Tuning the structure of the Josephson vortex lattice in Bi2Sr2CaCu2O8+δ single crystals with pancake vortices

In extremely anisotropic cuprate superconductors a lattice of stacks of pancake vortices nucleates when a magnetic field is applied perpendicular to the copper oxide layers, while an orthogonal lattice of highly elliptical Josephson vortices forms when the applied field is parallel to the layers. Under tilted magnetic fields these sublattices can interact in complex ways to form systems of vortex chains and composite vortex lattices. Here we have used high-resolution scanning Hall microscopy (SHM) to map the rich tilted-field vortex phase diagram in an underdoped Bi2Sr2CaCu2O8+δ single crystal. We find that the Josephson vortex lattice spacing has an unexpected non-monotonic dependence on the pancake vortex density reflecting the delicate balance between attractive and repulsive vortex interactions, and actually undergoes a field-driven structural transformation with increasing out-of-plane fields. We also identify particularly stable composite structures composed of vortex chains separated by an integer number of rows of interstitial pancake vortex stacks and are able to establish the precise evolution of vortex-chain phases as the out-of-plane field is increased at small in-plane fields. Our results are in good semi-quantitative agreement with theoretical models and could enable the development of vortex ratchets and lenses based on the interactions between Josephson and pancake vortices.

where ε s P S is the energy per unit length of an isolated pancake stack and γ E ≈ 0.5772 is the Euler constant. We also have to take into account the small attractive energy due to pancake-stack deformations with α = λ/γs and the interaction energy between straight pancake stacks in neighboring chains (we consider the regime c y > λ and neglect more remote interactions) Therefore the total pancake stack contribution is given by Note that the attraction term F a is only important at very small B z , while the inter-chain interaction term F i only becomes important at relatively large B z . Finally, is the JV-PS crossing energy.
B. Jump in JV spacing cy caused by penetration of PV stacks As the PV stacks attract at large separations, their concentration jumps to a finite value corresponding to the finite magnetic induction B z0 (B x ). Due to the crossing energy, penetration of the pancake stacks should lead to decrease of the equilibrium vertical separation between the Josephson vortices in comparison with the B z = 0 value. Correspondingly, the horizontal separation c y should increase. For accurate treatment of this problem we have to consider the energy for fixed H z and B x and find equilibrium B z by minimization of the energy where r x = cy γc is the parameter related to the aspect ratio of the Josephson vortex lattice. Subtracting irrelevant terms, we introduce the energy function as For small B z and a λ we can neglect interaction between pancake stacks in different chains and use exponential asymptotics for the repulsive interaction of straight stacks. In this caseG is explicitly given bỹ Instead of minimization with respect to B z , it is more convenient to minimize with respect to the reduced lattice parameterã = a/λ. Introducing notationsñ x = s 2 γB x /Φ 0 , "chemical potential" µ H = Φ0Hz 4πε0 − ε s P S ε0 and using the relations we rewrite the energy function in the reduced form as We have to minimize this function with respect toã and r x for different µ H . In the case when the optimal spacing is atã =ã m 1, the minimum conditions are Also, at the penetration point we haveG(ã m , r x ) = G L (r x0 ) with r x0 = √ 3/2 being the aspect ratio for B z = 0 or Using this equation, we can transform Eq. (13) into a more convenient form, Eqs. (12), (14), and (15) give three equations for three unknownsã m , r x , and µ H . Equation (15) allows us to obtain the change of the lattice aspect ratio in the limit α 1 whenã m 1 and r x − r x0 1. In this case we can use expansion G L (r x ) ≈ G L (r x0 ) + (r − r x0 ) 2 /6 and neglect the last term in Eq. (15) giving Correspondingly, the change of c y can be estimated as In more realistic case α 0.4, we have to minimize the energy (11) numerically. In this case one has to take into account that the analytic formulas for crossing energy and attractive interaction are valid only in the limit when the separation between the pancake stacks a is much larger than the maximum displacement of pancakes in the crossing region with the Josephson vortex, u 1 ≈ 2.2λ 2 γs ln(Cuγs/λ) . To account for this limitation, we introduce phenomenological cutoffs in the corresponding terms in Eq. (11) as 8α a ln(3.5/α) → 8α (ã+2ũ1) ln(3.5/α) and 16π 2 3 ln −2 Cu α α a 3 → 16π 2 3 ln −2 Cu α α a 3 +8ũ 3 1 withũ 1 = 2α/ ln (C u /α). Suppl. Fig. S2 shows B x dependences of the relative increase c y /c y0 at the penetration and minimum field B z0 = Φ 0 /c y a m . These curves have been numerically computed assuming s = 1.56 nm, γ = 850, and λ = 0.6 µm giving α = 0.45.

C. Dependences of the lattice parameters on Bz
In this section we compute the field dependences of lattice parameters for fixed B z larger than B z0 (B x ). For the fixed field B, due to the relation between the parameters (1), there is only one free lattice parameter, for which we again select the spacing between the pancake stacks in the chain a. We have to minimize energy with respect to this parameter.
To facilitate numerical minimization, we introduce the reduced energy as and the reduced parametersã = a/λ, v γ = a/(γc) = B x /γB z , andB z = B z λ 2 /Φ 0 . Then the reduced energy takes the following form We numerically minimized this energy with respect toã and computed the corresponding horizontal spacing as c y = Φ 0 /(B z λã). Figure 4(i) of the main text shows computed dependences c y (B z ) for different B x . All plots start with minimum B z at the pancake-stack penetration. At this field c y always exceeds its value at B z = 0 but rapidly decreases with increasing B z mostly due to the repulsive interaction between the pancake stacks in the same chain for a < a m . A noticeable feature of these dependences is the regions of quite sharp decrease of c y within the narrow ranges of B z . The origin of this phenomenon is the energy structure of the Josephson vortex lattice. It is well known that in the London limit the aligned Josephson vortex lattice is double-degenerate, the function G L (r x ) has two equal minima at r x = √ 3/2 and r x = 1/(2 √ 3). At small B z , due to the crossing energy, the energetically favorable lattice is with larger r x corresponding to larger c y . With increasing B z , due to repulsion between the pancake stacks, eventually the lattice with smaller r x becomes more favorable. Thus, the rapid decrease of c y (B z ) marks switching between the two stable configurations of the Josephson vortex lattice.   2607(1994) & A.N. Grigorenko et al., Phys. Rev. Lett. 89, 217003 (2002]. Calculated linescans along tilted vortices do exhibit an asymmetric modulation due to the periodic arrangement of pancake vortex chains. However, the signal measured at the Hall probe exponentially samples pancake vortices down to a depth of about ab=660 nm, and hence averages over many c-axis JV lattice spacings, cz~30 nm. As a consequence most of the asymmetry due to the tilted vortices is averaged out and the residual modulation of about 0.01 G lies below our measurement resolution (Bmin~0.1 G with the 100Hz measurement bandwidth used). .
Here Gi=2i/ach are the reciprocal lattice vectors of the chain (ach is the period of the repeating PV structure along the vortex chain), s=1.5nm is the spacing between CuO2 bilayers and n is summed over all CuO2 bilayers down through the thickness of the sample. The set of displacement vectors, un, defines the pancake vortex structure with respect to the origin at x=0 within a single period. at T=85K and the lowest out-of-plane field at which they can be resolved (Bc~0 Oe). The dashed line corresponds to a linear regression fit to anisotropic London theory with eff=840±20. Note the excellent linear behaviour exhibited by the data, supporting the assumption that deviation from linearity at higher out-of-plane fields arises due to interactions of Josephson vortices with pancake vortices.