Valley transport driven by dynamic lattice distortion

Valleytronics is an emerging field that uses the valley degrees of freedom, i.e., the local extrema of the electronic band structure, with the aim of developing innovative approaches to information processing, optoelectronic devices, and quantum computation Schaibley et al. (2016); Krasnok et al. (2018); Vitale et al. (2018). In 2D Dirac materials such as gapped graphene Semenoff (1984); Xiao et al. (2007) and monolayer transition-metal dichalcogenides (TMDCs) Xiao et al. (2012), the electrons carry valley-dependent orbital angular momentum that originates from the Berry curvature caused by inversion symmetry breaking Xiao et al. (2007, 2010). As a result, the valleys in these systems can be identified based on the intrinsic orbital angular momentum of the electrons, as illustrated in Fig. 1 (a). This means that the valleys can be controlled using an external field coupled to the orbital angular momentum. In fact, it has been demonstrated experimentally that circularly polarized light beams Cao et al. (2012); Mak et al. (2012); Zeng et al. (2012); Sallen et al. (2012); Guddala et al. (2021) and magnetic fields Li et al. (2014); Aivazian et al. (2015); MacNeill et al. (2015); Srivastava et al. (2015) can be used to manipulate the valley degree of freedom. Since these demonstrations, more versatile valley control methods have been desired. One candidate method taken from familiar phenomena is the gyroscopic effect, by which the angular momentum is coupled with mechanical rotation.

This gyroscopic coupling technique has been observed and used in a variety of systems. For example, gyroscopes are used to measure the gravitomagnetic field precisely in a curved space near the Earth Everitt et al. (2011). The quantum version of the gyroscopic effect is called the gyromagnetic effect, where both the orbital and spin angular momenta are coupled to the mechanical rotation. The gyromagnetic effect was discovered during an attempt to explore the origins of magnetism Einstein and de Haas (1915); Barnett (1915, 1935) and led to the discovery of electron spin angular momentum before quantum mechanics was established.

While the original targets in these gyromagnetic experiments were limited to ferromagnetic materials, gyromagnetic coupling itself is a universal phenomenon that occurs even in nonmagnetic materials. Indeed, gyromagnetic coupling has been observed in various branches of physics, including spintronics Chudo et al. (2014); Imai et al. (2018, 2019); Takahashi et al. (2016, 2020); Kazerooni et al. (2020, 2021); Kobayashi et al. (2017); Harii et al. (2019); Okano et al. (2019); Kurimune et al. (2020a, b); Mori et al. (2020); Tateno et al. (2020, 2021), ultrafast demagnetization processes Dornes et al. (2019), and quark-gluon many-body systems Adamczyk et al. (2017). These studies have shown that the coupling between angular momentum and rotation emerges universally in a variety of systems and continues to have a tremendous impact on a wide range of topics in fundamental physics.

In this work, we propose an alternative valley manipulation technique that uses gyroscopic coupling between the valley and the local rotational motion (or vorticity) excited by dynamic distortion in 2D Dirac materials. Figure 1 (b) shows schematic images of systems in which 2D Dirac materials are placed on a substrate and dynamic lattice distortion is then applied. We consider a Love wave, which is a type of horizontally polarized surface acoustic wave (SAW). This SAW leads to two effective magnetic fields: the strain-induced pseudo-magnetic field $B_{z}^{s}$ and the vorticity field $\omega_{z}$ as shown in Fig. 1 (c).

Both of these effective fields couple to the valley orbital angular momentum, thus allowing the valley to be manipulated using the SAW. We also consider electron doping of these 2D materials. In the presence of a standing SAW, valley transfer and dynamic valley polarization can be realized, depending on the Fermi energy $\varepsilon_{F}$ and the strengths of $B^{s}_{z}$ and $\omega_{z}$.

We also discuss the charge transport when an external DC electric field is applied to 2D Dirac materials in the presence of the SAW. We find that the conductivity is suppressed by the SAW and that a pulsed charge current is generated; this current can be used as a broadband microwave source. Our findings presented here pave the way towards valley device applications for 2D Dirac materials using the SAW devices Delsing et al. (2019).

The paper is organized as follows. The emergent gauge fields induced by the dynamic lattice distortion are introduced in Sec. II. The mechanism and procedure to incorporate the emergent gauge fields are explained. Finally, two kinds of the Zeeman like coupling are derived. The valley manipulation using the SAW are proposed in Sec. III. Furthermore, pulsed current generation and microwave radiation based on the proposed valley manipulation mechanism are discussed as application examples in Sec. IV and V, respectively. A conclusion is given in Sec. VI.

We consider 2D Dirac materials in which the dynamic lattice distortion is induced by the SAW in the substrate. The dynamic lattice distortion leads to two kinds of emergent gauge fields, the velocity fields due to the inertial effect Matsuo et al. (2017) and the strain-induced gauge field Suzuura and Ando (2002); Manes (2007); Vozmediano et al. (2010). They originate from the different mechanism. In the following subsections, we describe how each gauge field arises and how it is incorporated into the low-energy effective Hamiltonian of 2D Dirac materials.

In our setup, the Love wave is excited in the substrate by an external force, which leads to the dynamic lattice distortion in 2D Dirac materials. The detailed properties of the Love wave are explained in the Appendix. The lattice displacement vector in 2D Dirac materials is given by

which means that the vorticity field and the pseudo-magnetic field are given by

with $\omega_{0}=u_{0}\omega^{2}/c_{s}$ and

with $B_{0}^{s}=\omega_{0}E_{0}/evc_{s}$, respectively. As shown in the Appendix, the dispersion relation is approximately linear, so that we use the dispersion relation $\omega=c_{s}k$, where $c_{s}$ is the SAW velocity. The sum of the pVZC and the VVC can then be treated as a periodic potential

where the dimensionless material parameter $R$ is introduced

Different valley manipulations are possible, depending on the value of $R$. Note that the dynamic lattice distortion induced here is not directly related to the phonon modes in the 2D Dirac materials. We expect that the effect due to the pVZC and the VVC can be detected even in the presence of the thermally activated phonon modes. Indeed, phenomena related to our proposal, such as strain induced Landau level formation in graphene Nigge et al. (2019) and spin manipulation using the SAW device Weiler et al. (2011); Kobayashi et al. (2017); Kurimune et al. (2020a), have been observed at room temperature.

We consider three cases showing characteristic carrier distribution in the electron-doped system.

• •

  (i) When the Fermi energy is sufficiently smaller than both the pVZC and the VVC (i.e., when $\varepsilon_{F}\ll|l\gamma B_{0}^{s}|,|l\omega_{0}|$), the carriers are trapped at the bottom of potential and are transferred in the opposite direction for each valley, as shown in Fig. 1 (b)(i). This is a promising candidate mechanism to convey the valley information. This case can be realized using any value of $R$.

• •

  (ii) When the Fermi energy is sufficiently larger than the VVC but smaller than the pVZC (i.e., when $|l\omega_{0}|\ll\varepsilon_{F}\ll|l\gamma B_{0}^{s}|$), the carriers are then localized at the time $\omega t=(N+1/2)\pi,\ (N=0,\pm 1,\cdots)$, as shown in Fig 1 (b)(ii). In this case, the spatial distribution of the carriers is almost independent of the valley. This case can be realized when $R\gg 1$.

• •

  (iii) When the Fermi energy is sufficiently larger than the pVZC but smaller than the VVC (i.e., when $|l\gamma B_{0}^{s}|\ll\varepsilon_{F}\ll|l\omega_{0}|$), the carriers are then localized at the time $\omega t=N\pi,\ (N=0,\pm 1,\cdots)$, as shown in Fig. 1 (b)(iii). In contrast to the second case, each valley’s carrier is localized in a different location separated by $\pi/q$. This case can be realized when $R\ll 1$.

The valley transfer shown in Fig. 1 (b)(i) and the carrier localizations shown in Fig. 1 (b)(ii) and (iii) are detectable experimentally using time-resolved optical diffraction measurements. Because the carrier density pattern is periodic, diffraction patterns will appear under coherent illumination. The periods shown in Fig. 1 (b)(ii) and (iii) are different and thus the corresponding diffraction patterns are also different. Furthermore, the valley dependence of the carriers is also detectable using Kerr rotation microscopy, because the valleys carry an orbital magnetic moment Lee et al. (2016). The experimental signatures are listed in Table. 1.

Next, we estimate the strengths of the vorticity field and the pseudo-magnetic field. Using the parameters of graphene from the previous study Suzuura and Ando (2002), the relationship for the strengths of the vorticity field and the pseudo-magnetic field is given as $\omega_{0}/\gamma\sim 10^{-3}\times B^{s}_{0}$, and when the parameters of $\mathrm{MoS}_{2}$ from the preious study Rostami et al. (2015) are used, the relationship is given as $\omega_{0}/\gamma\sim B_{0}^{s}$. The experimentally feasible parameters for the SAW are $u_{0}\approx 100\ \mathrm{pm}$, $c_{t}\approx 10^{3}\ \mathrm{m/s}$, and $\omega\approx 30\ \mathrm{GHz}$, meaning the strength of the vorticity field can be estimated to be $\omega_{0}/\gamma\sim 0.01\ \mathrm{T}$.

In addition to the spatial distribution of the carriers, the carrier transport properties are also modulated by the SAW. Here, we discuss their longitudinal transport properties along the $x$ direction in the three cases described above based on the semiclassical Boltzmann transport theory. In case (i), the carriers are trapped at the bottom of the potential $U$ at any time, which means that the system becomes insulating. In cases (ii) and (iii), conversely, the system undergoes repeated carrier localization and delocalization when the amplitude of $U$ exceeds and falls below $\varepsilon_{F}$, respectively, because of the time dependence of $U$. To investigate the longitudinal transport properties in the latter cases, we use semiclassical Boltzmann transport theory. In the following discussion, we focus on the second case and omit the VVC for simplicity, which corresponds to the assumption that $R\gg 1$. This simplification does not affect the main results given below because the effect of the VVC is negligible due to the condition $|l\omega_{0}|\ll\varepsilon_{F}$. As shown in Eq. (9), although there is a phase difference between the pVZC and the VVC, their qualitative effects on the longitudinal transport properties are almost the same. Therefore, the results are also applicable to the third case.

The semiclassical Boltzmann kinetic equation is given by

where $f_{\xi}$ is the distribution function for the valleys $K_{+}$ and $K_{-}$, and the equations of motion are given by

where $\varepsilon_{\xi,\bm{p}}$ is the energy band in the presence of the pVZC, $\bm{\Omega}_{\xi}=\hbar\nabla_{\bm{p}}\times i\langle u_{\xi,\bm{p}}|\nabla_{\bm{p}}|u_{\xi,\bm{p}}\rangle$ is the Berry curvature for each valley, and $|u_{\xi,\bm{p}}\rangle$ is the periodic part of the Bloch wave function. The second term in Eq. (12) is the anomalous velocity Xiao et al. (2010), the second term in Eq. (13) is the Coriolis force that originates from the vorticity field of the lattice Matsuo et al. (2017), and the third term in Eq. (13) is the valley-dependent Lorentz force that originates from the pseudo-magnetic field.

Next, we consider the charge transport along the $x$-direction in the presence of the external electric field $\bm{E}=(E_{x},0,0)$. We use the following relaxation time approximation:

where $\tau$ is the valley independent relaxation time and $f^{(0)}_{\xi}=1/(1+\exp[(\varepsilon_{\xi,\bm{p}}-\varepsilon_{F})/k_{B}T])$ is the Fermi distribution function. The above expression of the relaxation time approximation is based on the three assumptions. First, the valley quantum number is treated as a conserved quantity. This assumption is reasonable if the sample is sufficiently clean that the effect of the atomic scale scatterers is negligible and there is no intervalley scattering process. In addition, the SAW rarely leads to intervalley scattering because the wavelength of the SAW is much larger than the lattice constant. Next, the relaxation time approximation is valid even in the presence of the time-dependent potential because the SAW period is much larger than the lifetime of the electron. In our setup, the SAW period is about a nanosecond, and the lifetime of the electron would be about a picosecond. Finally, the relaxation time is independent of the valley quantum number. This assumption is reasonable if the scatterers have no characteristic feature that breaks the valley degeneracy.

Assuming a weak vorticity field and a weak pseudo-magnetic field (i.e., $\omega_{0}m|\Omega_{\xi,z}|\ll 1$, $\omega_{0}\tau\ll 1$, and $\omega_{c}\tau\ll 1$ with $\omega_{c}=eB^{s}_{0}/m^{\ast}$), the anomalous velocity, the Coriolis force, and the Lorentz force then only give corrections for the longitudinal conductivity and have almost no effect on the qualitative behavior of the longitudinal conductivity. Based on this assumption, we can omit these three terms. Note here that there is no Hall voltage in the current setup because the net vorticity field is zero. The SAW dynamics are much slower than the dynamics of the electrons, which means that the adiabatic approximation is valid and it can be assumed that the steady state is achieved at each moment. Finally, the charge current can be calculated by deriving the steady state of the distribution function at each moment and using snapshot energy bands. We define $\delta f_{\xi}$ as $f_{\xi}=f^{(0)}_{\xi}+\delta f_{\xi}$, and $\delta f_{\xi}$ is given by

where $v_{x}=\partial\varepsilon_{\xi,\bm{p}}/\partial p_{x}$. Using $\delta f_{\xi}$, the charge current at $T=0$ can be given by

Figure 2 (a)-(c) show snapshot energy bands as a function of $p_{x}$ with a fixed $p_{y}=0$ at several values of $\omega t$. The pVZC is treated as a spatially periodic potential in each case, which means that the energy gap opens and the group velocity decreases as the amplitude $B_{z}^{s}$ increases. Therefore, the charge current is suppressed by the SAW. Figure 2 (d) shows the charge current as a function of $\omega t$ and the Fermi energy $\varepsilon_{F}$. At a fixed $\varepsilon_{F}$, the charge current changes with the period $\pi$ and reaches a maximum value at $\omega t=N\pi,\ (N=0,\pm 1,\cdots)$. The charge current is almost zero around $\omega t=(N+1/2)\pi,\ (N=0,\pm 1,\cdots)$ when the Fermi energy is sufficiently small in comparison to the pVZC. Figure 2 (e) shows the charge current as a function of $\omega t$ at several fixed $\varepsilon_{F}$. One can see that the sharp pulsed current is generated when $\varepsilon_{F}\ll|l\gamma B_{0}^{s}|$. At a finite temperature, the pulsed current is subject to thermal broadening. The sharp pulsed current can still be generated in a condition that the thermal broadening energy is much smaller than the pVZC (i.e., $k_{B}T\ll|l\gamma B_{0}^{s}|$). Using the parameters for graphene with energy gap $\Delta=0.1\ \mathrm{eV}$ Zhou et al. (2007, 2008), the pVZC is estimated as $|l\gamma B_{0}^{s}/2|\sim\ 30\mathrm{meV}$, which is the same order of the thermal broadening energy at room temperature. Therefore, the condition to generate the sharp pulsed current is expected to be experimentally feasible. Within the quadratic approximation, the Fermi energy $\varepsilon_{F}$ is proportional to the carrier density $n$ (i.e., $\varepsilon_{F}=2\pi\hbar^{2}n/m^{\ast}$), so that one can replace the Fermi energy in Fig. 2 with the carrier density by multiplying the factor $m^{\ast}/2\pi\hbar^{2}$.

To summarize the discussion of the transport properties above, when the Fermi energy is sufficiently smaller than both the pVZC and the VVC, the system becomes insulating, but when the Fermi energy is larger than either the pVZC or the VVC, the conductivity is then suppressed and a pulsed current can be generated, as listed in Table. 1.

The charge current obtained above can be divided into an AC component and a DC component using the form $j_{x}(t)=j^{\mathrm{ac}}_{x}(t)+j^{\mathrm{dc}}_{x}$, with $j^{\mathrm{dc}}_{x}=j_{x}(\pi/2\omega)$. The DC component $j^{\mathrm{dc}}_{x}$ corresponds to the minimum value of the charge current and becomes finite when $\varepsilon_{F}/\varepsilon_{0}\gtrsim 0.54$. The oscillating current $j^{\mathrm{ac}}_{x}(t)$ can be used for broadband and tunable microwave sources, which is useful for device applications Hu et al. (2021). A schematic image of the microwave radiation in our setup is shown in Fig. 3(a).

Figure 3 (b) shows the Fermi energy dependence of the full width at half maximum (FWHM) value of the pulse. When the Fermi energy increases, the FWHM also increases, and the range of Fourier components decreases. There is a kink structure of the FWHM at $\varepsilon_{F}/\varepsilon_{0}\approx 0.54$ because $j^{\mathrm{dc}}_{x}$ becomes finite and increases above this point.

The averaged radiation intensity of the microwave in the upper half space $I$ is given by

where $Z_{0}\approx 377\ \Omega$ is the characteristic impedance of free space, and $\epsilon_{+}$ and $\epsilon_{-}$ are permitivity of free space and substrate, respectively. Figure 3 (c) shows the averaged intensity as a function of the Fermi energy. The averaged radiation intensity becomes a local maximum at $\varepsilon_{F}/\varepsilon_{0}\approx 0.54$ above which $j^{\mathrm{dc}}_{x}$ becomes finite. Setting the charge current as $j_{0}\sim 1\ \mathrm{A}/\mathrm{m}$ and the permitivity as $\epsilon_{\pm}\sim 1$, the averaged radiation intensity is estimated as $I\sim 0.1\ \mathrm{mW}/\mathrm{cm}^{2}$ in the energy range considered here.

In this work, we have presented an investigation of the low-energy effective Hamiltonian of 2D Dirac materials in the presence of the dynamic lattice distortions. We have derived the valley-vorticity coupling (VVC), which is regarded as an inertial effect. In addition to the VVC, we also incorporated the strain-induced pseudo-magnetic field into the analysis. We have proposed mechanisms for both valley transfer and valley-dependent carrier localization using an SAW.

We have also investigated the electronic transport properties in the presence of the SAW. The conductivity along the direction of propagation of the SAW is suppressed by the dynamic lattice distortions. Periodic modulation of the conductivity results in a pulsed charge current when an electrostatic field is applied along the SAW. This pulsed charge current act as a broadband microwave source and can be used for optoelectronic device applications. These results pave the way toward the realization of next-generation valleytronics device applications using the SAW devices.

Acknowledgements.— The authors are grateful to Y. Nozaki and A. Yamakage providing valuable comments. D.O. is funded by the President’s PhD Scholarships at Imperial College London. This work was supported by the Priority Program of Chinese Academy of Sciences under Grant No. XDB28000000, and by JSPS KAKENHI for Grants (Nos. JP20H01863 and JP21H04565) from MEXT, Japan.