Mode-coupling model of Mott gap collapse in the cuprates: Natural phase boundary for quantum critical points

R. S. Markiewicz

Phys. Rev. B 70, 174518 – Published 22 November 2004

Abstract

A simple antiferromagnetic approach to the Mott transition was recently shown to provide a satisfactory explanation for the Mott gap collapse with doping observed in photoemission experiments on electron-doped cuprates. Here this approach is extended in a number of ways. Random phase approximation, mode coupling (via self-consistent renormalization), and (to a limited extent) self-consistent Born approximation calculations are compared to assess the roles of hot-spot fluctuations and interaction with spin waves. When fluctuations are included, the calculation satisfies the Mermin-Wagner theorem (Néel transition at $T = 0$ only—unless interlayer coupling effects are included), and the mean-field gap and transition temperature are replaced by pseudogap and onset temperature. The model is in excellent agreement with experiments on the doping dependence of both photoemission dispersion and magnetic properties. The magnetic phase terminates in a quantum critical point (QCP), with a natural phase boundary for this QCP arising from hot-spot physics. Since the resulting $T = 0$ antiferromagnetic transition is controlled by a generalized Stoner factor, an ansatz is made of dividing the Stoner factor up into a material-dependent part, the bare susceptibility and a correlation-dependent part, the Hubbard $U$ , which depends only weakly on doping. From the material-dependent part of the interaction, it is possible to explain the striking differences between electron and hole doping, despite an approximate symmetry in the doping of the QCP. The slower divergence of the magnetic correlation length in hole-doped cuprates may be an indication of more Mott-like physics.

43 More

Received 22 December 2003

DOI:https://doi.org/10.1103/PhysRevB.70.174518

Authors & Affiliations

R. S. Markiewicz

Physics Department, Northeastern University, Boston, Massachusetts 02115, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 70, Iss. 17 — 1 November 2004

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Susceptibility $χ_{0}$ near $\vec{Q}$ for several dopings near the $C$ point. Upper group at $μ = - 0.05 eV$ , middle at $μ = 0$ ( $C$ point), and bottom at $μ = + 0.05 eV$ . Temperatures are $T = 200 K$ (dotted lines), $100 K$ (short-dashed lines), $10 K$ (long-dashed lines), $1 K$ (solid lines). Horizontal line = $U_{eff} (μ = 0)$ .Reuse & Permissions
Figure 2
Plateau width $q_{c}$ , comparing Eq. (E3) (solid lines) and the measured widths (circles) from Fig. 32. Upper curve along $[q_{c}, 0]$ direction, lower along $[q_{c}, q_{c}] ∕ \sqrt{2}$ direction. Symbols $=$ experimental inverse correlation lengths $ξ^{- 1}$ from YBCO: large squares $=$ Ref. 35, triangles $=$ Ref. 36; LSCO: small squares $=$ Ref. 37. Diamonds $= T_{A}^{*} ∕ 5000 K$ . Dotted line: $ξ = 100 a$ . Inset $=$ plateau boundary for a series of chemical potentials $μ$ from 0 (smallest) to $- 0.359 eV$ (largest).Reuse & Permissions
Figure 3
(a) Susceptibility $χ_{0}$ at $\vec{Q}$ as a function of doping for several temperatures. From highest to lowest curves near $x = 0.1$ , the temperatures are $T = 1$ , 100, 300, 600, 1000, 2000, and $4000 K$ . Dotted line $= 1 ∕ U_{eff}$ , dot-dashed line $= 1.5 ∕ U_{eff}$ . (b) Density of states $N_{F}$ for the same temperatures. (c) Susceptibility $χ_{0}$ at $\vec{Q}$ as a function of doping for several frequencies at $T = 1 K$ : $ω = 0.01$ , 0.1, 0.3, 0.6, $1.0 eV$ . (d) Pseudo-VHS (peak of $χ_{0}$ ) as a function of temperature $T_{V}$ (circles) or scaled frequency $T_{c}^{-} = ω_{c}^{-} ∕ π$ (squares); triangles $= T_{incomm}$ .Reuse & Permissions
Figure 4
Mean-field magnetic transition temperatures determined from Stoner criterion using $U_{eff}$ of Fig. 3. Solid line: commensurate (at $\vec{Q}$ ); long-dashed line: incommensurate. Dot-dashed line $= 10 T_{N}$ , where $T_{N}$ is the onset of long range AFM order, from Refs. 39, 40 (with filled circles). Insets $=$ blowups near $C$ and $H$ points. Squares in inset (b) $=$ pseudogap data of Ref. 38.Reuse & Permissions
Figure 5
(a) $δ - δ_{0}$ (thin solid lines $= - δ_{0}$ ), (b) $ρ_{s}$ calculated from Eqs. (22, 14), (c) $- {\bar{δ}}_{0}$ , (d) $\bar{Δ}$ , Eq. (28). In all the plots, the solid curves correspond to $x = 0.0$ , dotted lines: $x = - 0.04$ , short-dashed lines: $x = - 0.10$ , long-dashed lines: $x = - 0.15$ .Reuse & Permissions
Figure 6
Temperature dependence of gap $\bar{Δ}$ for (from highest to lowest) $x = 0$ , $- 0.04$ , $- 0.10$ , and $- 0.15$ . Arrows show mean-field transition temperature $T_{N}$ .Reuse & Permissions
Figure 7
Imaginary part of the self-energy (25) assuming $1 ∕ C = 0.05 t$ , $δ = 0.002$ , $α_{ω} = 1$ , $T = 100 K$ . The branches are labeled $(k_{x}, k_{y})$ , in units of $π$ .Reuse & Permissions
Figure 8
SCR dispersion relations for electron doped materials, calculated at $T = 100 K$ : (a) $x = 0 (U ∕ t = 6)$ , (b) $x = - 0.04 (U ∕ t = 5)$ , (c) $x = - 0.10 (U ∕ t = 3.5)$ , and (d) $x = - 0.15 (U ∕ t = 2.9)$ . Linewidth indicates relative intensity; for $x = - 0.15$ all shadow features are extremely weak.Reuse & Permissions
Figure 9
Mean-field dispersion relations for electron doped materials, calculated at $T = 1 K$ : (a) $x = 0 (U ∕ t = 6)$ , (b) $x = - 0.04 (U ∕ t = 5)$ , (c) $x = - 0.10 (U ∕ t = 3)$ , and (d) $x = - 0.15 (U ∕ t = 2.6)$ . Linewidth indicates relative intensity.Reuse & Permissions
Figure 10
Mean-field dispersions in three-band model for electron doped materials, showing the two antibonding bands, assuming $m_{Q} = 0.3$ (a), 0.2 (b), 0.05 (c), and 0.01 (d). Other parameters are discussed in Appendix .Reuse & Permissions
Figure 11
Spectral functions for (a) $x = 0$ , (b) $x = - 0.04$ , (c) $x = - 0.10$ , and (d) $x = - 0.15$ , at $T = 100 K$ . Solid lines at $(π, 0)$ , and long-dashed lines at $(π ∕ 2, π ∕ 2)$ .Reuse & Permissions
Figure 12
Temperature dependence of (a) spectral function and (b) imaginary part of self-energy, for $x = 0.0$ at $(π, 0)$ . Temperatures are 100, 500, 1000, 2000, 3000, 4000, and $5000 K$ .Reuse & Permissions
Figure 13
Temperature dependence of (a) spectral function and (b) imaginary part of self-energy, for $x = 0.0$ at $(π ∕ 2, π ∕ 2)$ . Temperatures are 100, 500, 1000, 2000, 3000, 4000, and $5000 K$ .Reuse & Permissions
Figure 14
Temperature dependence of spectral function for $x = - 0.15$ at $(π, 0)$ , for $U ∕ t = 2.9$ (a) and 2.5 (c). Temperatures are 100 (solid line), 500 (long-dashed line), 1000 (short-dashed line), and $2000 K$ (dot-dashed line). (b) Imaginary part of self-energy at $T = 100 K$ , $U ∕ t = 2.9$ .Reuse & Permissions
Figure 15
$U$ dependence of spectral functions for $x = - 0.15$ at $T = 100 K$ near the $T = 0$ QCP, for $U ∕ t = 2.5$ (short-dashed line), 2.7 (long-dashed line), 2.9 (solid line), 3.0 (dot-dashed line), and 3.2 (dotted line).Reuse & Permissions
Figure 16
Fermi surface map for $x = - 0.10$ (a) and $- 0.15$ (b).Reuse & Permissions
Figure 17
(a) Calculated DOS for a series of hole-doped cuprates, assuming $U_{eff} = 2.3 t$ , with $x = 0.176$ (solid-line), 0.184 (long-dashed line), 0.202 (short-dashed line), 0.225 (dotted line), and 0.244 (dot-dashed line). (b) Comparison of shift of lower DOS peak (circles) from (a) with representative tunneling data (Ref. 38) (triangles).(c), (d) Band dispersion near the pseudogap for $x = 0.176$ (c) and 0.244 (d).Reuse & Permissions
Figure 18
Temperature dependence of correlation length $ξ$ in NCCO. Data are from Ref. 70: $x = 0$ (open diamonds) and $- 0.15$ (open squares); from Ref. 63: $x = 0$ (solid diamonds), $- 0.10$ (solid up triangles), $- 0.14$ (solid down triangles), and $- 0.18$ (solid circles). Fits are to Eq. (33), with parameters appropriate to $x = 0$ (solid line), $- 0.04$ (long-dashed line), $- 0.085$ (dotted line), and $- 0.10$ (short-dashed line). Temperatures are measured in units of $J = 125 meV$ . Inset: Plot of as-grown nominal doping $x_{g}$ vs reduced nominal doping $x_{r}$ for fixed values of $T_{N}$ (circles), $M$ (squares), and $ξ$ (diamonds). Solid line is $x_{g} = 1.2 x_{r} + 0.012$ .Reuse & Permissions
Figure 19
(a) Temperature dependence of correlation length $ξ$ in LSCO for $x = 0$ (open circles), 0.02 (inverted triangles), 0.03 (squares), and 0.04 (triangles) (from Ref. 37), compared with ${Nd}_{2} {CuO}_{4}$ (open and filled diamonds, as in Fig. 18). Thick solid curve $=$ fit for undoped material from Fig. 18). Thin solid lines $=$ fits to Eq. (31). Dotted line $=$ calculated value for $x = 0.10$ , as described in the text. (b) Comparison of magnetization extracted from $ξ (T)$ , Eq. (32) [filled triangles for LSCO (Ref. 37), circles for YBCO (Refs. 61, 60)], with that for NCCO, taken from ARPES fit, Ref. 9 (squares), and from magnetization (scaled to $M = 0.4$ at $x = 0$ ; open triangles: Ref. 73, inverted triangles: Ref. 63). All lines are simply drawn to connect the data points, except for that part of the dotted line connected with the ARPES data (squares) extrapolated beyond $x = 0.15$ . This represents a mean-field calculation, assuming that $U$ does not change with doping over this range, and using the band parameters of Ref. 9.Reuse & Permissions
Figure 20
Comparison of experimental Néel temperatures for NCCO and LSCO (solid line) and for the stripe (magnetic) ordering transitions observed in $Nd$ -substituted LSCO (Ref. 40) (solid line with squares) with the model of interlayer coupling with staggered stacking and $t_{z 0} = t ∕ 10 \sim 30 meV$ , plotted as $T_{N} ∕ 10$ (dot-dot-dash line). Also included is the approximate expression (39) (dotted line with circles). (Note that there is a range of hole doping for which $A$ is found to be negative; in this range $T_{N}$ was arbitrarily assumed to vanish in the staggered model, $T_{N} = 0$ .)Reuse & Permissions
Figure 21
(a) Mean-field transition temperatures for Néel $(T_{N})$ and ferromagnetic $(T_{C})$ orders and their difference $Δ T_{c} = T_{N} - T_{C}$ (upper dashed line). At any $U ∕ t$ the ratio of the shaded area to the total area below this line gives the fraction of the Brillouin zone which is significantly excited [ $T_{c} (\vec{Q}) ⩾ (1 - α_{0}) T_{N}$ , for $α_{0} = 0.01$ ]. (b) Replot of transition temperatures scaled to $U$ . (c) Plot of $T_{c}$ vs $\vec{q}$ , for $U = 6 t$ . The curve bears an uncanny resemblance to the (scaled) electronic dispersion of the LHB, long-dashed line.Reuse & Permissions
Figure 22
(a) Bare susceptibility $χ_{0} (\vec{Q}, 0)$ at $T = 1 K$ , for several values of $τ = - 0.0552$ , $- 0.110$ , $- 0.276$ , $- 0.552$ , $- 0.828$ , $- 0.9$ , $- 0.99$ (solid lines), and 0 (dashed line). (b) Crossover couplings as a function of $τ$ : $U_{V}$ (triangles), $U_{2}$ (squares), and $U_{1}$ (circles). (Dashed line $= U_{1}$ for electron doping.)Reuse & Permissions
Figure 23
(a), (b): Blowups of $Im (Σ)$ for $x = 0$ at $(π, 0)$ (a) and $(π ∕ 2, π ∕ 2)$ (b) at $T = 100 K$ (solid lines), $500 K$ (long-dashed lines), and $1000 K$ (a) or $750 K$ (b) (short-dashed lines). (c) Maximum of $Im (Σ)$ vs $T$ for $(π, 0)$ (squares) and $(π ∕ 2, π ∕ 2)$ (circles); 0.1/(full width at half maximum) for $(π, 0)$ (triangles) and $(π ∕ 2, π ∕ 2)$ (diamonds); solid line $=$ corresponding $ξ (T)$ , Eq. (20).Reuse & Permissions
Figure 24
Dispersion of all six bands in three-band model, assuming $m_{Q} = 0.3$ (a) and 0.01 (b).Reuse & Permissions
Figure 25
Effective magnetization $m_{eff} = m U ∕ 6 t$ for the three-band (circles) and one-band (squares) models. The one-band result has been multiplied by 3/4 to better agree with the three-band results.Reuse & Permissions
Figure 26
Renormalized parameters for lower (a) and upper (b) Hubbard bands: $Z$ (solid lines), $A_{1}$ (dot-dashed lines), and $A_{2}$ (dotted lines), in comparison with $Z A_{01}$ (long-dashed-short-dashed lines), and $Z A_{02}$ (short-dashed lines), assuming parameters $t = 0.326 eV$ , $t^{'} = - 0.375 t$ , and $t^{″} = 0.15 t$ . Horizontal lines $=$ experimental range for $A_{1}$ (long-dashed lines) and $A_{2}$ (solid lines), after Refs. 110, 111. Also shown are the individual renormalization factors $Z_{1}$ (long-dash-dotted line) and $Z_{2}$ (long-dash-dot-dotted line). Vertical lines delimit parameter values consistent with experiment.Reuse & Permissions
Figure 27
Parameter values consistent with ARPES data for $J$ (a) and $2 Δ$ (b). Broad range determined by $A_{1}$ ; narrow range [in both (a) and (b)] by $A_{2}$ . Circle $=$ parameters assumed in SCR result, square $=$ best SCBA approximation. Dashed line in (a) $J = 0.6 t - 1.5 t^{″}$ .Reuse & Permissions
Figure 28
Renormalized $U$ parameter as a function of $J$ : solid line $=$ bare $U = 4 t^{2} ∕ J$ in large gap limit; short-dashed line $=$ renormalized $U$ from Eq. (B24); long-dashed line $=$ bare $Δ$ corrected for the small gap limit, Eq. (B24); dotted line $=$ renormalized $U$ in the small gap limit from Eq. (B24); dot-dashed line $= A_{03}$ .Reuse & Permissions
Figure 29
Comparison of mean-field (dashed lines) and SCBA (solid lines) dispersions for $t^{'} = - 0.375 t$ , $t^{″} = 0.15 t$ , and two choices of $J$ , $J ∕ t = 0.42$ (a) or 0.33 (b).Reuse & Permissions
Figure 30
Mean-field band structure (solid lines) plus approximations involving Eqs. (B25, B26), with (short-dashed lines) or without (long-dashed lines) a finite $A_{03}$ , for $Δ ∕ t = 0.5$ (a), 1.0 (b), 2.0 (c), and 3.0 (d).Reuse & Permissions
Figure 31
Calculated $U_{eff}$ assuming (a) simple screening or (b) full vertex correction of Chen et al. (Ref. 117). In both cases, a bare $U = 6.75 t$ was assumed. Solid lines $=$ electron doping; long-dashed lines $=$ hole doping; triangles (squares) in (a) $=$ paramagnetic screening of $U$ , at $T = 1 K (2000 K)$ ; triangle in (b) $=$ undoped; circles $=$ data of Ref. 8.Reuse & Permissions
Figure 32
Susceptibility $χ_{0}$ near $\vec{Q}$ for a variety of dopings at $T = 100 K$ . From highest to lowest solid curves near $S \equiv \vec{Q}$ , the chemical potentials are $μ = - 0.35$ , $- 0.30$ , $- 0.25$ , $- 0.20$ , $- 0.15$ , $- 0.10$ , $- 0.055$ , $- 0.02$ , and $0 eV$ . For the dashed curves (top to bottom), $μ = - 0.352$ , $- 0.355$ , and $- 0.359 eV$ .Reuse & Permissions
Figure 33
Illustrating origin of plateaus (dotted line) from crossed hole pockets (short-dashed lines).Reuse & Permissions
Figure 34
(a) Expanded view of susceptibility $χ_{0}$ on the plateaus near $\vec{Q}$ for a variety of dopings at $T = 100 K$ (solid curves) or $1 K$ (dashed curves). From highest to lowest curves near $\vec{Q}$ , the chemical potentials are $μ = - 0.20$ , $- 0.15$ , and $- 0.05 eV$ (for both solid and dashed curves). All curves except $μ = - 0.20 eV$ have been shifted vertically to fit within the expanded frame. (b) Similar plateaus for the hole-doped materials $(T = 1 K)$ , with (from highest to lowest) $μ = - 0.359$ , $- 0.35$ , $- 0.3$ , $- 0.25$ , and $- 0.22 eV$ .Reuse & Permissions
Figure 35
(a) Calculated values of $A$ (circles for commensurate $\vec{Q}$ , diamonds for incommensurate $\vec{q} = \vec{Q} + {\vec{q}}^{'}$ ) and $C$ (squares). Solid line $=$ Eq. (F7). (b) Incommensurate wave vector $q^{'}$ in two different directions: circles along $(1, 0)$ , squares along $(1, 1)$ ; dashed line: $q^{'} \propto x$ .Reuse & Permissions
Figure 36
Calculated values of $χ_{\vec{Q}} ∕ ξ^{2}$ (solid line) and $ω_{S F} ξ^{2}$ (short-dashed line), assuming $U = 6 t$ . long-dashed line $=$ doping $x (μ) (\times 10)$ ; dot-dashed line $= ω_{SF} ξ^{2}$ corrected for incommensurate $\vec{q} \neq \vec{Q} (\times 1 ∕ 5)$ .Reuse & Permissions
Figure 37
Temperature dependence of $A^{'}$ for several dopings.Reuse & Permissions
Figure 38
(a) Calculated susceptibility $χ (q)$ for several values of overlap $δ$ . (b) Blowup of plateau region, for $χ_{q} - χ_{q = 0}$ . (c) Model of Fermi surfaces, defining $δ$ , $k_{1} (k_{⊥})$ , and $k_{2} (k_{‖})$ .Reuse & Permissions
Figure 39
(a) $Im χ (\vec{Q}, ω)$ , (b) $Re χ (\vec{Q}, ω)$ , (c) $Im χ ∕ ω \equiv \hat{C}$ , and (d) $d Re χ (\vec{Q}, ω) ∕ d ω$ , for (a), (c): $μ = 0$ (solid line), $- 0.05$ (long-dashed line), $- 0.10$ (dashed line), $- 0.15$ (dotted line), $- 0.20$ (dot-dashed line), $- 0.25$ (dot-dot-dashed line), and $- 0.30 eV$ (short- dashed line); (b), (d): $x = 0$ (solid line), $- 0.04$ (long-dashed line), $- 0.10$ (dashed line), and $- 0.15$ (dotted line).Reuse & Permissions
Figure 40
$\hat{C}$ calculated for several values of $μ$ : $μ = - 0.355$ (diamonds), $- 0.357$ (circles), $- 0.358$ (squares), $- 0.359 eV$ (triangles) $(μ_{v} = - 0.3599 eV)$ . Inset: Band dispersion $ϵ_{\vec{k}}$ (solid line), $ϵ_{\vec{k} + \vec{Q}}$ (dashed line), for $μ = 0$ . Arrow $= ω_{c}^{-}$ .Reuse & Permissions
Figure 41
Origins of critical cutoffs. Thick solid line $=$ FS; thin solid line $= Q$ -shifted FS; dashed lines $=$ Eq. (F9), for several values of $ω$ . Chemical potential $μ =$ (a) 0, (b) $- 0.1$ , (c) $- 0.14$ , (d) $- 0.2 eV$ . Horizontal arrows indicate $ω_{0}$ , vertical arrows $ω_{c}^{-}$ .Reuse & Permissions
Figure 42
(a) Normalized conductivity ratio, ${\hat{σ}}_{z z} ∕ σ_{x x}$ vs doping $E_{F}$ , for uniform (solid line) and staggered stacking [long-dashed line and short-dashed line $(\times 20)$ ] and (b) resulting normalized interlayer hopping ${\hat{t}}_{z 0}$ for staggered (solid line) and uniform stacking [long-dashed line and short-dashed line $(\times 4.5)$ ].Reuse & Permissions
Figure 43
$χ_{0} (\vec{Q}, Q_{z})$ at $T = 100 K$ vs chemical potential $μ$ , for uniform stacking and $Q_{z} = π$ (a), $π ∕ 2$ (b), and 0 (c). The various curves correspond to $t_{z 0} ∕ t = 0.01$ , 0.02, 0.05, 0.1, 0.2, and 0.5, with the peak in $χ_{0}$ shifting to the right with increasing $t_{z 0}$ . Inset (d): position of peak, $μ_{\max}$ , vs $t_{z 0}$ for $Q_{z} = π$ (solid line), $π ∕ 2$ (long-dashed line), and 0 (short-dashed line).Reuse & Permissions
Figure 44
$χ_{0} (\vec{Q}, Q_{z})$ vs chemical potential $μ$ , as in Fig. 43, but at $T = 10 K$ .Reuse & Permissions
Figure 45
$χ_{0} (\vec{Q}, Q_{z})$ vs chemical potential $μ$ , for uniform stacking and $t_{z 0} = 0.2 t$ , and $T = 10 K$ (a), or $100 K$ (b), with $Q_{z} ∕ π = 1$ (solid line), 0.75 (long-dashed line), 0.5 (short-dashed line), 0.25 (dotted line), 0 (dot-dashed line).Reuse & Permissions
Figure 46
(a) $χ_{0} (\vec{Q}, Q_{z})$ vs $Q_{z}$ for $t_{z 0} = 0.1 t$ , and $T = 10 K$ , and a variety of chemical potentials $μ = - 0.003559$ (solid line), $- 0.08898$ (long-dashed line), $- 0.1779$ (short-dashed line), $- 0.2669$ (dotted line), $- 0.2847$ (dot-dashed line), $- 0.2954$ (long-long-short-short-short-dashed line), $- 0.3025$ (long-short-short-dashed line), $- 0.3203$ (dash-dot-dot line), $- 0.3381$ (long-short-dashed line), and $- 0.3559 meV$ (long-short-short-short-dashed line). (b) $Δ χ = χ_{0} (\vec{Q}, Q_{z}) - χ_{0} (\vec{Q}, Q_{z} = 0)$ , where the curves have the same meaning as in frame (a).Reuse & Permissions
Figure 47
(a) $χ_{0} (\vec{Q}, Q_{z})$ vs $Q_{z}$ for $t_{z 0} = 0.02 t$ , and $T = 10 K$ , and a variety of chemical potentials $μ = - 0.003559$ (solid line), $- 0.08898$ (long-dashed line), $- 0.1779$ (short-dashed line), $- 0.2669$ (dotted line), $- 0.3025$ (dot-dashed line), $- 0.3381$ (long-long-short-short-short-dashed line), $- 0.3417$ (long-dashed-dotted line), $- 0.3452$ (long-short-short-dashed line), $- 0.3488$ (long-short-short-short-dashed line), and $- 0.3559 meV$ (long-dash-dot-dotted line). (b), (c) $A_{z}^{'} = A_{z} ∕ U c^{2}$ vs $μ$ for $t_{z 0} ∕ t = 0.02$ (squares, $A_{z}^{'} \times 25$ ) and 0.1 (triangles, circles). (d) $Q_{z m}$ vs $μ$ for $t_{z 0} ∕ t = 0.02$ (squares) and 0.1 (triangles).Reuse & Permissions
Figure 48
(a) $A_{z}^{'} = A_{z} ∕ U c^{2}$ vs chemical potential $μ$ for $Q_{z} = 0$ (solid lines) or $π$ (dashed lines), for a variety of values of $t_{z 0}$ and $T = 100 K$ . In order of increasing amplitude, the values are $t_{z 0} ∕ t = 0.01$ , 0.02, 0.05, 0.1, 0.2, and 0.5. (b) Scaling of $A_{z}^{' (0)}$ with ${(t_{z 0} ∕ t)}^{2}$ . Curves are $t_{z 0} ∕ t = 0.01$ (solid line), 0.02 (long-dashed line), 0.05 (short-dashed line), 0.1 (dotted line), 0.2 (dot-dashed line), and 0.5 (dot-dot-dashed line). (c) Comparison of $\max (A_{z})$ for staggered stacking (solid line) and uniform stacking (triangles, $\times 1 ∕ 20$ ) at $t_{z 0} ∕ t = 0.1$ .Reuse & Permissions
Figure 49
(a) $T_{3 D}$ vs $μ$ for $T = 10 K$ and uniform stacking with $t_{z 0} = 0.1 t$ (circles, $\times 1 ∕ 25$ ) or 0.02 (squares), or staggered stacking with $t_{z 0} = 0.02$ (triangles). (b) $T_{N}$ vs $x$ , comparing mean-field transition (solid line) with interlayer coupling models (uniform stacking) assuming $t_{z 0} ∕ t = 0.1$ (long-dashed line), 0.02 (short-dashed line), and $2 \times 10^{- 6}$ (dot-dashed line), and the staggered stacking model assuming $t_{z 0} ∕ t = 0.1$ (dot-dot-dash line).Reuse & Permissions
Figure 50
(a) Mean-field $T_{N}$ vs $x$ assuming paramagnetic $U_{eff}$ (Appendix ). (b) Corresponding $T_{N}$ vs $x$ , calculated using Eq. (G8). Squares $=$ staggered stacking with $t_{z 0} ∕ t = 0.1$ ; triangles $=$ uniform stacking with $t_{z 0} ∕ t = 0.02$ ; solid line and circles $=$ data, as in Fig. 20.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics