Diamagnetic susceptibility of spin-triplet ferromagnetic superconductors
H. Belich, Octavio D. Rodriguez Salmon, Diana V. Shopova, Dimo I. Uzunov
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Diamagnetic susceptibility of spin-tripletferromagnetic superconductors
H. Belich , , Octavio D. Rodriguez Salmon , Diana V. Shopova ,and Dimo I. Uzunov , † International Institute of Physics, Universidade Federal de Rio Grande do Norte,av. Odilon Gomes de Lima, 1722, 59078–400, Natal (RN), Brazil. Universidade Federal do Esp´ırito Santo (UFES), Departamento de F´ısica eQu´ımica, Av. Fernando Ferrari 514, Vit´oria, ES, CEP 29075-910, Brazil. Collective Phenomena Laboratory, G. Nadjakov Institute of Solid State Physics,Bulgarian Academy of Sciences, BG-1784 Sofia, Bulgaria. † Corresponding author: [email protected]
Key words : Ginzburg-Landau theory, thermodynamic property, superconductiv-ity, ferromagnetism, magnetization, phase diagram.
PACS : 74.20.De, 74.20.Rp
Abstract
We calculate the diamagnetic susceptibility in zero external magnetic fieldabove the phase transition from ferromagnetic phase to phase of coexistenceof ferromagnetic order and unconventional superconductivity. For this aim weuse generalized Ginzburg-Landau free energy of unconventional ferromagneticsuperconductor with spin-triplet electron pairing. A possible application ofthe result to some intermetallic compounds is briefly discussed.
In certain ferromagnetic unconventional superconductors the phase transition to su-perconductivity states occurs in the domain of stability of ferromagnetic phase (anexample is the itinerant ferromagnet UGe [1, 2, 3]). This seems to be a generalfeature of ferromagnetic superconductors with spin-triplet electron pairing [4, 5, 6](see also reviews [7, 8]). In such situation the thermodynamic properties near thephase transition line may differ from those known for the superconducting-to-normal1 T P c FM NFS T FS (P)T F (P) C Figure 1:
An illustration of the T − P phase diagram of UGe (details are omitted): N – normalphase, FM - ferromagnetic phase, FS - phase of coexistence of ferromagnetic order and supercon-ductivity, T F ( P ) and T F S ( P ) are the respective phase transition lines (solid line corresponds tosecond order phase transition, dashed lines correspond to first order phase transitions; 1 and 2 aretricritical points; P c ∼ . T F (0) ∼
53 K; T F S < .
22 K; the loop Cindicates a small domain (
T < . P ∼
16 GPa) where the shape of the phase diagram is notwell established by available experimental data. metal transition. We show this by using the example of diamagnetic susceptibilityabove the phase transition line of superconducting transition in spin-triplet ferro-magnetic superconductors. This is the line in the temperature-pressure ( T − P )phase diagram (Fig. 1), which separates the pure ferromagnetic phase (FM) and thephase (FS) of coexistence of ferromagnetic order and superconductivity. Here wepresent the result for diamagnetic susceptibility which follows from the Ginzburg–Landau theory for such type of superconductors [4, 5, 6]. We outline the main stepsof calculation of diamagnetic susceptibility in the Gaussian approximation. At theend we briefly discuss the possible application of our results to real systems.Following notations and results in Refs. [4, 7, 8], we present the GL free energy(fluctuation Hamiltonian) of spin-triplet ferromagnetic superconductors, which isessential in the present consideration, namely H = Z d x n ˆ H [ ψ ( x )] + ˆ H M [ ψ ( x )] o (1)by the energy densitiesˆ H = ~ m X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∇ − ie ~ c A (cid:19) ψ j (cid:12)(cid:12)(cid:12)(cid:12) + a s | ψ | (2)and 2 H M = iγ M · ( ψ × ψ ∗ ) + ρ M · ψ (3)In Eqs. (2)–(3), ψ ( x ) = { ψ j ( x ); j = 1 , , } is three dimensional vector field withcomplex components ψ j , which represents the superconducting order, M is the spon-taneous magnetization, the vector potential A is related to the magnetic inductionby B = H + 4 π M and obeys the Coulomb gauge ( ∇ · A = 0); a s = α s ( T − T s ), γ and ρ are positive material parameters, and 2 e and 2 m are the charge and the effec-tive mass of the electron Cooper pairs, respectively. We neglect the possible spatialanisotropy, which is usually represented in the gradient terms of the Hamiltonian H (see, e.g., Ref. [4, 7, 8]).Our task is to calculate the equilibrium free energy F = − β − ln Z Y x ∈ V D ψ ( x ) exp ( − β ˆ H ) , (4)in the volume V = L x L y L z of the superconductor and the diamagnetic susceptibilityper unit volume in zero external magnetic field, given by χ = [ − ∂ F/V ∂ H ] H =0 ; β − = k B T . In Eq. (4), the functional integral is taken over both real [ ℜ ψ ( x )] andimaginary [ ℑ ψ ( x )] parts of the complex field ψ ( x ), i.e., D ψ ( x ) ≡ d ℜ ψ ( x ) d ℑ ψ ( x ).Note that for temperatures near T F S ( P ) we can always set β ≈ β F S = 1 /k B T F S (see, e.g., [9]).As far as the behaviour in FM phase in a close vicinity of curve T F S ( P ) is of interestto our consideration, (see Fig. 1), the magnetization M has a magnitude | M | ≡ M ,given by M ( T, P ) = [ α f ( T − T F ) /b f ] / , i.e., the result from the standard Landautheory of ferromagnetic transitions with parameters a f = α f ( T − T F ) and b f [4] F m = a f M + b f M , (5)where a f = α f ( T − T F ), and b f >
0. Therefore, in our consideration M ( T, P ) is aknown thermodynamic quantity, which is established by the exhaustive thermody-namic analysis of the phases in the unconventional superconductor in [4].We choose the magnetization M = (0 , , M ) and the external magnetic field H =(0 , , H ) to lie along the ˆ z -axis. Then the first term in Eq. (3) takes the simpleform M ( ψ × ψ ∗ ) z = M ( ψ ψ ∗ − c.c. ). Under the supposition of uniform externalmagnetic field H , we take the gauge of the vector potential A as A = ( − By, , ψ j ( x ) by theseries 3 j ( x ) = 1 L x L z X q c j ( q ) ϕ j ( q, x ) (6)in terms of the eigenfunctions ϕ j ( q , x ) = 1( L x L z ) / e i ( k x + k z ) u n ( y ) (7)of the operator [ i ~ ∇ + (2 e/c ) A ] / m , corresponding to the eigenvalues E ( q ) = (cid:18) n + 12 (cid:19) ~ ω c + ~ m k z , (8)specified by the quantum number n = 0 , , . . . , ∞ , the wave vector components k x and k z , and the cyclotron frequency ω c = ( eB/mc ). In Eq. (6), the function u n ( y )is related to the Hermite polynomials H n ( y ) by u n ( y ) = A n e − ( y − y a H H n (cid:18) y − y a H (cid:19) , (9)where A − n = ( a B n n ! √ π ) / [13], y = a B k x , and a B = ( ~ c/ | e | B ) / ; B = | B | .Now the fluctuation Hamiltonian becomes H = P q ˆ H ( q ) withˆ H ( q ) = X j ˜ E ( q ) c j ( q ) c ∗ j ( q )+ iγ M [ c ( q ) c ∗ ( q ) − c.c.] , (10)where ˜ E ( q ) = E ( q ) + a s + ρM . (11)Applying the unitary transformation, c ( q ) = i √ − φ + ( q ) + φ − ( q )] (12a) c ( q ) = 1 √ φ + ( q ) + φ − ( q )] (12b)renders the fluctuation Hamiltonian as a sum of squares of field components c ( q ),and φ ± ( q ), and the free energy (4) can be calculated as usual Gaussian integralsover the same fields. 4ollowing approximations, justified in Ref. [12], we obtain the result FV = µB a / − + 1 a / + 1 a / ! , (13)where a ± ( γ ) = a s + ρM ± γ M, (14) a ≡ a ± (0), and µ = e k B T / π ~ c m / . Having in mind that ∂/∂H = ∂/∂B , thefluctuation diamagnetic susceptibility in Gaussian approximation takes the form χ ( T ) = − µ a / − + 1 a / + 1 a / ! , (15)In contrast to usual superconductors [12], where the contribution to the free energyfrom the diamagnetic currents is represented by a single term, here we have threeterms with labels 0, and ± which exactly correspond to the contributions of the fieldcomponents c , and φ ± , respectively.Now one should use known results [4, 5, 6, 7, 8] to analyze the singularities of freeenergy in a close vicinity (0 < T − T F S ≪ T F S ) to the phase transition curve T F S ( P )in the FM phase ( T F > T > T F S ), where M ( T, P ) = [ α f ( T F − T ) /b f ] / and, forsome real intermetallic compounds, for example, UGe , the condition ( T F − T F S ) ≫ ( T − T F S ) is satisfied. We shall briefly discuss the behaviour of the free energy (13)near the left-hand part of the curve T F S ( P ), where the phase transition FM-FS is ofsecond order. For this case the critical temperature T F S ( P ) is given in Refs. [5, 6].In the present notations T F S ( P ) is defined by the equation T F S = T s − ρα s ∆ + γ α s ∆ / , (16)where ∆ ≡ [ M ( T F S )] = α f ( T F − T F S ) /b f >
0. Expanding a ( T ), and a ± ( γ , T ) tofirst order in ( T − T F S ), one may easily check that a − ( T F S ) = 0 and a − ( T ) ≈ ˜ a − ( T − T F S ) , (17)where ˜ a − = α s − ρα f b f + γ α / f b f ( T F − T F S )] / , (18)5hile a and a + remain positive at T F S : a ( T F S ) = γ ∆ / , and a + ( T F S ) = 2 a ( T F S ).Therefore, only one of all three fluctuation diamagnetic contributions in Eqs. (13)and (15) will generate singularity of the free energy and the typical divergence ofsusceptibility. Keeping only the singular term in Eq. (13), we obtain that in a closevicinity of line T F S ( P ), where a − ≪ min( a , a + ), χ ( T ) = χ ( T − T F S ) / , (19a)( T > T
F S ), where the scaling amplitude χ is given by χ = − µ ˜ a / − . (19b)Note that a ± ( γ ) > a − is always positive for T F S ( P ) < T < T F ( P ).The formulae (19a) and (19b) are our main result. This scaling relation [9] is oftypical Gaussian type with an inverse root dependence on ( T − T F S ) whereas thescaling amplitude χ contains an essentially new information. Compared to knownresult for usual superconductors [12], the fluctuation diamagnetic susceptibility (19a)contains an extra factor (˜ a − ) − / , which depends on the material parameters of theunconventional ferromagnetic superconductor. The value of the new susceptibilityamplitude factor (˜ a − ) − / in Eq. (19b) should be taken at T F S ( P ) for any pressure P of interest. Thus in evaluating the parameter ˜ a − we may use the Eq. (16) for T F S ( P ).In some real systems the Eq. (18) can be simplified. For example, in UGe , T s ∼
0K [5, 6], T F ≫ T F S [1] and, therefore, one may use ˜ a − ≈ ( α s − ρα f / b f ). Thisresult is obtained with the help of Eq. (16). In itinerant ferromagnets with uniaxialanisotropy as, for example, UGe , both phases FM and FS may occur in two domainswith opposite magnetizations | M | = ± M . Here we have considered FM and FSwith M >
0. In the domains of FM, where
M <
0, the singular parts of the freeenergy and the susceptibility will be given by the terms, containing the quantity a + . Because of the invariance of the Eqs. (13) and (15) with respect to the change a ± → a ∓ , the results presented by Eqs. (13), (15), and (19a)–(19b) are valid in bothdomains of the FM and ψ -fluctuations corresponding to any domain ( M ≶
0) ofFS [4].We have used the Gaussian approximation, which is not valid in the critical re-gion [9] of anomalous fluctuations. However, the critical region of real ferromag-netic superconductors with spin-triplet electron pairing is often very narrow and,6ence, virtually of no interest. Therefore, the present results can be reliably usedin interpretation of experimental data for real itinerant ferromagnets, which exhibitlow-temperature spin-triplet superconductivity triggered by the ferromagnetic order.