Intrinsic nonlinear valley Nernst effect in the strained bilayer graphene

TL;DR

The paper addresses the absence of linear valley responses in centrosymmetric materials with both $\mathcal{P}$ and $\mathcal{T}$ symmetries and introduces a nonlinear valley Nernst effect (NVNE) as a second-order response to a temperature gradient. By developing a semiclassical Boltzmann framework, it identifies an intrinsic NVNE sourced from the quantum metric, independent of the relaxation time, and clarifies the symmetry constraints that govern its existence. The authors demonstrate a concrete realization in uniaxially strained, trigonal-warping bilayer graphene, deriving an effective Hamiltonian and showing that NVNE arises when the temperature gradient is perpendicular to the strain; they further show strain can flip the NVNE sign and that peaks correlate with Lifshitz-like changes in the Dirac-cone structure. An accompanying valley-contrasting orbital magnetization and a proposed magneto-optical Kerr detection scheme provide practical routes to observe NVNE, highlighting potential for strain-tunable valleytronics in centrosymmetric materials.

Abstract

We theoretically analyze the nonlinear valley Nernst effect (NVNE) as the second-order response of temperature gradient through the semiclassical framework of electron dynamics. Our study shows that an intrinsic nonlinear pure valley current can be generated vertically to the applied temperature in the materials with both inversion and time-reversal symmetries. This intrinsic NVNE has a quantum origin from the quantum metric and shows independence from the relaxation time. We find that the local largest symmetry near the valleys for the nonvanishing intrinsic NVNE is a single mirror symmetry in two-dimensional systems. We theoretically investigate the intrinsic NVNE in the uniaxially strained gapless bilayer graphene and find the intrinsic NVNE can emerge when applying the temperature gradient vertically to the direction of strain. Interestingly, a transition from the compressive strain to the tensile one results in the sign reversal of the intrinsic NVNE.

Figures (4)

• Figure 1: Schematic of the nonlinear valley Nernst effect. A transverse nonlinear valley Nernst current $\propto (\nabla T)^{2}$ is generated as a second-order response to the longitudinal temperature gradient in $\mathcal{P}-$ and $\mathcal{T}-$ symmetric materials, where the linear valley current and transverse linear (nonlinear) charge Nernst current disappear.
• Figure 2: Schematic of the energy contour [(a),(b)], the quantum metric elements $\mathcal{G}_{xx}$ [(c),(d)], $\mathcal{G}_{xy}$ [(e),(f)], $\mathcal{G}_{yy}$[(g),(h)] and the temperature-gradient-induced orbital magnetic moment $m_{z}^{\partial_{y} T}$ [(i), (j)] of the conduction band for different valleys of trigonal warping bilayer graphene. The up (bottom) ones are for $\mathrm{K}$ ($\mathrm{K}^{\prime}$) valleys, respectively. The green dash dot line (red dash line) and the solid lines in [(a),(b)] indicate energy band with the uniaxial tensile (compressive) strain and without strain, respectively. The color background in (a) and (b) represent the energy contour without strain ($w=0$). The strength of strain $w=1\varepsilon_{L}$ is fixed in (c)-(j). Momenta (energies) are measured in units of $k_{L}=m^{*}v_{3}/\hbar$ ($\varepsilon_{L}=m^{*}v_{3}^{2}/2$). The $\mathcal{G}_{ab}$ and $m_{z}^{\partial_{y} T}/\partial_{y} T$ are measured in units of $k_{L}^{-2}$ and $e/(\hbar k_{L}^{3})$, respectively.
• Figure 3: (a) The quantity $\alpha_{xyy}^{\mathrm{nl}}$ as a function of Fermi energy $E_{f}$ for different valleys. (b)The quantity $\alpha_{xyy}^{\mathrm{nl,valley}}$ versus the Fermi energy $E_{f}$ for different temperature $T$. (c) The quantity $\alpha_{xyy}^{\mathrm{nl,valley}}$ as a function of Fermi energy $E_{f}$ and strain parameter $w$. (d) The quantity $\alpha_{xyy}^{\mathrm{nl,valley}}$ versus $w$ for different $E_{f}$. Momenta are measured in units of $k_{L}=m^{*}v_{3}/\hbar\approx 0.035 ~ nm^{-1}$ and the effective mass $m^{*}=0.037m_{e}$M. Koshino. $T = 5~\mathrm{K}$ is fixed in (a) and (c). $w = 3~\mathrm{meV}$ is taken in (b).
• Figure C1: Schematics of the proposed device for the nonlinear valley Nernst effect (NVNE) in the strained graphene, showing the bilayer graphene placed on the flexible polymethyl methacrylate (PMMA) substrate and a local heater electrode generates a temperature gradient. NVNE can be detected through the magneto-optical Kerr method, namely focusing a linearly polarized probe beam onto the device under normal incidence and measuring the Kerr rotation angle $\delta\theta$ of the reflected beam.