Mean Field and Ginzburg-Landau Analysis of Two-Band Superconductors

Transcription

1 Mean Field and Ginzburg-Landau Analysis of Two-Band Superconductors Ethan Lae Dated: July 8, 15 We outline how to develop a mean-field theory for a generic two-band superconductor, and then apply our analysis to the case of the DEG with an external Zeeman field and Rashba SOC. We also examine the two-band superconductor problem in terms of a Ginzburg-Landau formulation, where a few interesting features arise. Finally, we mae use of the PRL latex template so that we can pretend to be super famous. Mean-Field Formulation We ve seen in the last set of notes how we can diagonalize the Hamiltonian by introducing operators that act on the two subbands that occur in the problem. Throughout, we will let and µ label these subbands. The Hamiltonian in our problem is H H f + H int, where H f ξ µa a and the interaction term is written as H int µ g µ e ij θ a a ζ a a e ijµθ a µa µ where j and j µ are the dominant angular momentum channels on each band and we have used the shorthand ζ ξ µ. The entries of the matrix g µ are the strengths of the intra-band and inter-band interactions in the given angular momentum channel. We let g 11 g 1, g g, g 1 g 1 g 3. To begin our mean-field analysis, we define the quantities A e ij θ a a and then write e ij θ a a A + e ij θ a a A We then substitute these into the interaction Hamiltonian, dropping terms quadratic in the fluctuation as usual: H int g µ A µ e ij θ a a µ + A e ijµθ a µ a µ A A µ We define the parameter as a linear combination of the anomalous averages on each band: µ g µ A µ and so we can write the interaction Hamiltonian as H int e ij θ a a + h.c. g µ A A µ µ The next step is to reformulate the Hamiltonian in a Nambu spinor representation Ψ h a, a T, ζ e ij θ e ij θ ζ The diagonal components pic up H f, but we are left with an extra ζ. In textboos we usually don t have the factors of on the off-diagonal entries these are needed since the spinor Ψ only has operators from a single band, as opposed to containing a and operator lie in standard BCS theory. Adding an extra term to eliminate the extra factor of ζ, the Hamiltonian can be recast into a nice bilinear form as H 1 ζ + Ψ h Ψ g µ A A µ µ We perform a unitary transform to diagonalize the Hamiltonian, denoting the new operators by γ: u v γ a v u γ a The square of the parameter v u has the physical meaning of giving the probability of a pair state being occupied unoccupied in the BCS wave function for each band: Ω u v a a Ω Since we have written the Hamiltonian as a sum over each band, we can do the diagonalization for each band separately. The procedure for doing so is well nown, and we get H E γ γ + 1 ζ E g µ A A µ µ

2 where the dispersion of the Bogoliubons is E ζ + 4 Since we usually do not have the factors of in h, the clumsy factor of 4 in the dispersion is usually absent. For this reason we will focus on, and not, as the quantity of physical interest. Defining a whole new symbol for would be overill, and so we will simply write instead. The coefficients u and v are found by substituting the expressions for the a operators in terms of the γ operators into the mean-field Hamiltonian, and requiring that the resulting expression is diagonal in the γ operators. This is straightforward, if a bit lengthy. Using the restriction that u + v 1, we find that v 1 u 1 1 ζ E Although the phases of u, v, and are each arbitrary, they are related by the requirement that e ij θ u /v E ζ is real. Without loss of generality we can let v be real, and assign u and e ij θ equal phases. The Self-Consistent Equations We now want to begin to solve the self-consistent equations for the order parameters. Taing the adjoint of our earlier relation for a in terms of the gamma operators, we obtain a v γ + u γ. This allows us to write the A s in terms of the γ operators. We also go over to integrals, and so Since A d π e ij θ v γ + u γ v u e ij θ u γ v γ 1 ζ E e ij θ E A becomes, after dropping the averages of γ γ and its hermitian conjugate, A d π 1 fe E We can now write down the self-consistent equations for : µ g µ d π µ E µ 1 fe µ If we mae the transformation d π ωc ρ ω c π dθ π and remember the identity 1 fx tanhx/t, we can write the self-consistent equations in a more useful form as µ g µ µ ρ µ ωc Which at T is just µ ζ + tanh ζ + T µ ωc g µ µ ρ µ ζ + µ Additionally, we should mention that we have implicitly assumed that the cutoff frequency ω c is the same for each band. It s worth outlining a slightly different way of deriving the self-consistent equations. The idea is to find that minimize the energy of the diagonalized Hamiltonian. That is, we see to minimize the function E 1 ζ E g µ A A µ µ Assuming that g µ is nonsingular, we can write g µ A A µ 1 g 1 +g 1 det g µ g with the factor of 1/ coming from the step when replaced with. Minimizing E over the two order parameters gives us the following system of equations: E,1 det g g 1 g 3 1 E, det g g 1 g 3 1 Multiplying the first by g 1 and the second by g 3 and adding, as well as multiplying the first by g 3 and the second by g and adding gives the following set of equations g1 1 E,1 + g 3 E, g E, + g 3 1 E,1 Going to the continuum recovers the self-consistent equations derived earlier.

3 3 Now we need to find the condensation energy E gs E in terms of. Looing at the form of the diagonalized Hamiltonian we derived, the ground state energy is relative to µ E gs 1 ζ E g µ A A µ µ while the energy of the normal state is E and so E cond 1 < F g µ A A µ µ < F, ζ ζ E + 1 > F,E ζ µ > F g µ A A µ ζ E The first sum is handled in the usual manner: ωc ρ ζ + ζ > F,E ζ ρ ωc ωc + ω c + sinh 1 ω c / ρ 1/ + sinh 1 ω c / Recalling our earlier expression for µ g µa A µ, we have E cond ρ 1 + sinh 1 ω c / 1 g 1 +g 1 det g g The sensitivity of the condensation energy to the relative phases of the order parameters on each band is not surprising, since it appears as a Josephson term. In the applications we consider, g 3 <. However, we will also have det g <, which means that states where the relative gauge-invariant phase difference of the two order parameters is π are energetically favored. Having found the condensation energy, we d also lie to compute the critical temperature T c for the system. We define ν µ g µ ρ µ, which means that we can write the finite-temperature self-consistent equations derived earlier as with the notation det 1 ν ωc 1 ν 1 ν 1 ν + 1 ζ + tanh ζ + T Note that this result holds for an arbitrary relative phase between 1 and, and so evidently the critical temperature T c is independent of the relative phase of the two order parameters. This is interesting, since we ve already seen that the condensation energy does depend on their relative phase. If we tae the limit as, then w The integral is w dz tanh z, z ζ/t, w ω c /T z dz tanh z z w ln z tanh z w dz ln z cosh z ln w dz ln z cosh z ln w + ln 4eΓ π which wors for w 1. Now define the quantity L lnω c e Γ /πt c. Solving the determinant form of the selfconsistent equations gives L ν 11 + ν ± ν 11 ν + 4ν 1 ν 1 det ν When the off-diagonal components of ν µ vanish, we have two solutions to this equation, namely 1/ν 11 and 1/ν, which agrees with the standard one-band result. In this case, each order parameter goes to zero at a different temperature. When we have interband coupling however, 1 and go to zero at the same critical temperature T c seen by looing at the self-consistent equations. In this case, we find the solutions to the above equation for L, and then see which one gives the highest critical temperature. An equivalent way to compute the critical temperature which requires a bit of algebra to see is to find the most negative eigenvalue ν of the matrix ν µ, which then gives T c ω ce Γ π e 1/ ν, ν < By evaluating the self-consistent equations at T T c, we obtain an expression for the ratio of the two gaps: 1 ν ν ν 1

4 4 which will turn out to be useful during our Ginzburg- Landau analysis later on. Also of interest is the fact that a superconducting ground state can exist even if all interactions in the system are repulsive i.e. if ν µ >, µ. We can see that in order for this to happen, we require ν 11 ν < ν 1 ν 1 since we must have a negative eigenvalue of ν µ. Searching the literature reveals that this has been noticed before, as far bac as 1963! by Kondo The mechanism for this is that virtual pairs can be excited from one band say, the 1 band to another and bac, which gives an attractive interaction in the 1 band with matrix element g 3/g. If ν 11 < ν 1 ν 1 /ν then we can have g 1 < g 3/g, and a superconducting ground state forms. Cool! The final thing to note is that in the case of inter-band scattering only, we recover the same T c as in regular oneband BCS theory, but with an effective state density of ρ1 ρ. Application to the Rashba + Zeeman Problem Our primary goal here is to solve the Zeeman + Rashba problem considered in the last set of notes. I ll repeat the main result for the coupling constants g µ for convenience. We label the + smaller band with the index 1 and the band with the index, in order to use consistent notation. For band 1, we have j 1 1 : E ξ 1 Φ 1 1 { m j 1 1 : dɛ Λ 1 Φ 1 1 } Λ 1 Λ Φ 1 π whereas for the larger band, j 1 : j 1 : m π m π E ξ 1 Φ 1 1 { dɛ U χφ 1 + Λ Φ 1 } Λ 1 Λ Φ 1 1 E ξ Φ 1 { } dɛ U χφ 1 E ξ Φ 1 where we ve remembered that on the smaller + band, χ l l. Superscripts on Φ indicate bands, while subscripts indicate angular momentum channels. We ve just written χ by itself, with the understanding that it is taen on the band and in the l 1 angular momentum channel. The simplest non-interacting choice is j 1, j 1 1, for which g 1, g 3, g U χ. In this case, the self-consistent equation at T is simply 1 g ρ ωc ln + ζ ωc ω c sinh 1 ω c and so, assuming ρ g is small, we can approximate ω c sinh[1/ g ρ ] ω ce 1/ρg which is just the usual expression for the gap for the single-band case note that for us, g <. In this scenario, we have A /g, and so the expression for the condensation energy is just E cond ρ 1 + sinh 1 ω c / or, using the last expression for sinh 1 ω c /, g E cond ρ 4 The other case of interest for us is when the + band has j 1 1 and the band has j 1. In this state, we have additional intra-band repulsion terms and also non-vanishing Josephson coupling. We now have a set of two self-consistent equations, which read ωc 1 1 ρ 1 Λ 1 1 ζ ωc + ρ Λ1 Λ ζ + U ωc ρ χ + Λ ζ + ωc + 1 ρ 1 Λ1 Λ 1 ζ The integrals are easy, as they all integrate to sinh 1 ω c /. However, the resulting equations are best solved numerically due to the transcendental nature of the resulting expressions. Various analytic approximations are possible, but the numerical wor is easy enough, and so we will stic with it. After solving the self-consistent equations, we plug the values for the order parameters into our expression for the condensation energy of the system. We find that the interacting case has the lower condensation energy, due to our freedom in choosing a relative phase difference of π between the two order parameters. Additionally, we would lie to compare the critical temperature in each of the two cases. For the noninteracting case, we have T c,noint ω ce Γ π e 1/ ρu χ

5 5 whereas in the interacting case, T c,int ω ce Γ π e 1/ ν where as before, ν is the most negative eigenvalue of the matrix ν µ. Note that a state will have a superconducting solution if ν µ has a negative eigenvalue. In the interacting case considered above det ν µ is always <, and so a superconducting solution is always possible, no matter what the values of α, B, F, or χ so long as χ <. However, the interacting case always has a lower critical temperature than the non-interacting case. If the interacting case had a higher critical temperature, then by looing at our expression for L we see that we would require lnt c,i /T c, ν 11 + ν det ν + ν11 + ν 4 det ν det ν > 1 ρ U χ which reduces after some algebra to ρ Λ <, which is never true for us. It s worth getting a rough understanding about how the critical temperature for each case evolves as the strength of SOC increases. As a proxy for this we examine the parameter lnt c,i /T c,, where T c,i and T c, are the critical temperatures for the interacting and noninteracting systems, respectively. Some algebra allows us to roughly separate the expression for lnt c,i /T c, in terms of powers of α F /B and U, which are the two fundamental energy scales that compete to determine the overall properties of the system. It s rather unwieldy, but for completeness, we have lnt c,i /T c, ρ 1 + ρ /η ρ 1 ρ U χ + 1 rη ρ 1 U 1 U χρ r ρ 1 η + ρ + ρ Ur χρ ρ 1 η + ρ U χ + where we ve defined r ρ 1 ρ rη U χ α F, 4B + α F, α F, 4B which is the ratio characterizing the strength of SOC, taen on the larger Fermi surface. To summarize, we ve seen that due to the dependence of condensation energy on the relative phase of the two order parameters, the interacting case can be made to have a lower condensation energy than the noninteracting case. However, the non-interacting case will always have the higher critical temperature, meaning that the interacting state is liely inaccessible to physical systems. A final note: that the non-interacting case always has the higher T c ultimately lies in the fact that when we have interaction, det ν is always negative. This fact also F /B FIG. 1: A schematic plot showing the log of the ratio of T c for interacting case over T c for the non-interacting case. Here we ve defined η F,1/ F,, and normalized ρ U 1, where ρ m/π. Since we ve treated the ratio α F /B perturbatively F is for the larger surface, our plot looses accuracy the further out on the x-axis we go. Note that the curve with η.5 thicest curve has the largest y-value for a given Rashba strength, since χ is most negative when η.5. warrants a bit of caution, since det ν acquires both positive and negative eigenvalues. The positive eigenvalues are not physical, since lnt c /ω c is not allowed to be positive. This means that care must be taen when minimizing the Ginzburg-Landau functional applied to this problem, since one must loo for saddle-point solutions rather than absolute minima. Path Integral Approach, Ginzburg-Landau Analysis The path integral approach is setched out here mostly for fun, as it is essentially the same as our meanfield analysis. However, it will motivate our extension to Ginzburg-Landau theory. The Hamiltonian is still H f + H int, with H int ψ ψ g µψ µψ µ µ Defining the operator η ψ ψ, the quantum partition function is Z Dη, η e S[η,η ] where the interacting part of the action is S int [η, η ] η g µη µ

6 6 with the summation over repeated indices implied. Introducing the bosonic variables and writing 1, T we eliminate the fermionic degrees of freedom by integrating over : { Z D, Dη, η exp 1 4 g 1 µ µ } + η g µη µ where the sums have been suppressed for simplicity. We can transform suitably in order to eliminate the η bilinear, at the expense of coupling η to the new field. This process results in matching our mean-field analysis S int [η, η,, ] g 1 µ µ + η + h.c Turing to the Nambu spinor formulation as usual, the second term in the action is made into a bilinear in Ψ. Being bilinear it is easily integrated, and we get with G 1 S int g 1 µ µ + Tr ln G 1 Hs σ z + σx + iσ y + σx iσ y where H s is the single particle Hamiltonian on the band. The whole point of this is to show that the G 1 part of the action is diagonal in the band index. This means that the standard technique of expanding G 1 over the order parameter field can be carried out as usual. This allows us to write down the Ginzburg-Landau functional for the theory, which is now woring in real space: { F dx α + β 4 +K 1 } g 1 +g 1 g 3 det g B 8π where α < for T < T c and β > to ensure that α/β >, and α 1 T/T c with β constant. We ve also used the notation + i π Φ A For what follows, it will simplify things a bit to write the GL function as { F dx a + b 4 +K + c B 8π } with the definitions a g 1 det g + α, b β, c g 3 det g Minimizing the free energy by varying the GL functional with respect to, we obtain a set of coupled equations: a + b K + c µ where µ denotes the opposite band as. Let τ 1 T/T c. We now that in the vicinity of T c there is a sign change in the quadratic action of the order parameter, and so we require that τ. The idea is then to tae one of the equations given by the above relation, solve for one order parameter, and plug it in to the other equation, retaining terms of order τ 3/ the order of τ in the usual one-band GL equation. Doing this gives, after a bit of algebra a 1 a c 1 a 1 K +a K b 1 a + b a 3 1/c 1 1 as well as another equation obtained by interchanging the 1 and subscripts in the above relation. Note that in zero field and in the vicinity of T c, we must have a 1 a c. Since c is constant, the a s must also have constant components, and cannot scale simply as τ. In light of this we write a a ατ with a a constant, since we still expect some linear dependence of a on τ. Now we need to connect this equation with the simpler GL equation derived earlier. We arrive at ατ + β K which is in the same form as a regular one-band GL equation. Here we have defined α α 1 a +α a 1, K a 1K +a K 1, β b a µ+b µ a 3 /c One interesting consequence of this equation is that it imposes constraints on the relative phase of the two order parameters. We can see by inspection that if satisfies the above equation with, then 1 β /β 1 satisfies the above equation with 1, implying that the phase difference of the two order parameters is either or π. This is useful since we saw earlier how coupling in the Rashba SOC problem favors a phase difference of π. Another interesting observation is that for zero field, ατ/β. So then at zero field, 1/ β 1 /β. Since β is a constant, the ratio of the squares of the order parameters is independent of temperature! The next step is to write the GL functional in terms of the various interaction constants in our theory. Recalling our result from earlier regarding the ratio of the two gaps at T c we can write, after a bit of manipulation, 1 T c T c ρ ρ 1 ν11 det νl ν det νl

7 7 where as before, L lnω c e Γ /πt c and ν µ g µ ρ µ. Comparing this with 1/ β 1 /β gives us the ratio of the coupling constants in the GL equation. Lucily, it s possible to derive the exact form of the one-band coupling constants from the BCS theory. We can use this result in combination with the above analysis to write the GL equations in a more transparent form. The original one-band coefficients are α ρ ln ω ce Γ πt, β 7ζ3ρ 16π T c For us, the effective coefficients are then a g µ det g ρ L τ, K b v F,, T 8π T c 7ζ3, K β v F, b ρ T, c g 3 det g, In the zero field case, we recall that the GL equations are satisfied when ατ/β. We can plug this in to the GL functional to find the equilibrium free energy for zero field, denoted by E. We then need to connect the coefficients in this equation with the ones presented in the above equation. This process is straightforward, but a bit of a mess. We won t write down all the algebra here, and will just quote the result. We get tentatively! E T τ { ν11 + ν } + 4V LV L ν ν 11 ν 11 V L /ρ 1 + ν V L /ρ where we ve used the abbreviation V det ν. To be continued...