Pis'ma v ZhETF, vol.86, iss. 10, pp.762-768
© 2007 November 25
Infrared catastrophe in two-quasiparticle collision integral
O. V. Dimitrov.E+*1), V. E. Kravtsov+* + L.D. Landau Institute for Theoretical Physics RAS, 117940 Moscow, Russia *The Abdus Salam International Centre for Theoretical Physics, P.O.B. 586, 34100 Trieste, Italy
Submitted 18 October 2007
Relaxation of a non-equilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. We show that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution which corresponds to the total electron spin magnetization M. = 0 and only for non-zero interaction in the triplet channel. In this case a non-perturbative treatment of the electron-electron interaction is needed to provide an effective infrared cut-off.
PACS: 71.23.An, 72.15.Rn, 72.25.Ba, 73.63.Nm
1. Introduction. The ground-breaking experiments by Pothier et al. of Ref. [1] have demonstrated that one can have a direct access to the non-equilibrium electron energy distribution function f(E) and through it to the inelastic collision integral /Ccoii (E) which enters the kinetic equation [2]:
dtf(E-,x,t)^DV2f(E-,x,t) = -KcM(E-,x,t). (1)
In turn, studying the collision integral gives one an important information on interaction and dynamics of qua-siparticles in a dirty metal. In this way the predictions of the theory of electron interaction in disordered metals [3, 4] were checked [1, 5] and an unexpected strong sensitivity of the energy relaxation to the presence of magnetic impurities [6] was established.
The main idea of Ref. [1] was to use the sharp features in the energy dependence of the density of states (DoS) fprobe(-E0 of a superconducting probe electrode, which enabled to extract f(E) by measuring the differential conductance of the tunnel junction between the normal metal sample and the probe electrode. Recently, the same idea has been suggested [7] to create a non-equilibrium spin-dependent electron energy distribution fcr{E) and thereby to obtain a spin-polarized current through the probe. The sketch of the experimental setup is shown in Fig. 1. The sample in a form of a quasi-one dimensional disordered normal-metal wire is connected through the weak tunnel junctions to the two superconducting leads. The non-equilibrium is created by applying the finite bias voltage U between the leads. The spin dependence of fcr{E) (a = ±1 for the spin projections f (4-)) is caused by magnetic field applied
e-mail: odimitro0ictp.it
to the superconducting leads with the DoS viR\E) = = vs(E+U/2±aEz) and vjL) {E) = vs(E-U/2+aEz) where vs{E) = v 0? [E/VE2 - A2J and the Zeeman shift Ez = imH is taken with the sign ± depending on whether the directions of the magnetic field in the right and left leads are parallel or anti-parallel. In the absence of relaxation fcr{E) is given by [7, 11]:
f(o)(F]_^L)(E)fF(E - U/2) + v^{E)fF{E+ U/2) U [ } +
(2)
where fw{E) is the Fermi distribution function. For a non-zero collision integral, the distribution function should be found from the equation that stands for the kinetic equation in the specific geometry of Fig.l:
U(E) - $\E) = -av(viLHE) + vWm^KconiM
(3)
where a = SLvRte2 is determined by the resistance lit of the tunnel contacts and the volume SL of the wire.
The measured quantity is the differential conductance with respect to the probe bias Fpr0be across the probe tunnel contact. The probe contact can act as a spin-analyzer provided an additional magnetic field is also applied to a superconducting probe electrode.
There are two distinct cases schematically shown in Fig.2a and b: (i) with parallel and (ii) with anti-parallel magnetic fields in the superconducting leads. In the former case a non-equilibrium state with a nonzero total spin polarization
M = j dE[f^E)-f^E)] (4)
Fig.l. The setup: a normal-metal wire connected through the weak tunnel junctions with resistance Rt to the two superconducting leads, biased at potentials —U/2 and V/2
(a)
h
Ädif
/ / / / t
7 7 7 7 /
V
\
Fig.2. Two distinct cases: (a) with parallel magnetic fields in the superconducting leads and nonzero total spin polarization M. / 0, and (b) with anti-parallel magnetic fields and M. = 0. The typical form of the difference ftdif = hf(E) — h±(E) is shown in both cases
is created, while in the latter case M. = 0. The typical form of the difference fnf = ff(E) — f±(E) that follows from Eq. (2) is shown in Fig.2 in both cases. Note that the magnetic field sets a preferred direction of the z-axis for the spin projections.
The aim of this work is to consider the relaxation of such a spin-dependent distribution caused by the electron-electron interaction. For this we derive the collision integral /Ccoii in the approximation of the two-quasiparticle collisions in the case where both the spin-singlet and the spin-triplet channel of the interaction are
present. The detailed derivation of the collision integral due to the electron-electron interaction has been recently carried out in Refs. [8, 9]. However, the analysis has been limited to the case of the spin-independent distribution functions f(E), while we are going to focus on the relaxation of the difference f&t(E). The results of calculation of the collision integral for the spin-dependent distribution function have been also recently reported in Ref. [10]. However, the authors considered a limited class of distributions with a very particular spin-dependence equivalent to a shift in the energy
IlHCbMa b ?K3T<I) Tom 86 Btm.9-10 2007
Fig.3. Three possible processes in the collision of two spinful quasiparticles
fl'(E) = fi(E + 5E). Such type of dependence does not hold e.g. in the experimental setup of Figs.l, 2.
The main qualitative result of our analysis is that there are three different contributions in the collision integral. Two of them are also present if only the singlet channel of the interaction is considered, with only their amplitudes depending on the triplet channel interaction constant F. The third contribution corresponding to the spin-flip process is only present when the triplet channel of the interaction is taken into account. Its magnitude depends essentially on the conserving total spin M. [10]. However more importantly, it is singular for the non-equilibrium spin-dependent distribution with M. = 0 which naturally arises in the experimental situation (ii) of Fig.2b. The existence of such a singularity which never occurs if only the singlet channel of the interaction is present, is the main qualitative result of this work.
2. Three contributions to the collision integral. For a generic two-quasiparticle collision in a disordered metal in the absence of spin-orbit interaction and magnetic impurities two quantities are conserved: the total energy £ and the total spin M.. The latter conservation law allows only three possible processes (see Fig.3): (i) in the initial state the quasiparticles have the same spin projections (l/2)a which remain unchanged during the collision, (ii) in the initial state the quasiparticles have opposite spin projections which do not change during the collision, (iii) in the initial state the quasiparticles have opposite spin projections and the collision results in a spin-flip of both quasiparticles. Each process corresponds to a certain term in the collision integral that contains combinations of the type fm(E + w) fm' (E1) [1 - /fin(-B)] [1 - /«"' (E1 + u,)] - [1 -- fin(E + w)] [1 - fin'(E')] fn(E) fn'(E' + w) which for the processes (i)-(iii) take, respectively, the forms:
= J dE1 [—(1 — hateha,e+w)(h<r,e'+w — h<r,e') +
+ (ha,E+w — hcr,E){ 1 — hcr^E'hcr^E'+w)] , (5)
1^ = J dE1 [ —(1 — hcr,ehcr,e+u.')(h-cr,e'+u.' — h-crie') +
+ (ha,E+w — hcr,E){ 1 — h-a,E'h-<T,£'+«)] , (6)
— J dE' [—(1 — h<r,eh-a,e+w)(h—<r,e'+w ~ h<r,e') + + (h-a,e+w ~ hcr,e){ 1 — hate'h-<r,£'+«)] , (7)
where hatE = 1 - 2fa(E).
The collision integral JCcon(a,E) can be represented as follows:
/Ccon(<T,E) = ]T / (8)
where v is the DOS (per spin direction) of the normalmetal sample at the Fermi level. The quantities K°{w) describe the strength of relaxation due to the corresponding processes (the quantities corresponding to the singlet channel iff 2(w) = A"ij2(o;) does not depend on M. and hence on <j). We obtained the following expressions for them valid in the limit ppi 1 (pf is the Fermi momentum, I is the elastic scattering length) and for the diffusive quasiparticle dynamics:
FY
(2£>q-
q
(D q
2\2
*»(«) = vE
1 I u2
4 + {2£>q'^
V^ W2 + (Dq2)2 (1 + F)2
, 0) (10)
(W
w = vE
F2
q
{Dq2)2 +
(H)
In Eqs. (9)-(ll) by F (F = corresponds to the Stoner instability) we denoted the Fermi-liquid constant corresponding to the triplet channel of the electron-electron interaction. The summation over q can be replaced by integration
V
ddq (2ttF
in the limit w Et h = D/L2 (D is the diffusion coefficient, L is the length of the disordered sample) which will be considered below.
IIncbMa b JK3T<t tom 86 bbin.9-10 2007
Note that at small F ■C 1 we obtain up to the linear in F terms: K?, = 0 and
(D q2
1±
2 F
CD?
(12)
In this limit the spin-spin interaction in the triplet channel results in only a small (and opposite in sign for the parallel and anti-parallel spins of the two quasiparti-cles in the initial state) change in the amplitudes of the processes (i) and (ii) which are dominated by the singlet channel of the electron-electron interaction. Note that under the restrictions on the form of the spin-dependence
of h,jtE adopted in Ref. [10] the combinations I^ and
¡2)
' appeared to be identical. This is why the result of Ref. [10] contained
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.