Hall Effect in Doped Mott–Hubbard Insulator

Abstract

We present theoretical analysis of Hall effect in doped Mott–Hubbard insulator, considered as a prototype of cuprate superconductor. We consider the standard Hubbard model within DMFT approximation. As a typical case we consider the partially filled (hole doping) lower Hubbard band. We calculate the doping dependence of both the Hall coefficient and Hall number and determine the value of carrier concentration, where Hall effect changes its sign. We obtain a significant dependence of Hall effect parameters on temperature. Disorder effects are taken into account in a qualitative way. We also perform a comparison of our theoretical results with some known experiments on doping dependence of Hall number in the normal state of YBCO and Nd-LSCO, demonstrating rather satisfactory agreement of theory and experiment. Thus the doping dependence of Hall effect parameters obtained within Hubbard model can be considered as an alternative to a popular model of the quantum critical point.

Appendices

APPENDIX

“BARE” ELECTRONIC DISPERSION AND ITS DERIVATIVES FOR BAND WITH SEMIELLIPTIC DENSITY OF STATES

Let us assume that electronic spectrum corresponding to density of states (5) is isotropic ε(p) = ε(|p|) ≡ ε(p). To calculate derivatives in (2) and (3) it is necessary to perform “angle” averaging of these derivatives by momentum components

$${{\left\langle {{{{\left( {\frac{{\partial \varepsilon ({\mathbf{p}})}}{{\partial {{p}_{x}}}}} \right)}}^{2}}} \right\rangle }_{\Omega }} = \varepsilon {{'}^{2}}(p){{\left\langle {\frac{{p_{x}^{2}}}{{{{p}^{2}}}}} \right\rangle }_{\Omega }} = \frac{1}{d}\varepsilon {{'}^{2}}(p) = \frac{1}{3}\varepsilon {{'}^{2}}(p),$$

(9)

where 〈…〉_Ω = $\int {\frac{{d\Omega }}{{4\pi }}}$… is solid angle averaging in three-dimensional system (d = 3) and ε'(p) = $\frac{{d\varepsilon (p)}}{{dp}}$ is derivative over the absolute value of momentum.

$$\begin{gathered} {{\left( {\frac{{\partial \varepsilon ({\mathbf{p}})}}{{\partial {{p}_{x}}}}} \right)}^{2}}\frac{{{{\partial }^{2}}\varepsilon ({\mathbf{p}})}}{{\partial p_{y}^{2}}} \\ = \varepsilon {{'}^{2}}(p)\left[ {\varepsilon ''(p)\frac{{p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}} + \frac{{\varepsilon '(p)}}{p}\frac{{p_{x}^{2}{{p}^{2}} - p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}}} \right], \\ \end{gathered} $$

(10)

where ε''(p) = $\frac{{{{d}^{2}}\varepsilon (p)}}{{d{{p}^{2}}}}$. Thus we have a problem of finding the angle average ${{\left\langle {\frac{{p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }}$. Let us introduce notations: ${{\left\langle {\frac{{p_{x}^{4}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }}$ ≡ a and ${{\left\langle {\frac{{p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }}$ ≡ b. First of all we have:

$$\begin{gathered} {{\left\langle {\frac{{{{{(p_{x}^{2} + p_{y}^{2} + p_{z}^{2})}}^{2}}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }} \\ = {{\left\langle {\frac{{{{{(p_{x}^{4} + p_{y}^{4} + p_{z}^{4})}}^{2}} + 2p_{x}^{2}p_{y}^{2} + 2p_{x}^{2}p_{z}^{2} + 2p_{y}^{2}p_{z}^{2}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }} \\ = d{{\left\langle {\frac{{p_{x}^{4}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }} + d(d - 1){{\left\langle {\frac{{p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }} = 3a + 6b = 1. \\ \end{gathered} $$

(11)

Similarly:

$$\begin{gathered} {{\left\langle {\frac{{{{{(p_{x}^{2} + p_{y}^{2})}}^{2}}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }} \\ = {{\left\langle {\frac{{p_{x}^{4} + p_{y}^{4} + 2p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }} = 2a + 2b = \frac{8}{{15}}. \\ \end{gathered} $$

(12)

As

$$\begin{gathered} \left\langle {\frac{{{{{(p_{x}^{2} + p_{y}^{2})}}^{2}}}}{{{{p}^{4}}}}} \right\rangle = {{\langle {{\sin }^{4}}\theta \rangle }_{\Omega }} \\ = \frac{1}{2}\int\limits_0^\pi {\sin \theta {{{\sin }}^{4}}\theta d\theta = \frac{1}{2}\int\limits_{ - 1}^1 {{{{(1 - {{\tau }^{2}})}}^{2}}d\tau = \frac{8}{{15}}} } \,{\text{,}} \\ \end{gathered} $$

(13)

where θ is an angle between vector p and z-axis, Then from Eqs. (11), (12) we immediately obtain a = ${{\left\langle {\frac{{p_{x}^{4}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }}$ = 1/5 and b = ${{\left\langle {\frac{{p_{x}^{2}p_{y}^{2}}}{{{{p}^{4}}}}} \right\rangle }_{\Omega }}$ = 1/15, so that we have:

$${{\left\langle {{{{\left( {\frac{{\partial \varepsilon ({\mathbf{p}})}}{{\partial {{p}_{x}}}}} \right)}}^{2}}\frac{{{{\partial }^{2}}\varepsilon ({\mathbf{p}})}}{{\partial p_{y}^{2}}}} \right\rangle }_{\Omega }} = \frac{{\varepsilon {{'}^{2}}(p)}}{{15}}[\varepsilon ''(p) + 4\varepsilon '(p){\text{/}}p]\,.$$

(14)

To find derivatives ε'(p), ε''(p) for the spectrum determined by semi-elliptic density of states (5) we can use the approach developed in [13]. Equating the number of states in a phase volume element d³p and the number of states in an energy interval [ε, ε + dε], we obtain differential equation determining ε(p):

$$\frac{{4\pi {{p}^{2}}dp}}{{{{{(2\pi )}}^{3}}}} = {{N}_{0}}(\varepsilon )d\varepsilon \,.$$

(15)

Assuming the quadratic dispersion of ε(p) close to lower band edge we obtain the initial condition for (15): p → 0 for ε → –D. As a result:

$$p = {{\left[ {6\pi \left( {\pi - \varphi + \frac{1}{2}\sin (2\varphi )} \right)} \right]}^{{\frac{1}{3}}}},$$

(16)

where φ = arccos$\left( {\frac{\varepsilon }{D}} \right)$ and momentum is given in units of inverse lattice parameter. This expression implicitly defines the dispersion law ε(p) on electronic branch of the spectrum ε ∈ [–D, 0].

We can determine characteristic momentum p₀ corresponding to ε = 0:

$${{p}_{0}} = p(\varepsilon = 0) = {{(3{{\pi }^{2}})}^{{\frac{1}{3}}}}.$$

(17)

We are interested in calculating two derivatives of this spectrum over the momentum. From (15) we get:

$$\varepsilon '(p) = \frac{{d\varepsilon }}{{dp}} = \frac{{{{p}^{2}}}}{{2{{\pi }^{2}}}}\frac{1}{{{{N}_{0}}(\varepsilon )}},$$

(18)

where p is defined by (16).

$$\begin{gathered} \varepsilon ''(p) = \frac{d}{{dp}}\frac{{d\varepsilon }}{{dp}} = \frac{1}{{2{{\pi }^{2}}}}\frac{{2p{{N}_{0}}(\varepsilon ) - {{p}^{2}}\frac{{d{{N}_{0}}(\varepsilon )}}{{d\varepsilon }}\frac{{d\varepsilon }}{{dp}}}}{{N_{0}^{2}(\varepsilon )}} \\ = \frac{1}{{{{N}_{0}}(\varepsilon )}}\left[ {\frac{p}{{{{\pi }^{2}}}} - \varepsilon {{'}^{2}}\frac{{d{{N}_{0}}(\varepsilon )}}{{d\varepsilon }}} \right], \\ \end{gathered} $$

(19)

where $\frac{{d{{N}_{0}}(\varepsilon )}}{{d\varepsilon }}$ = –$\frac{2}{{\pi {{D}^{2}}}}\frac{\varepsilon }{{\sqrt {{{D}^{2}} - {{\varepsilon }^{2}}} }}$, ε'(p) is determined from (18), while p is defined by (16).

On the hole branch of the spectrum (ε ∈ [0, D]), to obtain quadratic dispersion law close to the upper edge of the band (ε → D) we introduce a hole momentum $\tilde {p}$ = 2p₀ – p and equate the number of states in a phase volume element d³$\tilde {p}$ and in energy interval [ε, ε + dε]:

$$\frac{{4\pi {{{\tilde {p}}}^{2}}d\tilde {p}}}{{{{{(2\pi )}}^{3}}}} = - {{N}_{0}}(\varepsilon )d\varepsilon \,.$$

(20)

Demanding $\tilde {p}$ → 0 at the upper band edge ε → 0, we obtain:

$$\tilde {p} = {{\left[ {6\pi \left( {\varphi - \frac{1}{2}\sin (2\varphi )} \right)} \right]}^{{\frac{1}{3}}}}.$$

(21)

For the velocity on the hole branch of the spectrum we get:

$$\varepsilon '(p) = - \frac{{d\varepsilon }}{{d\tilde {p}}} = \frac{{{{{\tilde {p}}}^{2}}}}{{2{{\pi }^{2}}}}\frac{1}{{{{N}_{0}}(\varepsilon )}}.$$

(22)

Equations (18), (22) determine the dependence of velocity ε'(p) on energy. One is easily convinced that velocity is even in energy and goes to zero at band edges. The second derivative over momentum in this approach is explicitly defined on electronic branch of the spectrum (ε ∈ [–D, 0]), but on the hole branch it is more difficult to do. However, we can require full electron-hole symmetry of the model, which reduces to demanding the square of velocity, entering Eq. (2), being even in ε(p), while Eq. (14) entering Eq. (3) for Hall conductivity being odd (sign change under change of the type of charge carriers).With the account of such symmetry the results obtained in this Appendix allow to replace summation over momenta in Eqs. (2), (3) by integration over energy.