Table Of ContentDeformation Electron-Phonon Coupling
in Disordered Semiconductors and Nanostructures
A. Sergeev∗
Research Foundation, University at Buffalo, Buffalo, New York 14260
5
0 M. Yu. Reizer
0 5614 Naiche Rd. Columbus, Ohio 43213
2
n V. Mitin
a Electrical Engineering Department, University at Buffalo, Buffalo, New York 14260
J
3 Westudytheelectron-phononrelaxation (dephasing)rateindisorderedsemiconductorsandlow-
1 dimensional structures. The relaxation is determined by the interference of electron scattering via
thedeformationpotentialandelasticelectronscatteringfromimpuritiesanddefects. Wehavefound
] thatincontrasttothedestructiveinterferenceinmetals,whichresultsinthePippardineffectiveness
l
l conditionfortheelectron-phononinteraction,theinterferenceinsemiconductingstructuressubstan-
a tiallyenhancestheeffectiveelectron-phononcoupling. Theobtainedresultsprovideanexplanation
h
toenergy relaxation in silicon structures.
-
s
e PACSnumbers: PACSnumbers: 72.10.D
m
.
t Elastic electronscattering from impurities and defects Pippard ineffectiveness condition, obtained for metals.
a
m drastically changes the electron-electron and electron- However,inthe temperaturerangesinvestigatedinRefs.
phonon (e-ph) interaction and modifies temperature de- [12, 13] and [14], DP is strongly screened and the re-
-
d pendencies of the relaxation/dephasing rate. As a re- laxationrate in pure 2D and quasi-2D-structuresfollows
n sultofdiffusivemotionofelectrons,theelectron-electron to T5-dependence [15, 16]. Therefore, the ineffectiveness
o interaction is significantly enhanced in bulk and low- would result in the T6, rather than the T4-dependence.
c
[ dimensionalconductors [1]. Recent theoretical[2, 3] and Itisnotsurprisingthatthetheorydevelopedformetals
experimental [2, 4] studies have shown that the piezo- [5, 6, 7] fails to describe semiconductors. Indeed, DP in
1
electrice-phcouplingisalsoenhancedinsemiconductors metals and semiconductors has different nature [17]. In
v
with short electron mean free path. Effects of disorder metals, DP is associated with electron gas compressibil-
6
2 on the deformation e-ph coupling are more complicated. ity,whileinsemiconductorsthiscontributionisnegligible
3 In impure metals, the deformation coupling originates due to small carrier concentrations. DP in semiconduc-
1 from ”pure” electron-phonon scattering, electron scat- tors results mainly from a shift of the conduction-band
0
teringfromvibratingimpurities,andvariousinterference edge under the deformation, while in metals such con-
5
processes. If electron scatterers vibrate in the same way tribution is small because of strong screening. It is im-
0
/ as host atoms, the destructive interference of scattering portantthatDPhasdifferenttensorstructuresinmetals
t
a mechanisms [5, 6] results in the Pippard ineffectiveness and semiconductors [17, 18], and this difference clearly
m condition[7],whichmeanssuppressionofthee-phrelax- manifestitselfeveninweaklydisorderedconductors[19].
- ation. In this case, at low temperatures the relaxation Here we show that the tensor structure of DP plays a
d rate modifies from T3-dependence in the pure materials crucialroleinkinetics ofstronglydisorderedconductors:
n to T4-dependence in the impure metals. However, even in contrast to the destructive interference in metals, the
o
c smallamountofstaticscatterers(e.g. toughboundaries) electron-phonon-impurityinterferenceinsemiconducting
: or incomplete drag of impurities and defects increases structures substantially enhances e-ph coupling.
v
thee-phrelaxation[8]. Disorder-suppressedrelaxationis
i Here we report results on the e-ph relaxation in dis-
X observed in disordered metallic films [9, 10], while alloys
ordered bulk semiconductors, two-dimensional electron
r commonlydemonstratethedisorder-enhancedrelaxation
structures, and multi-channel one-dimensional conduc-
a
with T2-dependence of the relaxation rate [11].
torsinteractingwith3Dphonons. Effects ofdisorderare
Recently,therehasbeensignificantinteresttotheelec- described by the dimensionless parameter ql, where q is
tron relaxation in disordered semiconductors and struc- the characteristic momentum transferred to the electron
tures, where the electron relaxation is determined by due to e-ph scattering, l =v τ is the electron mean free
F
electron-phonon scattering via the deformation poten- path due to scattering from impurities, v is the Fermi
F
tial (DP). The relaxation rate has been measured in Si velocity. In the impure limit, ql 1, a phonon interacts
≪
crystals containing δ(Sb)-layer [12, 13] and in Si films with an electron that diffuses in the interaction region,
[14]. Experimental results, including T4-dependence of L 1/q l. In bulk conductors, q is the wavevector of
∼ ≫
the electron relaxation rate, were associated with the a thermal phonon, q = T/u (u is the sound velocity),
T
2
and the crossoverto the impure limit occurs at T u/l. For 1D conductors we should take into account the dy-
∼
In low-dimensional conductors, the characteristic mo- namical character of electron screening. If q r 1 (r is
k
≫
mentum q isdetermined bythe phononwavevectorcom- the conductor radius), the dielectric function is
ponent q , which is parallel to the conductor. In two-
dasiminenbsuiolknkaslesmyisctoenmdsu,cqtkoriss,otfhtehecroorsdseorveorfiqsTdaenscdr,ibaesdwbelyl ǫR =1+e2ν1 ln(q1r)2 · (q l)2 (q(kωlτ)2)2 iωτ. (5)
k k
− −
the parameter q l = Tl/u. In 1D channels the trans-
T
The e-ph relaxationrate is calculatedas a variationof
ferred momentum q is (u/v )q and effects of dis-
k ∼ F T the collision integral τ−1 = δI /δn . In equilib-
order become important at significantly higher tempera- e−ph − e−ph e
rium,N =Neq(T)andn =neq(T), andthe relaxation
tures, Tτ qkl < 1. The same parameter Tτ describes ω ω ǫ ǫ
∼ rate of electrons at the Fermi surface (ǫ=0) is
modification of the electron-electron interaction [1].
Investigating the electron energy relaxation, we focus
1 dq γ2 ζ (ω )
our attention on the time scale much longer than the =4τ (Neq+neq) n q . (6)
τ Z (2π)3 ǫR 2 ωq ωq ℜ1 ζ (ω )
electron momentum relaxation time. In this time do- e−ph | | − n q
main,electron-phononkineticsisdescribedbythe angle-
We also calculate the heat flux from hot electrons with
averaged electron and phonon distribution functions, n
ǫ the temperature θ to phonons with the temperature T.
and N . We consider interference processes, which are
ω The heatflux may be presentedthroughthe energy con-
characterized by the momentum transfer much smaller
trol function F(T) as
than the Fermi momentum. In this case, the interfer-
enceofelectron-phononandelectron-impurityscattering
P(θ,T)=ν dǫ ǫI (θ,T) =F(θ) F(T), (7)
is takeninto accountby the electronself-energydiagram nZ ǫ −
shown in Fig. 1. The corresponding electron-phonon dq γ2 ζ (ω )
collision integral is [6, 8] F(T)=4τνnZ (2π)3 ǫR 2 ωq2Nωeq ℜ1 nζ (qω ). (8)
n q
| | −
dqdω γ2 ζ
I = 4τ DR(q,ω) n First we calculate the relaxation rate in a bulk semi-
ǫ − Z (2π)4 ǫR 2ℑ ℜ1 ζ
| n| − n conductor. Substituting ζ3 (Eq. 3) and ǫR3(q) (Eq. 4)
Nωnǫ(1 nǫ+ω) (1+Nω)(1 nǫ)nǫ+ω, (1) into Eq. 6 we find
× − − −
where DR(q,ω) is the phonon Green function
DR(q,ω)=(ω−ωq+i0)−1+(ω+ωq+i0)−1, (2) τe−1ph = Dρ02uν23(pFTu3)2 F(qTl,qT/κ3), (9)
γ = D0(q e)/(2ρωq)1/2 is the vertex of the electron- F(y,z) = ∞dx x2 xyarctan(xy)
·
phonon interaction, D0 is the constant of DP, and e is Z0 xy arctan(xy)
−
the phonon polarization vector. (xz)2 2
In the collision integral Iǫ, ζn is an integral over the × (cid:18)(xz)2+1(cid:19) (Nxeq +nexq). (10)
impurity-averagedelectron Green functions [6],
These formulas in limiting cases are summarized in Tab.
1 dp
ζn = πνnτ Z (2π)nGA(p,ǫ)GR(p+q,ǫ+ω) Ir.esIunlttshe[1p7]u:reinlitmhiet,cTasle/uof≫we1a,kwsecrreeepnroindgu,cTe w>elul-κk3n,otwhne
arctan(ql) n=3, relaxation rate is proportional to T3; for screened DP,
(ql) T < uκ3, the relaxation rate changes as T7. In the im-
= 1+1(qkl)2 n=2, (3) upnusrcerleiemnietdthDePrealanxdattoionT6r/altefoisrpthroepsocrrteieonneadltDoPT.2T/lhufosr,
p 1 iωτ contraryto the Pippardineffectiveness conditioninmet-
− n=1,
where ν is the two-s(p1in−eilωeτct)r2o+n(dqeknls)i2ty of states in n- ahlasn[c5e,d6b,y7]a, tfahcetorerlaoxfaut/io(Tnlr)adteueintoseemlaisctoicndeuleccttorrosnissceant--
n
tering. The energy control function may be estimated
dimensional electron system. We limited our considera-
tion by the condition ql>u/v 10−2, which allowsus as F(T) CeT/τe−ph, where Ce is the electron heat
F ≃
∼ capacity. In Tab. I we present F(T) with exact coeffi-
to put ω =0 in ζ3 and ζ2.
cients,becausemeasurementsofF(T) arewidely usedto
ThescreeningofDPisdescribedbythedielectricfunc-
tion ǫR(q,ω). Further calculations show that in the 3D obtained D0.
n
Now we consider the e-ph relaxation in two-
and 2D electron systems the characteristic frequencies
ω T are small compared with Dq2 (D is the diffusion dimensional electron gas. Using Eqs. 3 and 4, we find
∼ that the relaxation rate (Eq. 6) may be presented as
coefficient). In this limit the dielectric function is
ǫRn(q)=(cid:26)11++((κκ32//qqk)2),, κκ232==42ππee22νν32 32DD;. (4) τe−1ph = 2πD2ρ02vTF3u4 Φ(qTl,qT/κ2), (11)
3
Φ(y,z) = ∞dx x2 π/2dθ xy F(T)= 105ζ(9/2) D02ν1T9/2. (15)
Z0(xz)2sin3Zθ0 p1+(xysinθ)2−1 128√π(cid:18)1−e2ν1ln(qcr)2(cid:19)3/2 √τρvFu4
(Neq +neq). (12)
× (xzsinθ+1)2 x x AsseenfromEq. 15,screeningsubstantiallychangesval-
ues of τ and F(T), but just weakly affects the tem-
These formulas in limiting cases are summarizedin Tab. e−ph
peraturedependencies. CompareEqs. 13and15,wefind
II. In the pure limit, we reproduce well-known results
that in the impure limit the electron phonon interaction
[15]. In the temperature range T ≫ κ2u, where DP is is enhanced by the factor of 1/√Tτ.
weakly screened, the relaxation rate is proportional to
The electron-phonon-impurity interference in metals
T3; for stronglyscreenedDP the relaxationrate changes
and semiconductors may be qualitatively understood in
as T5. In the impure limit, in the case of weak screen-
the following way. First, elastic electron scattering ef-
ing, the relaxation rate is proportional to T2lnT and
fectively averages DP over the Fermi surface. Second,
inversely proportional to l. At low temperatures, where
the diffusive motion holds an electron in the interaction
DP is strongly screened, the relaxation rate is propor-
region and increases the interaction time. In metals the
tional to T4/l. Thus, in heterostructures elastic electron
Fermisurfaceaverageofthedeformationpotentialequals
scattering significantly enhances the e-ph interaction.
to zero [17, 18]. As a result of this averaging the effec-
Finally, we consider the e-ph interaction in the mul-
tive e-ph vertex is substantially decreased (see Ref. [6]).
tichannel 1D system. Channels may be associated with
In metals this effect prevailsoverthe modificationof the
wires, shells, and electron subbands. Variations of the
interaction time and strongly suppresses the e-ph relax-
multichannel model are applied to one-dimensional or-
ation. In semiconductors, DP weakly depends on the
ganic conductors, CuO-chains in high-T superconduc-
c electron momentum and the DP tensor is usually ap-
tors, and multi-wall carbon nanotubes [20]. For sim-
proximated by a constant. Therefore, elastic scattering
plicity we consider identical channels and neglect the
insemiconductorsenlargestheinteractiontime,whichin
Coulomb interaction between channels. We suggest
turn enhances the e-ph relaxation.
that electrons are scattered between channels and in-
Recently the e-ph relaxation rate has been directly
terchannel scattering prevails over backscattering in the
measured in 2D electron gas in Si with MBE-grown Sb
same channel, so the system is in the conducting state.
δ-layer [12, 13]. Because of lack of the theory for semi-
Electron-phononscatteringkeepsanelectroninthesame
conducting materials and structures, the observed T4-
channel and, therefore, it is screened by electrons in this
dependence was associated with the Pippard concept of
channel. UsingEqs. 3and6,wefindthatwithoutscreen-
the ineffectiveness of the e-ph interaction. According to
ing the relaxation rate in the pure conductor is given by
our results, the T4-dependence in 2D structures origi-
1 7ζ(3) D2T3 nates from disorder-enhanced screened DP coupling (see
= 0 . (13) Tab. II).Analogousdata withT4-dependence havebeen
τ 8π ρv u4
e−ph F
obtainedinheavilydopedquasi-two-dimensionalSifilms
Calculating the integrant in Eq. 6 in the general case, at subKelvin temperatures [14]. Note, that e-ph relax-
note that for a 1D conductor q =qcosφ=qx (φ is the ationrateis often evaluatedfromthe electrondephasing
k
angle between q and a wire ) and within the logarithmic rate. Such data also give evidence in favor of significant
accuracy the integral over the direction of q is given by enhancementof e-phcoupling in disorderedsemiconduc-
tors. For example, in 3D Si:P layers with l 5 nm the
1 dx (qlx)2 relaxationtimeat4.2Kwasfoundtobe 10p∼s[21]which
Z0 2 [(1 2e2ν1lnqrx)(qlx)2 (ωτ)2]2+(ωτ)2 is significantly shorter than that in pure materials.
− −
To conclude, we calculate the e-ph relaxation rate
1 π/2 ωτ 1;
= 2ql(1−2e2ν1lnqcr)3/2 (cid:26)π/√8ωτ ωτ≫≪1, (14) idnimdeinsosirodnearled(Esqe.m1ic1o,nTdaubc.to2r)sa(nEdq.one9-,diTmaebn.sio1n)a,l t(wEoq-.
Eq. 14showsthatthecrossovertotheimpurelimitisde- 15) semiconducting structures. Our results show that
scribedbytheparameterωτ,whichisoftheorderofTτ. the e-phrelaxationis stronglyenhanced due to disorder.
In the impure limit, Tτ 1, the characteristic value of The research was supported by the ONR and MRCAF
the transferredmomentu≪mqc is l−1√ωτ/(l√1+e2ν1) grants.
l−1√Tτ/(l√1+e2ν1). In this case, the relaxation rat≃e
andenergycontrolfunctionfortheone-dimensionalmul-
tichannel conductor are
∗
1 3 8 √2 ζ(5/2) D2T5/2 Electronic address: [email protected]ffalo.edu
= − 0 [1] B. L. Altshuler and A.G. Aronov, Electron-Electron In-
τ (cid:0) (cid:1) 3/2√τρv u4
e−ph 64√2π(cid:18)1−e2ν1ln(qcr)2(cid:19) F taenrdacMtio.nPoilnakD(iNsoorrdtehr-eHdoSlaynstde,mAsm, setdeirtdedamb,y1A98.5L).. Efros
4
[2] E.Chow,H.P.Wei,S.M.Girvin,andM.Shayegan,Phys. [16] R. Fletcher, V.M. Pudalov, Y.Fenget al., Phys. Rev.B
Rev.Lett. 77, 1143 (1996). 56, 12422 (1997).
[3] D.V. Khveshchenko and M. Reizer, Phys. Rev. B 56 [17] V.F.GantmakherandY.B.Levinson,Carrier Scattering
15822 (1997). in Metals and Semiconductors, North-Holland, Amster-
[4] R.Fletcher,Y.Feng,C.T. Foxon,and J.J. Harris,Phys. dam, Oxford, New York,Tokyo(1987).
Rev.B 61, 2028 (2000). [18] InisotropicmetalstheDPtensorisG(δij−3ninj),G=
[5] A.Schmid,Z. Physik 259, 421 (1973). 2ǫF/3, n=p/p, p is the electron momentum (see Refs.
[6] M.Yu.ReizerandA.V.Sergeev,Zh.Exsp. Teor. Fiz.90, 5, 6 and 17); in isotropic semiconductors the DP tensor
1056 (1986) [Sov. Phys. JETP 63, 616 (1986)]. is a constant.
[7] A.B. Pippard, Philos. Mag. 46, 1104 (1955). [19] A. Sergeev, M.Y. Reizer, and V. Mitin, Phys. Rev. B.
[8] A.SergeevandV.Mitin,Phys.Rev.B.61,6041(2000). 69, 075310 (2004).
[9] M.E. Gershenson, D. Gong, T. Sato et al., Appl. Phys. [20] E. G. Mishchenko, A. V. Andreev, and L. I. Glazman
Lett.79, 2049 (2001). Phys. Rev.Lett. 87, 246801 (2001).
[10] J.T. Carvonen, L.J. Taskinen, and I.J. Maasilta, Phys. [21] D.R.HeslingaandT.M.Klapwijk,SolidStateCommun.
Stat.Sol. (c) 1, 2799 (2004). 84, 739 (1992).
[11] J.J. Lin and J.P. Bird, J. Phys. Cond. Matt. 14, R501
(2002) FIG. 1: Electron self-energy diagram. Wavy line stands for
[12] V.Y. Kashirin, Y.F. Komnik, A.S. Anopchenko et al., e-ph scattering, a dotted line stands for to elastic electron
Low Temp. Phys.23, 303 (1997). scattering from random potential, and a straight line stands
[13] S.Agan,O.A.Mironov,E.H.Parkeretal.,Phys.Rev.B for the electron Green function.
63 075402 (2001).
[14] P. Kivinen, A. Savin, M. Zgirski et al., J. Appl. Phys.
94, 3201 (2003).
[15] P.J. Price, J. Appl. Phys. 53, 6863 (1982).
5
TABLE I:Electron-phonon energy relaxation time and energy control function in a bulksemiconductor.
T >uκ3 (weak screening) T <uκ3 (strong screening)
T >u/l T <u/l T >u/l T <u/l
−1 7πζ(3)D02ν3 T3 3π2D02ν3 T2 5715πζ(7)D02ν3 T7 3π6D02ν3 T6
τ
e−ph 4 ρu2 (pFu)2 4 ρu2 p2Flu 8 ρu2 p2Fκ43u6 4 ρu2 p2Fκ43lu5
D02ν32 T5 π4D02ν32 T4 D02ν32 T9 4π8D02ν32 T8
F(T) 6πζ(5) 10080πζ(9)
ρu2 (pFu)2 10 ρu2 p2Flu ρu2 p2Fκ4u6 5 ρu2 p2Fκ43lu5
TABLE II: Electron-phonon energy relaxation time and energy control function in two-dimensional electron structures.
T >uκ2 (weak screening) T <uκ2 (strong screening)
T >u/l T <u/l T >u/l T <u/l
−1 7ζ(3) D02T3 D02T2 T 93ζ(5) D02T5 π2 D02T4
τ ln
e−ph 4π ρvFu4 ρ lvFu3 κ2u 8π ρκ22vFu6 4 ρκ22 lvFu5
6ζ(5)D02ν2T5 2π2D02ν2T4 T 90ζ(7)D02ν2T7 8π4 D02ν2T6
F(T) ln
π ρvFu4 15 ρ lvFu3 κ2u π κ22vFu6 63 ρκ22 lvFu5
Fig.1
