∼
Meissner-like Effect on Normal-superfluid Interface of Imbalanced Fermi Gas
N. Ebrahimian∗ and M. Mehrafarin
Physics Department, Amirkabir University of Technology, Tehran 15914, Iran (Received 5 June 2012, in final form 5 November 2012)
We examine the N-SF interface of a polarized Fermi gas with two spin species a and b, in the presence of a weak external magnetic field. In our analysis we shall, therefore, consider the possibility of the Meissner effect too. We use perturbation theory to solve the Bogoliubov equations and obtain the wave functions. We consider the various scattering regions of the BCS regime and analytically obtain the transmission coefficients and the heat conductivity across the interface. We describe how the heat conductivity is affected by the Meissner effect and the species imbalance. It suffices to remark that the leading order term in transmission coefficients are independent of energy E. Also the additional heat conductivity is found to be proportional toλ2 (λis penetration depth). The corresponding graphs is also plotted and discussed.
PACS numbers: 03.75.Hh, 03.75.Ss, 68.03.Cd
Keywords: Imbalanced Fermi gas, Normal-superfluid interface, Heat conductivity, Meissner effect DOI: 10.3938/jkps.62.2218
I. INTRODUCTION
Much of the interest in ultra-cold Fermi gases comes from their amazing tunability. In balanced mixtures of fermions, this tunability is exploited to study the crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) super- fluid [1]. In the case of imbalanced mixtures of spin- polarized Fermi gases, since the BCS-phase cannot sup- port polarization at zero temperature, one typically ex- pects, in a harmonic trap, phase separation into an un- polarized superfluid core, surrounded by the polarized normal Fermi gases. Hence much research has been done about this subject [2–13]. One of the interesting results in this connection is the appearance of a temperature difference between the two phases as a consequence of the blockage of energy transfer across the N-SF inter- face. This blockage is due to a SF gap, which causes low-energy normal particles to be reflected from the in- terface. By studying particle scattering of the inter- face, the heat conductivity has been also calculated [14–
16]. In Ref. 14, heat conductivity of normal-superfluid interface of imbalanced Fermi gas for mass-symmetric case at fixed the average chemical potential was theo- retically calculated in absence of any magnetic field. In Ref. 15, in the mass symmetric case, linear response of heat conductivity of normal-superfluid interface of a po- larized Fermi gas to orbital magnetic field was investi- gated and dependence of heat conductivity versus tem- perature across the interface, the strength of magnetic
∗E-mail: [email protected]
field, and the average and the species-imbalance chem- ical potentials were determined. In Ref. 16, by consid- ering a mass-asymmetric polarized Fermi system in the presence of Hartree-Fock (HF) potentials and in absence of the orbital magnetic field, heat conductivity was in- vestigated on relevant physical parameters. In this arti- cle, we note that in the presence of the weak magnetic field, Meissner effect may occur at the N-SF interface.
The occurrence, like in superconductors, depends on the sample size among other factors [17]. In our analysis we shall, therefore, consider the possibility of the Meissner effect. We use Bogoliubov-de Gennes (BdG) or Blonder- Tinkham-Klapwijk (BTK) equations [18–20]. We use perturbation theory to solve the Bogoliubov equations and obtain the wave functions. We consider the various scattering regions of the BCS regime associated with dif- ferent energies of the incident particle and analytically obtain the transmission coefficients and the heat conduc- tivity across the interface.
II. PERTURBED BOGOLIUBOV EQUATION We consider a Fermi gas consisting of two imbalanced species aand b, with masses m, and chemical potential µi (i = a, b). The Bogoliubov equations of the system [18]
Haua,k+ ∆νb,k=Ekua,k
∆∗ua,k−Hb∗νb,k =Ekνb,k (1) Where, the Hamiltonian,Ha,b, of the system in the pres- ence of the vector potentialAis (neglecting the coupling -2218-
of the additional weak magnetic field with spin) Ha,b =−(∇ −ieA)2
2m −(µ)±(h) (2)
where µ = µa+µb
2
and h = µa−µb
2
are the average and the species-imbalance chemical potentials, respec- tively. Denoting the particle-like and hole-like solutions of the Bogoliubov equations are (using the first order perturbation theory)
Ui=u(0)i +u(1)i , νi =νi(0)+νi(1) (3)
whereu(0)i andνi(0)are the solutions to the unperturbed Bogoliubov equations. We use the approximation that, the pair potential, ∆, is independent of r. To proceed, let us take the N-SF interface to be in thex= 0 plane.
From the unperturbed Bogoliubov equations we obtain k(h) and k(p,h), where q = p, h refers to particle, hole [15]. Introducing the superscript s(n) for the solutions in the SF(N) phase, we expandu(n/s)i , ν(n/s)i in terms of the unperturbed eigenfunctions. Thus, for theN and SF phase, we write [15]
u(n)ak(r) =
σ=±Uk(p)σ φσk(p)(r) + perturbation terms(PT) νbk(n)(r) =
σ=±Vk(h)σ φσk(h)(r) + perturbation terms(PT) (4)
⎧
⎨
⎩
u(S)ak(0)(r) =
σ=±Ukσ(q)φσ(q)k (r), νbk(S)(0)(r) =
σ=±Vkσ(q)φσ(q)k (r) u(S)ak(1)(r) =
k′,q,σaσ(q)kk′ φσ(q)k′ (r), νbk(S)(1)(r) =
k′,q,σbσ(q)kk′ φσ(q)k′ (r)
(5)
where
φ±(q)k (r) = exp i(k
·r±k(q)x)
, φ±k(q)(r) = exp i(k
·r±k(q)x)
(6) Vkσ(p)=BUkσ(p), Vkσ(h)=BUkσ(h) , B=(E+h−
(E+h)2−∆2)
∆ (7)
It is noted that k
(k(g), k(q)) is the component of the wave vector k parallel (perpendicular) to the interface.
Substituting (5) into (1), we obtain aσ(q)kk′ and bσ(q)kk′ in terms of the amplitudesU andV. These amplitudes are to be determined by matching the wave functions and their derivatives atx= 0, of course [15].
III. DIFFERENT SCATTERING REGIONS Consider an incoming a-particle from theN side with energyE. Because there is no correction to the energy to lowest order inA, in view of solutions of the Bogoliubov equations, there are different scattering regimes involved in the E−ξ(p) plane where ξ(p) = k(p)2 /2m [14]. We concentrate on two regions:
1) For ∆< ξ(p)<2∆, particle-like and hole-like exci- tations both occur in theSF side but Andreev reflection is forbidden.
2) For 2∆ < ξ(p) <∆ +µ we have particle-like and
hole-like excitations as well as normal and Andreev re- flections.
Because in the other regions the particle has in suf- ficient energy to excite the SF side, the transmission coefficients in these regions vanish. Moreover, our focus is on energies slightly above the transmission threshold (E ≈∆−h), because we are considering low tempera- tures.
IV. HEAT CONDUCTIVITY AND MEISSNER-LIKE EFFECT
We concentrate on regions (1) and (2). We proceed to obtainU andV by matching the wave functions and their derivatives. The transmission coefficient is given by W =jxT/jxI, where the superscriptsT andI refer to the transmitted and incident quasi-particle current densities, respectively. The general form ofj is [19]
j(r, t) = −i
2m[u∗a(r, t)∇ua(r, t)−ua(r, t)∇u∗a(r, t)−νb∗(r, t)∇νb(r, t) +νb(r, t)∇νb∗(r, t)]
−e
m[u∗a(r, t)ua(r, t) +νb∗(r, t)νb(r, t)]A (8)
Using (3) and (4) and
k′ → d3k′
(2π)3 =4πm2 dE′
dk′(p) we obtain, to first order in the perturbation W =W(0) + 1
k(p)
k′
ηkk(1)′(k′(h)−k(p)) +η(2)kk′(k′(p)−k(h)) +ηkk(3)′(k′(h)+k(h)) +η(4)kk′(k′(p)+k(p))
(9) where in region (1)
W(0)= 4k(p)k(p)|k(h)−k(h)|2(B−2−1) + 4k(h)k(p)|k(h)+k(p)|2(−B2+ 1)
|B(k(h)−k(p))(k(p)+k(h))−B−1(k(h)−k(h)(k(p)+k(p))|2 (10) and in region (2)
W(0)= 4k(p)k(p)|k(h)−k(h)|2(B−2−1) + 4k(h)k(p)(k(h)+k(p))2(−B2+ 1)
|B(k(h)−k(p))(k(p)+k(h))−B−1(k(h)−k(h)(k(p)+k(p))|2 (11) Also we have defined
ηkk(1)′ = Uk+(p)∗Uk−(h)
Uk(p)+2 (Ykk′ ′−Xkk′), ηkk(2)′ = Uk+(p)Uk−(h)∗
U+2k(p) (Ykk′−Xkk′ ′) ηkk(3)′ = U−(h)
2
k
Uk(p)+2 (B−2Ykk′ ′−Xkk′), η(4)kk′ =U+(p)
2
k
Uk(p)+2 (Ykk′−B2Xkk′ ′) (12) We consider a constant magnetic field with A = c′xˆj in the N side and anticipate the possibility of the Meissner effect in theSF side. Therefore, we writeA=ce−x/λˆjfor theSF side, wherecis constant coefficient (indicating the strength of external magnetic field) andλis the penetration depth. Equation (7) leads to
W+ = W+(0)+meλ2√
∆(∆−hs)
(µs−χ)(∆ +χ) 3√
2√χ(χ−
(∆ +χ)(−∆ +χ)) × (20−8√
2)∆ + 3√ 2(7
χ−∆−√χ)√
∆ + 4((−5 + 2√
2)χ+ (−17 + 8√ 2)
χ(χ−∆))
(13) W− = W−(0)+meλ2√
∆(∆−hs)
(µs−χ)(∆ +χ) 3√χ(√
χ−∆ +√
χ+ ∆)2 × (8−10√
2)∆ + 3(7
χ−∆ +√χ)√
∆ + 2((−4 + 5√
2)χ+ 2(−16−17√ 2)
χ(χ−∆)
(14)
where χ = ξ(p)−∆ and W+ and (W−) refer to region (1) and (2).
It suffices to remark that the leading order term inW±
are now independent of E (depending only on χ), and not proportional toEas in the previous case in Ref. 15.
However, only a tiny portion of region (1) contributes to W+. On the other hand, the heat conductivity is given by [15]
κ= m 4π2
∂
∂T
q=p,h
dξq
dEEf(E)W(E, ξq) (15)
where f(E) is Fermi-Dirac distribution, which, for suf- ficiently low temperatures reduces to f(E) ∝ e−Tm/T where 32Tm= ∆−h[16].
Using (13) we find κ=κ(0)+κ(1), where κ(0) is the heat conductivity in the absence of the external field and κ(1)is the additional term due to the vector potential in the SF side. The effect of the vector potential in the N side is of second order and, therefore, negligible. Our final expression for the additional heat conductivity is
k(1) = meλ2c
12π2 (µ+ ∆)(∆−h)4 −
(64−68√
2)∆ log((1 +√ 3)√
∆) + (−4 + 5√
2)∆ log(∆)
−6√
2∆ tan−1 1
√2
+ (34√
6−32√
3 + 6)∆
+ 42√
2∆−42 µ∆
+2 (3√
∆ + (−4 + 5√ 2)
µ−∆ + (−16 + 17√ 2)
µ+ ∆) µ−∆
−3√
2∆ tan−1 √
µ−∆
√2∆
+ (−4 + 5√
2)∆ log(µ−2∆) + (32−34√ 2)
√
∆e−(∆−h)/T
√T (16)
Fig. 1. (Color online) Additional heat conductivity versus c across the interface in BCS regime for various temperatures.
The additional heat conductivity is found to be propor- tional toλ2, which is very small.
V. CONCLUSION AND DISCUSSION We obtained the transmission coefficients and the heat conductivity across the interface by considering the pos- sibility of occurrence of the Meissner-like effect. The leading order term in transmission coefficients are now independent of energy E (it is noted that in the previ- ous case given in Ref. 15, the transmission coefficients are proportional to E). Also, only a tiny portion of region (1) with ∆ < ξ(p) < 2∆ contributes to the cal- culation of transmission coefficient. Also the additional heat conductivity is found to be proportional toλ2. The corresponding graphs are in Figs. 1 and 2.
Figure 1 shows the additional heat conductivity versus c across the interface for various temperatures (κN = 2mµT /π2, heat conductivity in the normal phase). As seen, at fixedc, the larger the value ofT /TF, the larger is the heat conductivity. Also, in the specified temperature range (T /TF > 0.03), the additional heat conductivity does not change significantly withc.
Figure 2 shows the dependence ofκ(1)/k2Fλ2κN onµ at fixed temperature for various values ofT /TF. As seen, κ(1)/k2Fλ2κN decreases as the temperature are raised.
Fig. 2. (Color online) Additional heat conductivity versus average chemical potential across the interface in BCS regime forc= 0.01.
In summary, we obtained heat conductivity of normal- superfluid interface of imbalanced Fermi gas in the pres- ence of Meissner-like effect for mass-symmetric case and also dependence of heat conductivity to the physical parameters such as the average chemical potential and strength of magnetic field was determined. It was shown that heat conductivity of interface is proportional to square of penetration depth of magnetic field. Up to now, there is no any experimental data for heat conduc- tivity of normal-superfluid interface, when the system is subject to an additional weak magnetic field. Now, we want to compare the results of this paper with those of Ref. 14. It should be noted that in Ref. 14, heat conduc- tivity of normal-superfluid interface of imbalanced Fermi gas for mass-symmetric case at fixed the average chem- ical potential was theoretically calculated in absence of any magnetic field. The obtained theoretical results in this paper in comparison with the obtained theoretical results in Ref. 14 show that calculated additional heat conductivity due to Meissner-like effect is very small than the heat conductivity without considering Meissner-like effect.
Here, we want to compare the obtained results from this paper with our obtained results in Ref. 15 that is for mass-symmetric case. First, it should be noted that there is an essential difference between the present paper and our previous paper (refer to Ref. 15). This discrep-
ancy is due to different kinds of additional weak magnetic fields applied to the system (that causes to have different results about additional heat conductivity). In Ref. 15, we consider a particular additional weak magnetic field without the occurrence of Meissner-like effect. In this paper, we consider an additional weak magnetic field corresponding to occurrence of Meissner-like effect and thereby magnetic field in superfluid phase has exponen- tial form with respect to coordinate space and magnetic field in normal phase is constant. One can see that heat conductivity by considering Meissner-like effect (which calculated in this paper) is very small than heat conduc- tivity without considering Meissner-like effect [15], due to depending on penetration depth.
REFERENCES
[1] G. B. Patridge, W. Li, R. I. Kamar, Y. A. Liao and R.
G. Hulet, Science311, 503 (2006).
[2] M. W. Zwerlein, A. Schirotzek, C. H. Schunck and W.
Ketterle, Science311, 492 (2006).
[3] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek and W. Ketterle, Phys. Rev. Lett. 97, 9030401 (2006);
G. B. Patridge, W. Li, Y. A. Liao, R. G. Hulet, M. Haque and H. T. C. Stoof, Phys. Rev. Lett.97, 190407 (2006).
[4] W. V. Liu and F. Wilczek, Phys. Rev. Lett.90, 047002 (2003).
[5] P. F. Bedaque, H. Caldas and G. Rupak, Phys. Rev.
Lett.91, 247002 (2003).
[6] H. Caldas, Phys. Rev. A69, 063602 (2004).
[7] J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 (2005).
[8] T. Mizushima, K. Machida and M. Ichioka, Phys. Rev.
Lett.94, 060404 (2005).
[9] F. Chevy, Phys. Rev. A96, 130401 (2006).
[10] M. Haque and H. T. C. Stoof, Phys. Rev. A74, 011602 (2006).
[11] T. N. De Silva and E. J. Mueller, Phys. Rev. A 73, 051602 (2006).
[12] D. T. Son and M. A. Stephanov, Phys. Rev. A74, 013614 (2006).
[13] D. E. Sheehy and L. Radzihovsky, Ann. Phys. (NY)322, 8 (2007).
[14] B. V. Schaeybroeck and A. Lazarides, Phys. Rev. Lett.
98, 170402 (2007).
[15] N. Ebrahimian, M. Mehrafarin and R. Afzali, Physica B 407, 140 (2012).
[16] N. Ebrahimian, M. Mehrafarin and R. Afzali, Physica C 480, 49 (2012).
[17] J. V. Delft, Ann. Phys.10, 219 (2001).
[18] P. G. de Gennes,Superconductivity of Metals and Alloys (Addison-Wesley, New York, 1966).
[19] J. B. Ketterson and S. N. Song,Superconductivity(Cam- bridge University Press, U.K., 1995).
[20] G. E. Blonder, M. Tinkham and T. M. Klapwijk, Phys.
Rev. B25, 4515 (1982).
