Ginzburg-Landau-type theory of spin superconductivity

Abstract

Spin superconductivity is a recently proposed analogue of conventional charge superconductivity, in which spin currents flow without dissipation but charge currents do not. Here we derive a universal framework for describing the properties of a spin superconductor along similar lines to the Ginzburg-Landau equations that describe conventional superconductors, and show that the second of these Ginzburg-Landau-type equations is equivalent to a generalized London equation. Just as the GL equations enabled researchers to explore the behaviour of charge superconductors, our Ginzburg-Landau-type equations enable us to make a number of non-trivial predictions about the potential behaviour of putative spin superconductor. They enable us to calculate the super spin current in a spin superconductor under a uniform electric field or that induced by a thin conducting wire. Moreover, they allow us to predict the emergence of new phenomena, including the spin-current Josephson effect in which a time-independent magnetic field induces a time-dependent spin current.

ARTICLE

Received 5 Jun 2013 | Accepted 18 Nov 2013 | Published 16 Dec 2013

Spin superconductivity is a recently proposed analogue of conventional charge super-conductivity, in which spin currents ow without dissipation but charge currents do not. Here we derive a universal framework for describing the properties of a spin superconductor along similar lines to the GinzburgLandau equations that describe conventional superconductors, and show that the second of these GinzburgLandau-type equations is equivalent to a generalized London equation. Just as the GL equations enabled researchers to explore the behaviour of charge superconductors, our GinzburgLandau-type equations enable us to make a number of non-trivial predictions about the potential behaviour of putative spin superconductor. They enable us to calculate the super spin current in a spin superconductor under a uniform electric eld or that induced by a thin conducting wire. Moreover, they allow us to predict the emergence of new phenomena, including the spin-current Josephson effect in which a time-independent magnetic eld induces a time-dependent spin current.

DOI: 10.1038/ncomms3951

GinzburgLandau-type theory of spin superconductivity

Zhi-qiang Bao1, X.C. Xie2,3 & Qing-feng Sun2,3

1 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 2 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China. 3 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China. Correspondence and requests for materials should be addressed to X.C.X. (email: mailto:[email protected]

Web End [email protected] ) or to Q.-f.S. (email: mailto:[email protected]

Web End [email protected] ).

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a0s p E a0p E s, where a0

e2m 2c2 and p i hr.

Here we consider the Hamiltonian being 1

2m

p a0s rj2

p2 2m

a0p E s m

a20

Superconductivity was discovered about a century ago1. It has attracted worldwide attention owing to its fascinating properties and many applications. It is still one of the

central subjects in condensed matter physics. The mechanism of superconductivity can be understood by the BardeenCooper Schrieffer (BCS) theory2, which shows that electrons in the superconductor can form Cooper pairs and condense into the BCS ground state. Each Cooper pair contains electric charge 2e, and usually is spin singlet. Recently, a new quantum state, named the spin superconductor, was proposed3,4. The spin superconductor is the counterpart of the charge superconductor. It is formed by condensed bosons in sufciently low temperatures. The bosons are electrically neutral and their spins are non-zero. On one hand, the spin superconductor allows dissipationless ow of spin current and the spin resistance is zero. On the other hand, the charge current cannot ow through it and it is a charge insulator. Moreover, an electric Meissner effect against a spatial varying electric eld exists in the spin superconductor3. The spin superconductivity may exist in spin-polarized triplet exciton systems of the graphene310, some three-dimension ferromagnetic materials, BoseEinstein condensate of magnetic atoms, 3He superuidity and so on. Furthermore, the BCS-type theory, London-type equations and spin-current Josephson effect in the spin superconductor have also been presented3.

In the history of superconductivity, another well-known theory, named the GinzburgLandau (GL) theory11, has also had an important role. The GL theory gives the phenomenological description of the superconductivity. In this Article, we derive the GL-type equations of the spin superconductor. Moreover, we show that the second GL-type equation is the generalized London-type equation, and analyse the characteristic parameters of the spin superconductor. Furthermore, we use the GL-type equations to calculate the super spin current in a spin superconductor under a uniform electric eld and the super spin current where a thin charged wire is brought into the vicinity of a quasi two-dimensional (2D) spin superconductor. Finally, we use the GL-type equations to study the spin-current Josephson effect of the spin superconductor. We show the DC spin-current Josephson effect, the AC spin-current Josephson effect and the effect of an external electric eld.

ResultsThe GL-type equations of spin superconductors. To derive the GL-type equation, we should rst write out the free energy of the system. For the spin superconductor under an external electric eld E, the free energy can be represented as Fs R

d3rfs, where

2 E s2 with a0 m a0, in which the

rst and second terms are the kinetic energy and the spinorbit coupling, respectively. The third term m

a20

2 (E s)2 is very small

and is usually ignored. However, this term cannot be omitted in our derivation, because it contains the electric eld E and its variation is not small. In fact, if we start with the Dirac equation of an electron in a potential j and vector potential A, and take the non-relativistic limit, the canonical kinetic momentum can be written as13 pcanonical i:r ecA a0s rj, thus

H p

2

canonical

2m

1

2m (p a0s rj)2 for the charge-neutral carrier.

Similar with the vector potential A, the s rj term has a role of

a SU(2) gauge vector potential. In addition, if we start from the Hamiltonian H

1

2m (p a0s rj)2 m

. B, and use the Hamilton canonical equations, we can get the correct equation of motion of a magnetic moment in the external electric eld and magnetic eld. In above expression, the eld E has been written as

rj. This is reasonable because r E 0 is always tenable in

our system1416. The last term 12E0 (rj)2 in equation (1) is the

energy of the electric eld. Note that the electric eld E contains both the external eld and the eld induced by the super spin current. Next, we can use the variational method to obtain the GL-type equations.

If we minimize the free energy with respect to the complex conjugate of the wave function, we get equation (2):

ac b j c j2 c

1

2m i hr a0s rj

2c 0 2

In this derivation, we use the boundary condition:

i hr a0s rj n cr 0

3

If we minimize the free energy with respect to the electric potential j, we get equation (4):

r r

i ha02m c rc crc

m j c j 2 s rj s 4

where r E0r2j is the charge density.

Equations (2) and (4) are the centre results of this paper. They are the rst and second GL-type equations of the spin superconductor. This GL-type equations are universal to all spin superconductors and can also be used to study most of their properties and behaviours, including the electric Meissner effect, the spin-current Josephson effect, the proximity effect and their responses under an applied electromagnetic eld. To see the physical meaning of equation (4) clearly, we make a simple transformation. We use the standard spin-current denition15,1719

js Rec ^vc

i h2m crc c rc

a20

fs fn aT j cr j2

1

2 bT j cr j 4

1

2m

12 E0rj2 1 In equation (1), cr is the quasi-wave function of the spin

superconductor. fs and fn are densities of free energy of the superconducting state and normal state, respectively.

a(T)j cr j2 and 12 bT j cr j4 are the low-order terms in the

series expansion of the free energy fs, which are the same as the terms in the free energy of the charge superconductor11. The specic expressions of a(T) and b(T) can be obtained using the method proposed by Leggett12. Similar calculations manifest that they are also the same as the charge superconductor.

1

2m j i hr a0s rjcr j2 can be seen as the kinetic energy

and the spinorbit coupling term, where m* is the effective mass of the carriers and a0 is the coefcient of the spinorbit coupling.

The usual form of the spinorbit coupling term is

j i hr a0s rjcr j2

m j c j2 s rj 5

with the velocity operator ^v. Equation (4) is written as r r (js a0s). Here js is the super spin current. As the

spin is a vector, the spin current is a tensor product of the carrier current and the spin vector15,1719. However, the direction of the spin is xed in the spin superconductor; thus, we can use a vector js, which describes the carrier current to represent the super spin current. The GL equations in equations (2) and (4) are time independent and we can obtain r js 0 from them. It is the

special case of the continuity equation when the system is time independent. As the usual spin current, the super spin current js can generate an electric eld E in space1416, which is the same as

a0

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equation (7), we get

1

2m ^px

that generated by the electric dipole moment. js a0s is the

equivalent electric dipole moment Pe. Next, r r Pe can be

seen as the equivalent charge induced by the spin current. Thus, the second GL-type equation describes the equivalent charge induced by the super spin current. This is different from the second GL equation of the charge superconductor, which describes the super current itself11.

The relation between the GL-type and London-type equations. By substituting cr

nsr

2 E02 ^p2y ^p2z

c ac 8

The form of equation (8) is similar to the Schrdinger equation, and the left part can be regarded as ^

Hc. As ^px, ^py and ^pz are

commutative with ^

H, the eigenfunction can be chosen as eipx x= heipy y= heipz z= h. The magnitudes of px, py and pz are determined by the boundary conditions and external conditions. We can choose the proper condition that makes px py pz 0,

that is, we consider the case that the super spin current is zero when E 0. Under this case, we use c0 to represent the wave function,

which is uniform in the space. According to equation (5), we obtain js a0m j c j2 s E ha02m c20E0ex. It should be noted that the

super spin current js is non-zero in the existence of the electric eld E. The equivalent dipole moment is Pe / js a0s ha

20

ha0

p

eiyr into equation (5), we get

6

Here, nsr j cr j2 is the spin superuid carrier density.

Equation (6) can be seen as the generalized London-type equation. This is manifested as follows. If ns is independent of r, and take the curl of equation (6), we can get r js

a0nsm [(r E)s (s r)E)]. In addition, the derivation of the

second London-type equation in ref. 3 does not consider the effect of electric charge in the system. Thus, we can take r E 0,

then we obtain r js a0nsm (s r)E. It is the same as the second

London-type equation deduced in ref. 3.

The characteristic lengths of the spin superconductor. As we know, the charge superconductor has three characteristic parameters: the GL coherence length x, the penetration depth of the magnetic eld l and the GL parameter k. In the following, we will analyse the characteristic parameters of the spin superconductor from the GL-type equations. First, if we set E 0, we can nd that

the GL-type equations of the spin superconductor are the same as the charge superconductor20 when A 0. Therefore, the

denition of the GL-type coherence length is also the same, that is x2(T)

js

nsrm hry a0s rj

h22m aT. Second, in the charge superconductor,

the super current is jepA. It means that the rst derivative of je is proportional to the magnetic eld B. Considering the Maxwell equation r B m0je, we get r2BpB. From this equation, we

can see that the magnetic eld B decays exponentially from the surface to the interior. For the spin superconductor, however, the super spin current is jspE; it is proportional to the electric eld.

As a result, the penetration depth l is impossible to dene, so is the GL-type parameter k.

The super spin current induced by a uniform electric eld. As an application of the GL-type equations, we use them to calculate the super spin current when a spin superconductor confronts a uniform electric eld. When the eld is sufciently high, the spin superconductivity is destroyed. If we continuously decrease the eld, at a certain eld E Ec, spin superconductivity begins to

appear. The Ec is called the critical electric eld. In the region where the eld is very close to the critical eld Ec, the spin superconductivity is just beginning to appear; therefore, j c j is

very small. As a result, we can linearize the rst GL-type equation2022 as:

1

2m i hr a0s E2c ac 7

Suppose the surface of the spin superconductor lies in the xy plane and the uniform electric eld is parallel to the surface, pointing to the y axis direction. The electric eld can be written as

E E0ey, and we have s E h2 ez E0ey hE02 ex. Here we have

assumed that the direction of the spin s is xed at z direction by the external eld. By substituting s E hE02 ex into

2m c20E0ey;

thus, the equivalent charge is Q r Pe 0. Q 0 illustrates

that the spin superconductor does not screen the electric eld. This result is corresponding to the electric Meissner effect of the spin superconductor3, which says what the super spin current screens is the gradient of the electric eld rather than the electric eld itself.

We can compare it with the case in the charge super-conductor20. Suppose there is a uniform magnetic eld parallel to the surface of the superconductor. The momentum p should be replaced by p ec A. The magnetic eld B is the rst-order

differential of A; therefore, if B is uniform, A is a linear function of r. By substituting it into the rst GL-type equation of the charge superconductor, we get a potential energy that is a quadratic function of r. This equation is similar to the Schrdinger equation of the harmonic oscillator; thus, the solution is a local wave function. As a result, the super current is distributing near the surface. This super current can screen the magnetic eld, producing the Meissner effect in the charge superconductor. In the spin superconductor, however, the momentum is changed to p a0s E. When the electric eld is

uniform, and by substituting it into the rst GL-type equation, we get a constant potential. This potential cannot cause any nontrivial modication and the super spin current cannot be localized near the surface. In fact, the fundamental cause of this difference is that the electric charge is monopole but magnetic moment is dipole. It is the major asymmetry between electricity and magnetism. This cause the different Meissner effects for the charge superconductor and the spin superconductor.

The super spin current induced by a thin charged wire. Next, we calculate the super spin current when a quasi 2D spin superconductor confronts a eld induced by a thin charged wire. As shown in Fig. 1a, a thin charged wire is on the top of a quasi 2D spin superconductor. The spin superconductor is a thin lm and its thickness of the z direction is less than the GL-type coherent length x. Thus, the wave function is uniform in the z direction, which can be ignored. Furthermore, the thin lm is innite in the y direction, whereas the length in the x direction is nite. Suppose the charge density of the wire is r0, then the magnitude of the induced electric eld is E

r02pE0r. As shown

in Fig. 1, the electric eld can be written as E

r0 2pE0

2 ez. By substituting it into equation (7),

and considering equations (4) and (5), we obtain the super spin current js ha02m r02pE0

x z2 x

x z2 x

2

ex

z z2 x

2

ey, the equivalent dipole moment

Pepjs a0sB

x z2 x

2 ex and the equivalent charge Q

z2 x2

z2 x

2

2.

The variation of the electric eld induced by the super spin current along z direction, @E

induced

z

@z , is proportional to the equivalent charge

Q

z2 x2

z2 x

2 2. Note that

@Ez @z

x2 z2

z2 x

2 2, thus, the electric eld

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B1 B2

100

Q,jand zE z

(with no unit)

50

j

Q

zEz

O

r

0

z

Ex

Z y

x

x

50

Ez

E

100

2 1 0 1 2

E

x (m)

Figure 1 | The quasi 2D spin superconductor under a thin charged wire. (a) The schematic diagram for the device consisting of a thin charged wire and a thin lm of spin superconductor. The red arrows describe the ow direction of the super spin current. (b) The variation @zEz of the electric eld, the induced super spin current j and the equivalent charge versus x.

Figure 2 | The spin-current Josephson junction. (a) The schematic diagram for the device of Josephson junction (spin superconductor-insulator-spin superconductor junction). The magnetic elds imposed on both sides are unequal B1aB2. (b) The schematic diagram for the device showing the effect of electric eld on the spin-current Josephson effect.

induced by the super spin current cancels out @Ez@z. As a result, we can say that the super spin current screens the variation of the electric eld. This conclusion, which is according to the electric Meissner effect of the spin superconductor, is also clearly claried in Fig. 1b. It should be noted that the altitudes of the js, Q and @Ez@z in Fig. 1b are

meaningless, because we have ignored the coefcients. What Fig. 1b reects is the tendency that the super spin current screens the variation of the electric eld. We can also see that the ow direction of the super spin current in x40 is opposite to that in xo0. In x 0,

js 0.

Spin-current Josephson effect of the spin superconductor. We know that the Josephson effect is another highlight of the charge superconductor23. In the following, we use the GL-type equations to discuss the spin-current Josephson effect of the spin superconductor. The schematic diagram for the device is shown in Fig. 2. We suppose that the magnetic elds, used to polarize the spins of electrons, are the same on both sides of the junction. When E 0, the GL-type equations of the spin superconductor

are the same as the charge superconductor when A 0.

Therefore, a spin superconductor has the same DC spin-current Josephson effect as a charge superconductor23. The super spin current is j j0 sin g0, where g0 g1 g2. j0 is the Josephson

critical super spin current, and g1 and g2 are phases of the spin superconductors on both sides of the Josephson junction. Next, consider the AC spin-current Josephson effect. As shown in Fig. 2a, the magnetic elds on both sides are different, B1 B1ez,

B2 B2ez, B1aB2. The super spin current ows from side 1 to

side 2, and the change of phase is g g0 a0 h R

2 1

s E exdx.

2 j c j4 Z d3rbc2c dc 10

For the fourth term, we get

d

Z d3r

1

2m j i hr a0s rjcr j2

Z d3r

Thus, we have @g

@t

a0 h R

2 1

s @E@t exdx. By substituting

@E

@t

1m0E0 r B into the above equation, we get @g@t

a02m0E0 B2 B1

and g g0

a02m0E0 B2 B1t. Therefore, the super spin current is

j j0 sin(g0 o0t), where o0

e4m B2 B1. It can be seen that the super spin current is an alternating current. If the difference between the magnetic elds on both sides B2 B1 is

0.01T, we can estimate the frequency o0, and its order of magnitude is 1 GHz. On the basis of this AC spin-current Josephson effect, we can very sensitively detect the spatial variance of the magnetic eld by measuring the the frequency of the spin current. A difference of the magnetic elds, 2 10 7

Gauss, can induce the spin current with its frequency around 1 Hz, which can be easily measured using the present technology. Next, we consider the effect of an external electric eld shown in Fig. 2b. The uniform electric eld is along the lateral direction of the Josephson junction and is written as E E0ey. The phase is

g g0 a02 R

ez E0ey exdx g0 a02 E0d, where d is the width

of the junction. Thus, the spin current is j j0 sina02 E0d g0.

It is changed by the sine rule according to the electric eld.

DiscussionIn conclusion, we have derived the GL-type equations of the spin superconductor. We show that the second GL-type equation is the generalized London-type equation. In addition, we analyse some characteristic parameters of the spin superconductor by using the GL-type equations. Moreover, as the applications of the GL-type equations, we use them to calculate the super spin current in a spin superconductor under a uniform electric eld and the super spin current where a thin charged wire is brought into the vicinity of a quasi 2D spin superconductor. Our result veries the electric Meissner effect of the spin superconductor. We also discuss the spin-current Josephson effect of the spin superconductor from the GL-type equations, including the DC spin-current Josephson effect, the AC spin-current Josephson effect and the effect of an external electric eld. To see the differences between the charge superconductor and the spin superconductor clearly, we summarize them in Table 1.

Methods

The derivation of the GL-type equations. At rst, we minimize the free energy shown in equation (1) with respect to the complex conjugate of the wave function c . For the second term of equation (1), we have

d

Z d3ra j c j2 Z d3racdc 9

For the third term of equation (1), we have

d

Z d3r

b

a02m0E0 B2 B1

i h2m rdc i hr a0s rj

c

f g

1

2m

Z d3ra0s rj i hr a0s rj

c

f gdc

11

Note that

i h 2m

Z d3rrdc i hr a0s rj

c

f g

i h 2m

I dS dc i hr a0s rj

c

f g

i h 2m

Z d3rdc r i hr a0s rj

c

f g

12

and considering equations (9), (10), (11) and (12) together, we can obtain

ac b j c j2 c

1

2m i hr a0s rj

2c 0 13

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Table 1 | The comparison of the charge and spin superconductors.

The charge superconductor The spin superconductor Current carrier Charge Spin

Basic phenomenon Zero electric resistance Zero spin resistanceMeissner effect Against the magnetic eld Against the variation of the electric eld

The rst GL equations 1

2m

i hr ec A2c ac b j c j2 c 0

1

2m

i hr a0s rj 2c ac b j c j2 c 0 The second GL equations je i he2m c rc crc e mc j c j2 A r r (js a0s), where

js

i h 2m

crc c rc am j c j2 s rj

ddt js / s rB r js / s rE

London equations d

dt jepE

r jepB

GL coherence length x x2T

h2m aT x2T

h2m aT

Penetration depth l l2 m c4pne Difcult to dene GL parameter k k lx Difcult to dene

DC Josephson effect DC electric current through the Josephson junction DC spin current through the Josephson junction AC Josephson effect The difference between the electric elds on bothsides of the junction induces the AC electric current

The difference between the magnetic elds on both sides of the junction induces the AC spin current

and

ncr 0 14 Equation (13) is the rst GL-type equation of the spin superconductor.

Equation (14) is the boundary condition for the rst GL-type equation, where the subscript n denotes the component perpendicular to the surface. On one hand, the boundary equation (14) can be viewed as the demand of the variational principle. The similar demand was adopted in the original paper written by Ginzburg and Landau11. On the other hand, the physical reason for choosing this boundary condition will be discussed shortly. If the wave function c satises equation (13), it is easily shown that the super spin current is source free24,25, that is, r js 0. In

addition, if the wave function c satisfy the equation (14) together, the spin superconductor sustains no net force.

Next, we minimize the free energy with respect to the electric potential j. For the fourth term of equation (1), we have

d

Z d3r

1

2m j i hr a0s rjcr j2

Z d3r

i hr a0s rj

choose the external condition of the total super spin current being zero), we get

h2 2m

@2@x2 fx

a20 h2

8m

r20

4p2E20

x2

z2 x2 2

fx afx 19

By multiplying equation (19) by f* and subtracting the product between the complex conjugate of equation (19) and f, we have @@x f @@x f f @@x f 0. By

considering the boundary condition i hr a0s E

ncr 0, we get

f @@x f f @@x f j

boundary

0. Therefore, f @@x f f @@x f 0 at any point. It is easy to get, from the second GL-type equation, jx 0 and jy ha2m

r 2pE

xz x .

Therefore, we have js ha2m

r 2pE

xz x ey, the equivalent dipole moment

Pe / js a0s

xz x ex, and the equivalent charge Q r Pe

z x z x .

References

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1

2m i ha0s rdjc rc

a20s rdj s rjcc

c:c:

Z d3r

i ha02m r c rc crEc s

dj

Z d3r

a20

m r s rjcc s

dj

15

For the fth term of equation (1), we get

d

Z d3r

1

2 E0rj2 E0 Z d3rr2j dj 16

Note that the total electric led contains the external eld and the eld induced by the super spin current. The external eld is xed; thus, its result of variation is zero. We only need to minimize the free energy with respect to the eld induced by the super spin current. Considerequations (15) and (16) together, we can obtain equation (17):

r r

i ha02m c rc crc

s

17

where r E0r2j. Equation (17) is the second GL-type equation of the spin

superconductor.

Solution of the super spin current under a thin charged wire. By substituting the electric eld E r2pE

xz x ex

a20

m j c j2 s rj

zz x ez into equation (7), we get

1

2m h2

ac

18

The left part of equation (18) can be regarded as ^

Hc and ^py is commutative

with ^

H. Therefore, the eigenfunction can be chosen as c eip y= hfx. By sub

stituting it into equation (18) and taking py 0 (the value of py is zero when we

@2@x2 c

@2@y2 c i h2a0

r0 2pE0

x z2 x

a20 h2

r20

4 4p2E20

2

@@y c

x2

z2 x2 2

c

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3951

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Acknowledgements

This work was nancially supported by NBRP of China (2012CB921303, 2009CB929100 and 2012CB821402) and NSF-China under grants numbers 11274364, 11074174 and 91221302.

Author contributions

Z.-q.B., X.C.X. and Q.-f.S. performed the calculations, discussed the results and wrote this manuscript together.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Bao, Z.-q. et al. GinzburgLandau-type theory of spin super-conductivity. Nat. Commun. 4:2951 doi: 10.1038/ncomms3951 (2013).

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