A Singlet-pairing superconductor is always also a super-spin-current-conductor
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov A Singlet-pairing superconductor is always also a super-spin-current-conductor
Chia-Ren Hu ∗ Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA (Dated: November 12, 2007)It is shown that, in a conductor carrying a moderate spin current, singlet pairing can still takeplace almost as effectively as without a spin current. The system will still be fully gapped forall spin-up and -down single-particle excitations, with no depaired electrons. All thermodynamicproperties will be practically the same as without the spin current. All universality relations andlaws of corresponding states found in the theory of Bardeen, Cooper, and Schrieffer should remainvalid. Thus a singlet-pairing superconductor can always carry a persistent and dissipation-less spin-current. A heuristic argument to support this conclusion is also given, as well as two possibleexperimental tests of this prediction.
The Bardeen, Cooper, Schrieffer (BCS) theory ofsuperconductivity[1] is often naively introduced as Bose-Einstein condensation (BEC) of bound pairs of electronsin the spin-singlet channel, which is represented by anantisymmetric spin wave-function of one spin-up electronand one spin-down electron. Breaking such a bound pairwould cost twice of an energy gap, so it seems impossi-ble for the two electrons to go separate ways withoutfirst obtaining this energy. If this naive picture werecorrect, it would be inconceivable for the condensate ofa singlet-pairing superconductor (SPSC) to be able tocarry a dc spin current, which is defined as the spin-up and -down electrons flowing in opposite directions(relative to some spin quantization axis), with no netcharge flow. Fortunately the BCS state is much moresubtle than that. Thus, as an important difference be-tween a singlet-pairing BCS state and a BEC state oftightly bound, spin-singlet fermion pairs, we show belowthat in a conductor carrying a moderate spin current,spin-singlet Cooper pairing[2] of electrons can still takeplace almost as effectively as without the spin current.An energy gap still opens up on both the spin-up- and-down Fermi surfaces, which are now centered at ± q / n dancingcouples arranged in a big circle with each couple circlinga small circle at diametrically opposite positions like abound state of two electrons. If each couple is also ad-vancing along the big circle, it would be the analog of asuper(-charge-)current (SChC). If all dancers now agree that, in every five minutes, say, changing partners coher-ently should take place, with each man stepping forwardand each woman stepping backward, along the big circle,until new dancing partners can be formed, one then findsthat a new kind of flow is occurring in the floor, with allmen advancing in the counterclockwise direction, and allwomen in the clockwise direction. This is an analog ofa SpC, with a man (woman) the analog of a spin-up (-down) electron. If the intra-pair distance were smallerthan half of the the inter-pair distance, the pairs wouldhave to be broken before new pairs can be formed. Inthe BCS state of a SC, however, the intra-pair distance— which is of the order of the coherence length [2] ξ — is much larger than the inter-pair distance, so thatCooper-pair orbits are in strong overlap, allowing thispartner-changing process to take place without the needto break the pairs, and thus a SSpC can flow in the sys-tem. This partner-changing process would be drasticallysuppressed in a BEC state of tightly bound pairs, wherethe intra-pair distance is much smaller than the inter-pairdistance, although quantum mechanical wave-functionsdo not cut off abruptly, so the probability that this pro-cess can happen is never zero. Thus we surmise that onlyan extremely weak SSpC can exist in such a BEC state,to the extend of being negligible.For a simple theory, we begin with the second-quantized BCS Hamiltonian: K ≡ H − µN = X k ,s ξ k ˆ c † k ,s ˆ c k ,s − ( λ/ X k , k , q ,s ,s ˆ c † k + q ,s ˆ c † k − q ,s ˆ c k ,s ˆ c k ,s , (1)where Ω the total volume of the system, λ > ξ k ≡ ¯ h k / m ∗ − µ , with m ∗ the electron effective mass, and µ the chemical potential.Adding to it the Zeeman Hamiltonian: H = − h X k ,s,s ′ ˆ c † k ,s σ zss ′ ˆ c k ,s ′ , (2)(where σ z is the third Pauli matrix, and h is the elec-tron magnetic moment times an external magnetic fieldin the z direction, and can be viewed as a Lagrange mul-tiplier,) would favor a spin imbalance along z in the spinspace. Similarly, to favor a ChC, one can introduce avector Lagrange multiplier v ch , whose direction definesthe direction of the charge flow, and add to H : H = − v ch · X k ,s ¯ h k ˆ c † k ,s ˆ c k ,s +(1 / m v ch X k ,s ˆ c † k ,s ˆ c k ,s , (3)where the second term is a correction to µ . Thus to favora SpC in the system (with the same spin quantizationdirection), we introduce a vector Lagrange multiplier v sp ,and add to H : H = − v sp · X k ,s,s ′ ¯ h k ˆ c † k ,s σ zss ′ ˆ c k ,s ′ +(1 / m v sp X k ,s ˆ c † k ,s ˆ c k ,s . (4)For discussing a superconductor carrying no SChC,BCS neglected all interaction matrix elements that in-volves electron-pairs of non-zero momentum. Then amean-field approximation changes K to: K , M F = X k ,s ξ k ˆ c † k ,s ˆ c k ,s − ∆ ∗ X k ˆ c − k , ↓ ˆ c k , ↑ − ∆ X k ˆ c † k , ↑ ˆ c †− k , ↓ + (Ω /λ ) | ∆ | , (5)∆ ≡ ( λ/ Ω) X k << ˆ c − k , ↓ ˆ c k , ↑ >> T , (6)with << · · · >> T denoting grand-canonical ensembleaverage at temperature T . The Hamiltonian K , M F ≡K , M F + H can be diagonalized in the same way as K ,MF : by using a Bogoliubov-Valatin transformation:ˆ c k , ↑ = u ∗ q , k ˆ γ q , k , ↑ + v q , k ˆ γ † q , − k , ↓ (7)ˆ c †− k , ↓ = − v ∗ q , k ˆ γ q , k , ↑ + u q , k ˆ γ † q , − k , ↓ , (8)where q ≡ m ∗ v sp / ¯ h . The result is: K , M F = X k E q , k (ˆ γ † q , k , ↑ ˆ γ q , k , ↑ + ˆ γ † q , − k , ↓ ˆ γ q , − k , ↓ )+ X k ( ξ q , k − E q , k ) + (Ω /λ ) | ∆ q | , (9) u q , k = (cid:20)
12 (1 + ξ q , k E q , k ) (cid:21) (10) v q , k = ∆ q | ∆ q | (cid:20)
12 (1 − ξ q , k E q , k ) (cid:21) , (11) ξ q , k ≡ ξ ( k − q / , (12) E q , k = q ξ q , k + | ∆ q | , (13) where ∆ q is defined similar to ∆ of Eq. 6. That E q , k is positive definite for all q shows that a SpC does notinduce pair breaking in the system.Many physical quantities can now be evaluated as inthe original BCS theory: The difference between thesuperconducting and normal ground-state energies at T = 0 is:∆ E | T =0 ≡ << K ( S )3 ,MF >> T =0 − << K ( N )3 ,MF >> T =0 = − (1 / N (0)Ω | ∆ q , | , (14)where N (0) is the density of states per spin at the q = 0Fermi energy, and the usual weak-coupling approxima-tion has been assumed; ∆ q , ≡ ∆ q | T =0 . Let the cut-offenergies be still ± ¯ hω c for energy integration around the( q = 0) Fermi energy, the zero-temperature gap equation1 = (1 / N (0) λ Z − dµ (cid:20) ln 2(¯ hω c − ¯ hv F qµ )∆ q , +ln 2(¯ hω c + ¯ hv F qµ )∆ q , (cid:21) (15)gives: ∆ q , ≃ ∆ , exp " − (cid:18) ¯ hv F q/ hω c (cid:19) , (16)with ∆ , the T = 0 gap in the BCS theory. The gapequation at T = 0:1 = λ Ω X k E q , k tanh (cid:18) E q , k k B T (cid:19) (17)implies T c, q = T c, exp " − (cid:18) ¯ hv F q/ hω c (cid:19) , (18)where T c, is the transition temperature in the BCS the-ory. Thus the universality relation of the BCS theory isobtained: 2∆ q , k B T c, q = 2 πe C = 3 . , (19)with C = 0 .
577 the Euler constant. From Eq. 17, onecan derive the same BCS universality relation between | ∆ q / ∆ q , | and T /T c, q . The fractional specific-heat jumpat T c, q , or ∆ C/C n | T c, q , is also found to be that of theBCS theory. Other thermodynamic quantities have notbeen investigated, but we expect all universality relationsand laws of corresponding states in the BCS theory toremain valid. Evaluating the ensemble average of theSpC density: hh j sp ii T ≡ (1 / Ω) hh X k (¯ h k /m ∗ )(ˆ c † k , ↑ ˆ c k , ↑ + ˆ c †− k , ↓ ˆ c − k , ↓ ) ii T = 1Ω X k ¯ h k m ∗ (cid:18) − ξ q , k E q , k tanh E q , k k B T (cid:19) , (20)which, in the weak coupling approximation, is alwaysequal to its normal-state value n v sp at all T < T c, q (andabove), where n is the electron density, and v sp is thespin-current velocity. This finite SpC must be all SSpCat T = 0, because at T = 0 there is no normal fluidcomponent and the system is fully gapped, and it must beall normal SpC at T = T c, q , because we have a continuousphase transition to the normal state at this temperature.At any T between 0 and T c, q it must be partly superand partly normal, but we have not yet split apart thetwo components. The important point established hereis that a SSpC exists and that a SPSC can carry a SSpC.If the corresponding ChC problem K ,MF + H is solvedby the same approach, and the ensemble-average of theChC density is calculated, one would obtain hh j ii T = 2 e ¯ hm X k k f ( − ¯ h k m ∗ · q E k ) , (21)( q ≡ m ∗ v ch / ¯ h ) which vanishes at T = 0 and is equal tothe normal-state value ne v ch at T = T c . This ChC is allnormal at any T : The Hamiltonian K ,MF + H is just K ,MF in the frame moving with the velocity v ch . In thatframe the normal fluid component is stationary, whereasthe pair condensate is moving with velocity − v ch . Thus hh j ii T in that frame is all SChC. Translating back to thelab frame, we obtain only normal current, since the paircondensate is now stationary. Such a trick of going toa moving frame can not be played to the SpC problem.Our mean-field solution for K ,MF + H is also not amean-field solution of K ,MF alone, which might bothersome readers, as it seems that the “fictitious field” v sp cannot be realized in the laboratory. [4] Our reply is:We think that a superconducting sample subject to anexternal SpC (in the x direction, say) can be modeledas the Hamiltonian K together with the boundary con-dition ˆ ψ σ ( x + L ˆ e x ) = e − i ( q/ Lσ ˆ ψ σ ( x ) for the field op-erator ˆ ψ σ ( x ). If one then makes a gauge transforma-tion ˆ ψ σ ( x ) = e − i ( q/ xσ ˆ ψ ′ σ ( x ) , one would obtain K + H with v sp along x and the usual periodic boundary con-ditions, which we have solved. This gauge transforma-tion changes the eigen-wave-functions, but not the eigen-energies, nor ∆ q .We expect v sp to be comparable in magnitude to v ch , and is of the order of 1 mm/s or less. Thus¯ hv F ( q/ / ¯ hω c = m ∗ v F v sp / ¯ hω c appearing in Eqs. 16 and18 is of the order of 10 − or smaller. Thus ∆ q , / ∆ , and T c, q /T c, are exceedingly close to unity. We con-clude that all thermodynamic properties of a supercon-ductor are practically not affected by the presence of alaboratory-realizable SpC in the system.Increasing the magnitude of a SChC velocity can leadto two critical values: At the first, Landau critical ve-locity, depairing begins, and at the second one the orderparameter is suppressed to zero [5] (in three dimensions— In < hv F q > ¯ hω c , butits value would be so large that it is practically irrelevant.The Landau critical velocity is already quite large, butfor SChC, there exist lower critical velocities due to thecreation of phase-slip centers in one dimension and super-current vortices (or flux lines) in higher dimensions. Thedecay mechanisms for SSpC carried by a SPSC remainto be investigated,[7] but it appears that a SPSC can atleast carry comparable SSpC as SChC (after artificiallyintroducing a factor of | e | to the definition in Eq. 20 sothey can be compared). This is potentially useful to thedevelopment of SpC circuits and for spin injection, as itappears that SpC can not flow in most normal conductorsfor macroscopic distances.If the length L is that of the circumference of a ringsample, then ( q/ L = 2 nπ for an integer n , so thatˆ ψ σ ( x ) can satisfy the usual periodic boundary conditions.The SSpC is then quantized as a SChC in such a situ-ation. Thus line singularities in SSpC can also exist, asvortices (flux lines) in SChC, even as phase gradient ofthe pair wave function is no longer a relevant concept toSSpC.The ultimate confirmation of this prediction must bevia experimentation. We thus propose the following twoexperimental tests of our prediction:In the first proposed test, one should simply connecta SPSC to a battery except that at least one section ofthe connection wiring should be made of a half-metallicconductor (HMC) [8], allowing, say, only spin-up cur-rent. If the SPSC can carry a dc SSpC (referred to as“scenario A”), then it can also carry a dc super-spin-up-current, since an equal mixing of a SpC and a ChC isjust a pure spin-up current. No voltage drop will occuracross the SC, so the whole voltage drop will exist in theleads, driving the same amount of pure spin-up currentin the leads. On the other hand, if the SPSC can notcarry a dc SSpC (referred to as “scenario B”), then spin-up current cannot flow through the SC, and will initiallygenerate spin-up charge accumulations at the interfacesbetween the leads and the SC. Spin-neutral SChC willthen be induced in the SC, until the spin-up charge ac-cumulations at the said interfaces are all converted topure spin-density accumulations. The SChC will thenstop. The “SpC voltage” [9] established in the leads willthen cancel out the effect of the battery voltage on thespin-up charges in the lead, (but they will add up on spin-down charges which, however, can not flow in the leads,)so that no net electro-chemical-potential gradient will beacting on the spin-up electrons in the leads, and no netvoltage will be acting across the SC. The final state isa zero current state. The difference between these twoscenarios can be easily differentiated by a current meterin the circuit, or by the magnetic field generated by thespin-up current in the scenario A only. If the HMCs areonly nearly perfect, then some small ChC will always flowin the circuit, which becomes SChC in the SC, but theactual total current flowing in the circuit will be muchlarger in scenario A than in scenario B. In scenario Athe proper ratio of spin-up and -down currents will flowin the circuit, as if the SC does not exist in the circuit,whereas in scenario B, the final state is such that only asmall amount of spin-neutral ChC will flow in the circuit,which becomes pure SChC in the SC. Spin-density accu-mulations will exist at the said interfaces, to reduce theelectro-chemical-potential gradient of the spin-up elec-trons, and enhance that of the spin-down electrons, byso much so that spin-up and -down current densities inthe circuit can be equal. This is a self consistency prob-lem which will be investigated in the future.In the second proposed test, a SC strip is shaped intoa ring with a narrow gap, which is filled with a HMCallowing only spin-down current, with good contacts withboth ends of the SC strip to form a closed circuit of lowresistance. A battery with two HMC leads connected tothe two ends of the SC strip are used to send a pure spin-up current through the SC strip. In scenario B no ChC orSpC will flow through the SC. In scenario A the batterywill succeed in sending a spin-up current through the SCstrip, but to reduce the magnetic energy associated withthe magnetic field induced in the loop, we think that aspin-down current will be induced in the ring, so that inthe SC there is only a SSpC but no SChC. To distinguishbetween these two scenarios one can use either a currentmeter to detect the spin-up current in scenario A only, ora thermometer placed in contact with the HMC filling thegap of the ring in order to detect the heat generated bythe spin-down current in it in scenario A only. In eitherscenarios there is little magnetic field in the ring to bedetected. Imperfect HMCs can be similarly analyzed.Hirsch has pointed out [10] that if the BCS wave func-tion is altered to break parity but not the time-reversalsymmetry, it can then carry a dissipation-less SSpC. Thusthe ground-state wave-function obtained here as a SpC-carrying wave-function is not new, but as the mean-fieldground-state wave-function of a meaningful Hamiltonianassociated with a SpC in the system it is new. In addi-tion, the main purpose of Ref. [10] is to discuss when asymmetry-breaking SCing (i.e., pairing) condensate willcarry also a spontaneous SpC. Thus it has to find specialmodel Hamiltonians to achieve this goal. On the otherhand, the main purpose of the present work is to dis-cuss what happens if one attempts to send a pure SpCthrough an ordinary, SPSC. We conclude that all SPSCscan let a SpC flow through without dissipation, and theSCing properties of such a system is little affected by anymoderate SpC flowing in it. Furthermore, a quantized,persistent, dissipation-less SSpC can exist in any ringsample of any ordinary SPSC. No special Hamiltonian isneeded to achieve either goals. We have also proposed two currently feasible experimental tests of this predic-tion. Spin-orbit interaction has not yet been consideredin this work. This interaction led Hirsch [11] to proposea spin Meissner effect as a SC response to ionic electricfields.The concept of SSpCC (or SpC SC) has also been in-troduced in a very different context in spin-orbit-splithole-bands of semiconductors such as Si or GaAs, with-out invoking pairing at all. [12, 13] An off-diagonal trans-port coefficient was considered there, which is naturallydissipation-less. But when diagonal transport coefficientsare considered at the same time, the SpC may no longerbe dissipation-less. The SSpC introduced in the presentwork is associated with a diagonal transport coefficient— SpC conductivity: If a dc SpC can exist persistentlyand without dissipation, this diagonal transport coeffi-cient must diverge in the zero-frequency limit, just likethe dc (ChC-)conductivity in a SC. Thus the conceptof SSpCC introduced here is genuine. Its potential forpractical applications should be enormous.The author wishes to thank J. H. Ross, Jr., and YongChen for useful comments. The author acknowledgessome summer support from the Texas Center for Super-conductivity at the University of Houston in 2007. ∗ Electronic address: [email protected][1] J. Bardeen et al. , Phys. Rev. , 1175 (1957).[2] L. N. Cooper, Phys. Rev. , 1189 (1956)[3] An s -wave pairing is assumed here. Extention to non-s-wave singlet pairing is straightforward. We have alsonot yet investigated the possibility of a small admixtureof higher waves (such as d-wave) to an originally s-wavepairing state before a spin-current is introduced.[4] Actually, the spin-orbit interaction can create a veryweak v sp from an external electric field applied perpen-dicular to the spin axis and the desired direction for v sp .See C.-M. Ryu, Phys. Rev. Lett. , 968 (1996).[5] A. M. Zagoskin, Quantum Theory of Many-Body Sys-tems, Techniques and Applications , (Springer-Verlag,New-York,1998), Sec. 4.4.5.[6] D. G. Zhang et al. , Phys. Rev. B , 172508 (2004).[7] Triplet up-up and down-down pairings can easily carry aspin current, but phase-slip-centers or vortices can alsoexist independently in each spin species, leading to simi-lar magnitudes for the critical SpC and ChC velocities.[8] R. A. de Groot et al. , Phys. Rev. Lett. , 2024 (1983).See also, I. ˇZuti c et al. , Rev. Mod. Phys. , 323 (2004).Page 327 and the references cited therein.[9] The “SpC electric field” is the sum of a chemical potentialgradient that is opposite for spin-up and -down electrons,and, possibly, a Zeeman-field gradient.[10] J. E. Hirsch, Phys. Rev. B , 184521 (2005).[11] J. E. Hirsch, cond-mat.supr-con/0710.0876.[12] S. Murakami et al. , Science , 1348 (2003).[13] J. Sinova et al. , Phys. Rev. Lett.92