
to Maxwell-Boltzmann distribution,
 −cθsθsφ −cθsθsφ 1−c2θ 
4π1/2¯h3ǫ2 15π 2/3 8λ2m∗k T 3/2  
µ⊥ ≃ em∗2Q2Ndλ4 "π1/3+(cid:18) 8 (cid:19) π¯h2B # Nofetghaetievleecstigrincifineltdhedeovffeldoipaegdo.nals indicates the direction
(15)
and when the carrier gas is fully degenerate
F. Angular Distribution of Dislocations
µdeg ≃ 3h¯5ǫ2 (4/3)2/3+(5π/16)2/34k2λ2 3/2
⊥ 4m∗3Q2N λ4 F TF
d TF h (16i) Letus nextconsiderthe casewhenallthe dislocations
arenot parallelto eachother but areinstead distributed
withauniformdistributionofangles. Onecan,ofcourse,
average over the angles[24] appearing in equation (19).
D. Hall Factor
This averaging over the angles in the rotated resistivity
tensoramountstheuseofMatthiessen’sruleandwillnot
In most experiments it is not the drift but the Hall
change the temperature dependence of mobility.
mobility which is measured. Under the assumption that
For a better approximation, we again start from the
the scattering rate does not alter in presence of a mag-
linearized Boltzmann equation, equation (3). In the
neticfield,Bandwhenthe magneticfieldisalignedwith
present case, the relaxation time must be isotropic and
the dislocations’ axis, only the in-plane relaxation time
therefore let the ansatz for the distribution function be
comes into the picture. Using the same line of argu-
ments, it is easy to again establish its existence for ar- ¯he∂f
φ(k)=− 0kτ (k)k·F (20)
bitrarily strong non-quantizing magnetic fields. Then, if m ∂E iso
k
5
We shall further assume incoherent scattering such the Since theaveragingoverthe dislocationorientationsis
scattering rates due to different dislocation lines add. If equivalenttoanaveragingovertheelectronwavevectors,
thescatteringrateduetoanithdislocationisWi ,then the expression for the relaxation time becomes
k,k′
the total rate is Wi .
i k,k′
Withoutlossofgenerality,one canchoosethe electron 1 N Q2e2λ4m∗N π sinθ(1−cos2θ)
A d
wofatvheevietchtdorislkoctaoPtiboen,adloinigstahteanz-aaxnigsl,ek(θ=,φk)zˆw.iItfhtrheespaexcist τiso(k) = 4π ¯h3ǫ2 Z0 dθ[1+k2λ2sin2θ]3(/224)
to the z-axis, then the unit vector along the dislocation where the φ integral has been performed and we have
axis is given by dˆi = sinθsinφxˆ+sinθcosφyˆ+cosθzˆ. 2π 2π
notedthat sinφdφ= cosφdφ=0. Fromhereon,
The component of the wave vector perpendicular to the 0 0
it is straightforward to calculate the isotropic mobility,
dislocation axis is given by R R
although it must be done numerically [25].
ki⊥ =k−(k·di)dˆi
Or
V. SUMMARY
ki⊥ =k[(1−cos2θ)zˆ−cosθsinθcosφyˆ
Within the framework of the conventional transport
−cosθsinθsinφxˆ] (21)
theory, we have shown that a relaxation time can be
Substituting back in the Boltzmann equation, we get defined for scattering of carriers by charged disloca-
tions. Difference between quantum and classical scat-
1 tering times was discussed and it was pointed out that
F k =−τ (k)F· −ki (22)
z z iso ⊥τ(k,ki ) the anisotropy necessitates an appropriate angular av-
Xi (cid:20) ⊥ (cid:21) eraging. A new approximate formula for mobility was
Converting the sum into an integral, derived and it was shown be within 5% of the exact re-
sultatalltemperatures. ThevalueoftheHallscattering
N 1 factor and the effect of dislocation tilt on resistivity was
Fzkz =τiso(k) 4πAF· dΩki⊥τ(k,ki ) (23) determined. Finally we derived a new expression for the
Z ⊥ relaxation time when the angular orientation of disloca-
Here N is the dislocation density per unit solid angle. tions is isotropic.
A
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