The Splitting of SDW State into Commensurable and Incommensurables Ones and the Peculiarities of the Behavior of Thermodynamic Quantities in a Magnetic Field Arbitrarily Oriented to Magnetization in Quasi Two-Dimensional Systems

Abstract

The quasi-two-dimensional system in which magnetism is caused by spin density wave (SDW) with an anisotropic energy spectrum (with defined impurity concentration x) is examined. The wave vector $\vec{Q}$ is supposed to be different from 2k _F and the umklapp scattering (U-processes) is taken into account. The system is placed in a magnetic field arbitrarily oriented with respect to the vector $\vec{M}_{Q}$. The basic equations for order parameters $M_{Q}^{z}, M_{Q}', M_{z}, M^{\sigma}$ are obtained and the system of these equations is transformed taking into account the U-processes. The particular cases $( \tilde{H} \Vert \vec{M}_{Q} )$ and $( \tilde{H} \bot \vec{M}_{Q} )$ and the case of small arbitrarily oriented magnetic fields $\vec{\tilde{H}}$ are examined in detail. The conditions of the system transition to commensurable and incommensurable SDW state are analyzed. The phase diagram (T,x) at H=0 is traced. The influence of the magnetic field $\vec{\tilde{H}}$ on the temperature of magnetic transition is researched and the aspect of the phase diagram in magnetic field in the cases H ^z H ^σ=0 is presented. The longitudinal magnetic susceptibility χ _∥ which demonstrates that at x<x _c the temperature behavior is similar to the case when the system has a gap, and at x>x _c to a gapless case. At x∼x _c in the dependence X _∥(T) a local maximum appears. The influence of the energy spectrum anisotropy on the system’s properties is researched. Also the angular anisotropy of the quantity χ _∥ at different values of T and x is determined.

Appendix

Here the results of transformation of system of equations (3)–(6) are given taking into account the transfer processes.

The system of equations for order parameter $\vec{M}_{Q}$ and spontaneous magnetization $\vec{M}$ at an arbitrary direction of magnetic field to magnetization and taking into account the U-processes has the form

$$\begin{aligned} M_{Q}^{Z} =&g \sum_{\alpha, \beta} \int _{- \tilde{W}}^{\tilde{W}} N ( \varepsilon )\, d \varepsilon \biggl[ M_{Q}^{Z} I_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \\ &{} + \sum_{\sigma} M_{Q}^{\sigma} \tilde{H}_{Z} \tilde{H}^{-\sigma} \mathcal{I}_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \biggr], \end{aligned}$$

(44)

$$\begin{aligned} M_{Q}^{\sigma} =&g \sum_{\alpha, \beta} \int _{- \tilde{W}}^{\tilde{W}} N ( \varepsilon ) \,d \varepsilon \bigl[ M_{Q} I_{2} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \\ &{}+ \tilde{H}^{\sigma} \bigl[ 2 \tilde{H}_{Z} M_{Q}^{Z} + M_{Q}^{-\sigma} \tilde{H}^{\sigma} \bigr] \mathcal{I}_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \bigr], \end{aligned}$$

(45)

$$\begin{aligned} M_{Z} =&g \sum_{\alpha, \beta} \int _{- \tilde{W}}^{\tilde{W}} N ( \varepsilon ) \,d \varepsilon \biggl[ \tilde{H}_{Z} I_{3} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \\ &{}+ \sum _{\sigma} M_{Q}^{Z} M_{Q}^{\sigma} \tilde{H}^{-\sigma} \mathcal{I}_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \biggr], \end{aligned}$$

(46)

$$\begin{aligned} M^{\sigma} =&g \sum_{\alpha, \beta} \int _{- \tilde{W}}^{\tilde{W}} N ( \varepsilon ) \,d \varepsilon \bigl[ \tilde{H}^{\sigma} I_{4} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \\ &{} + M_{Q}^{\sigma} \bigl[ 2 \tilde{H}_{Z} M_{Q}^{Z} + M_{Q}^{\sigma} \tilde{H}^{\sigma} \bigr] \mathcal{I}_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr) \bigr], \end{aligned}$$

(47)

$$\begin{aligned} x =&2 N_{0} \sum_{\alpha\beta \sigma ' \sigma^{\prime\prime}} \int _{0}^{\tilde{W}} \frac{N ( \varepsilon )}{N_{0}} \biggl\{ \frac{th \frac{\beta ( \tilde{\varphi}_{\sigma ' \sigma^{\prime\prime}} - \mu_{\alpha\beta} )}{2}}{2 \tilde{\varphi}_{\sigma ' \sigma^{\prime\prime}} \tilde{H} A_{k}} \\ &{}\times \bigl[ ( \varepsilon+ \tilde{\varphi}_{\sigma ' \sigma^{\prime\prime}} ) \bigl( \tilde{H}^{2} + \sigma^{\prime\prime} \tilde{H} A_{k} -\sigma \tilde{H}_{Z} ( \varepsilon+ \tilde{\varphi}_{\sigma ' \sigma^{\prime\prime}} ) \bigr) \\ &{} + ( \varepsilon- \tilde{\varphi}_{\sigma ' \sigma^{\prime\prime}} ) \tilde{H}^{2} + \tilde{\psi} \bigr] \\ &{}- \frac{\mathit{th} \frac{\beta ( \tilde{\varphi} '_{\sigma ' \sigma^{\prime\prime}} + \mu_{\alpha\beta} )}{2}}{2 \tilde{\varphi} '_{\sigma ' \sigma^{\prime\prime}} \tilde{H} A_{k}} [ \bigl( -\varepsilon+ \tilde{\varphi} '_{\sigma ' \sigma^{\prime\prime}} \bigr) \bigl( \tilde{H}^{2} + \sigma^{\prime\prime} \tilde{H} A_{k} \\ &{} -\sigma \tilde{H}_{Z} \bigl( -\varepsilon+ \tilde{\varphi} '_{\sigma ' \sigma^{\prime\prime}} \bigr) \bigr) + \bigl( -\varepsilon+ \tilde{\varphi} '_{\sigma ' \sigma^{\prime\prime}} \bigr) \tilde{H}^{2} + \tilde{\psi} \biggr\} \,d\varepsilon. \end{aligned}$$

(48)

Also we have

$$\begin{aligned} &{I_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr)} \\ &{\quad = \frac{1}{8 \tilde{H} A ( \varepsilon )} \sum _{\sigma ', \sigma ''} \sigma ' \sigma^{\prime\prime} \frac{2 \tilde{H}_{Z}^{2} - \sigma^{\prime\prime} \tilde{H} A ( \varepsilon )}{\xi_{\sigma ''} ( \varepsilon )} \mathit{th} \frac{E_{\alpha \sigma ' \sigma^{\prime\prime}}^{\beta}}{2 T},} \\ &{\mathcal{I}_{1} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr)} \\ &{\quad = \frac{1}{8 \tilde{H} A ( \varepsilon )} \sum _{\sigma ', \sigma ''} \sigma ' \sigma^{\prime\prime} \frac{1}{\xi_{\sigma ''} ( \varepsilon )} \mathit{th} \frac{E_{\alpha \sigma ' \sigma^{\prime\prime}}^{\beta}}{2 T},} \\ &{I_{2} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr)} \\ &{\quad = \frac{1}{8 \tilde{H} A ( \varepsilon )} \sum _{\sigma ', \sigma ''} \sigma ' \sigma^{\prime\prime} \frac{\eta_{\sigma ' \sigma^{\prime\prime}} ( \varepsilon )}{\xi_{\sigma ''} ( \varepsilon )} \mathit{th} \frac{E_{\alpha \sigma ' \sigma^{\prime\prime}}^{\beta}}{2 T},} \end{aligned}$$

(49)

$$\begin{aligned} &{I_{3} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr)} \\ &{\quad = \frac{1}{8 \tilde{H} A ( \varepsilon )} \sum _{\sigma ', \sigma ''} \sigma ' \sigma^{\prime\prime} \frac{\varphi_{\sigma ' \sigma^{\prime\prime}} ( \varepsilon )}{\xi_{\sigma ''} ( \varepsilon )} \mathit{th} \frac{E_{\alpha \sigma ' \sigma^{\prime\prime}}^{\beta}}{2 T},} \\ &{I_{4} \bigl( \varepsilon, M_{Q}, \tilde{H}, \mu_{\beta}^{\alpha} \bigr)} \\ &{\quad= \frac{1}{8 \tilde{H} A ( \varepsilon )} \sum _{\sigma ', \sigma ''} \sigma ' \sigma^{\prime\prime} \frac{\psi_{\sigma ' \sigma^{\prime\prime}} ( \varepsilon )}{\xi_{\sigma ''} ( \varepsilon )} \mathit{th} \frac{E_{\alpha \sigma ' \sigma^{\prime\prime}}^{\beta}}{2 T},} \\ &{\eta_{\sigma ' \sigma^{\prime\prime}} ( \varepsilon ) = \frac{1}{4} \bigl[ \bigl( \xi_{\sigma ''} ( \varepsilon ) \bigr)^{2} - 4 \varepsilon^{2} \bigr] - M_{Q}^{2} - \tilde{H}_{Z}^{2},} \\ &{\varphi_{\sigma ' \sigma^{\prime\prime}} ( \varepsilon ) = \frac{1}{4} \bigl[ 2\varepsilon- \sigma ' \xi_{\sigma ''} ( \varepsilon ) \bigr]^{2} - \tilde{H}^{2} + \bigl( M_{Q}^{Z} \bigr)^{2} - \vert M_{Q}^{\sigma} \vert^{2},} \end{aligned}$$

(50)

$$\begin{aligned} &{\psi_{\sigma ' \sigma^{\prime\prime}} ( \varepsilon ) = \frac{1}{4} \bigl[ 2\varepsilon- \sigma ' \xi_{\sigma ''} ( \varepsilon ) \bigr]^{2} - \tilde{H}^{2} - \bigl( M_{Q}^{Z} \bigr)^{2},} \\ &{A ( \varepsilon ) =2 \bigl[ M_{Q}^{2} \cos^{2} \psi + \varepsilon^{2} \bigr]^{{1} / {2}},} \\ &{\xi_{\sigma^{\prime\prime}} ( \varepsilon ) = \bigl[ \bigl( A ( \varepsilon ) - 2 \sigma^{\prime\prime} \tilde{H} \bigr) +4 M_{Q}^{2} \sin^{2} \psi \bigr]^{{1} / {2}},} \end{aligned}$$

(51)

$$\begin{aligned} &{E_{\alpha \sigma ' \sigma^{\prime\prime}}^{\beta} = - \mu_{\beta}^{\alpha} - \frac{1}{ 2} \sigma ' \xi_{\sigma^{\prime\prime}} ( \varepsilon ),} \\ &{\mu_{\beta}^{\alpha} =\mu+\alpha \frac{w_{1} q_{x}}{2} +\beta \frac{w_{2} q_{y}}{2},} \\ &{\tilde{\varphi}_{\sigma ' \sigma^{\prime\prime}} =2 \sqrt{\tilde{H}^{2} + M_{Q}^{2} + \tilde{H} A_{k} + \varepsilon^{2}}, }\\ &{\tilde{\varphi} '_{\sigma ' \sigma^{\prime\prime}} =2 \sqrt {\tilde{H}^{2} + M_{Q}^{2} - \tilde{H} A_{k} + \varepsilon^{2}}} \\ &{\tilde{\psi} = \sigma \tilde{H}_{z} \bigl( \tilde{H}_{z}^{2} - M_{Q}^{z^ 2} \bigr) + \sigma \tilde{H}_{z} \bigl( | M_{Q}^{\sigma} |^{2} +| H^{\sigma} |^{2} \bigr)} \\ &{\phantom{\tilde{\psi} =}{} - \sigma M_{Q}^{z} \sum _{\sigma ' = \sigma, - \sigma} M_{Q}^{\sigma '} \tilde{H}^{- \sigma '}.} \end{aligned}$$

(52)

The summation over α,β=±1 is determined by the consideration of normal and transfer processes.