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Description
We compared polaron masses obtained from a Bose–Einstein condensation (BEC) based estimate of the superconducting transition temperature in two cuprates: LSCO and YBCO. The method assumes that the experimentally measured superconducting critical temperature T_c can be associated with the BEC temperature T_BEC of preformed intersite bipolarons (or, more generally, bosonic charge carriers). From the BEC relation one obtains an estimate of the boson mass and — taking into account lattice anisotropy — the polaron masses both in three dimensions and inside the CuO2 planes. We use published experimental inputs for lattice parameters, anisotropy factors, and T_c from the literature [1-7]. The BEC temperature for an ideal boson gas is: T_BEC=(3.31ℏ^2 n^(2⁄3))⁄(k_B m), where m is the boson mass and n is density. For quasi-two-dimensional systems the scaling changes, but for the dilute, screened cuprate case we assume T_c ≈ T_BEC as a working hypothesis [7,8]. Consequently, the boson mass (and thus the polaron mass) can be estimated from the experimental T_c and carrier density. In YBCO, the boson density is related to hole content p and unit-cell volume V_a by: n_b=p⁄((2V_a)). The oxygen–hole mapping used here is [9]: p(x) = { x³ ,0 ≤ x ≤ 0.5,(x - 0.5)³ + 0.125 ,0.5 < x ≤ 1. Because YBCO is strongly anisotropic, the 3D polaron mass m_p is related to the in-plane (m_(p,ab)) and out-of-plane (m_(p,c)) masses via: m_p=m_(p,ab)^(2⁄3)∙m_(p,c)^(1⁄3). Introducing the mass anisotropy parameter γ_m^2= m_(p,c)/ m_(p,ab), this becomes: m_p=γ_m^(2⁄3)∙m_(p,ab). Equating experimental T_c (x) to T_BEC (Eq. above) with n=n_b and m=2m_p, we solve for m_p and then use the anisotropy relation to extract m_(p,ab). The YBCO unit-cell volume is fit approximately by: V_a (x)≈176.58703- 3.32613∙x. An empirical anisotropy form is: γ_m (x)≈γ_(m,0) + γ ̃m exp[-x/x ̃], where γ(m,0)=-0.186708 , γ ̃m=924.44601 and x ̃=0.18492 . The resulting YBCO 3D polaron mass m_p (p) shows an inverse correlation with T_c (p): m_p (p) ∝T_c^(-1) (p). Numerically, m_p decreases from very large values in the strongly underdoped region to values below ~9 m_e near optimal doping, then increases again in the heavily overdoped regime. The in-plane mass m(p,ab) rises monotonically with oxygen/hole doping — from ~0.5 m_e in deeply underdoped samples to >10 m_e in fully oxygenated YBCO — mainly because γ_m decreases sharply with p. The obtained results are in good agreement with the experimental data of works [10,11].
For LSCO, we used the Presland et al. [12] empirical dome formula: T_c (x)=T_(c,max) [ 1 - 82.6 (x - 0.16)² ]. We also compared to the Marino et al. [7] theoretical T_c (x) expression (given in their text). The input lattice parameters — pseudo-tetragonal ã, unit-cell volume V_a=ã²c, and anisotropy γ_m (x) — were taken from Radaelli et al. [1] and various magnetotransport/penetration-depth studies [2-5]. An empirical fit of LSCO anisotropy is: 〖γ̃〗m (x) = 11.28421 + 275.64904 exp(-19.84499 x) The unit-cell volume fit is: V_a (x) = 189.5474 - 13.6024 x + 13.7927 x² (extrapolated from [1]). Applying the same BEC inversion (Eq. above) yields LSCO 3D polaron masses m_p (x) in the range ~20–30 m_e for underdoped and optimally doped compositions — systematically lower (by about a factor of two) than [6], but in good agreement with modern cyclotron and optical mass measurements [13, 14] for in-plane m(p,ab). LSCO shows the same inverse correlation m_p∝T_c^(-1), with divergences near the superconducting dome edges x_SC^±, two quantum critical points where the superconducting dome starts at T=0, consistent with the BEC model.
Both cuprates exhibit the same BEC-driven fingerprint: larger polaron (bipolaron) masses correspond to lower T_c, and vice versa. However, doping trends differ — YBCO shows nonmonotonic 3D mass behavior (minimum near optimal p), while LSCO’s 3D masses are moderate and span a narrower range. In-plane masses for both systems agree better with experimental effective masses when γ_m (x) is included. Differences with earlier theoretical estimates can be traced to carrier density assumptions, anisotropy treatment, and whether T_c was equated to T_BEC. This comparison underscores the utility of a common BEC framework for unifying mass determinations and isolating material-specific effects such as anisotropy, unit-cell volume, and hole mapping.
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