Phonon Hall effect in four-terminal nano-junctions

Lifa Zhang¹, Jian-Sheng Wang¹ and Baowen Li^1,2,3

¹ Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, 117546 Singapore
² NUS Graduate School for Integrative Sciences and Engineering, 117456 Singapore

³ Author to whom any correspondence should be addressed.

E-mail: phylibw@nus.edu.sg

Received 24 July 2009
Published 20 November 2009

Contents

• 1. Introduction
• 2. Method and formulae
• 3. Results
• 4. Conclusion
• Acknowledgments
• References

1. Introduction

In parallel to the study of electronics, phononics has been a very active and hot topic recently [1]. Several conceptual phononic (thermal) devices such as thermal rectifier/diodes [2], transistors [3], logic gates [4] and even thermal memory [5] have been proposed to control phonons and process information with phonons. Most strikingly, the thermal rectifier has been realized experimentally in nanostructures [6]. The magnetic field is another degree of freedom, which could be potentially used to control thermal transport [7]. Indeed, a novel phenomenon—the phonon Hall effect (PHE)—has been discovered experimentally by Strohm et al  [8], which is an analogue of the electrical Hall effect for the heat flow in dielectrics. The authors found a temperature difference up to 200 μK between the sample edges in the direction perpendicular to both the heat flow and the magnetic field. This effect has been confirmed in [9]. The electronic Hall effect is well understood in terms of the Lorentz force. However, since no charges are involved and phonons cannot couple to the magnetic field directly, it is thus not straightforward at all to understand the PHE. The magnetic field can polarize the paramagnetic ions; the subsystem of isolated ions carrying magnetic moment M couples to phonons, and this spin–phonon interaction (SPI) determines the PHE [10, 11].

In order to understand the physics underlying the experiments [8, 9], theoretical models for PHE have been proposed in [10, 11], in which phonons are treated ballistically. However, according to Strohm et al  [8], the mean free path (1 μm) is far less than the system size (15.7 mm); therefore, it is not appropriate to treat the diffusive PHE with a ballistic theory. Moreover, in all previous theoretical work, the SPI were considered by a perturbation theory, and they are not consistent with each other.

As we know, the thermal transport in nanoscale structures can be regarded as ballistic because the size is far less than the mean free path; therefore we can use a ballistic theory to study the PHE in nanosystems. In this paper, we address two questions: (1) whether the PHE exists in nanoscale systems; (2) whether there is an exact non-perturbative theory for PHE in the ballistic region. The first question is in fact not trivial, since many physical laws valid at the macroscopic scale are not necessarily true at the nanoscale. For example, Fourier's law of heat conduction is broken down at the nanoscale [12]. As for the second question, we will take the nonequilibrium Green's function (NEGF) approach [13]–[15]. The NEGF is an elegant and powerful method to treat nonequilibrium and interacting systems in a rigorous way. The NEGF is widely applied to electronic and thermal transport, and is successful for the study of the spin Hall effect in junctions [16].

2. Method and formulae

To develop a nonperturbative theory for PHE in nanoscale four-terminal junctions, we consider a model shown in figure 1 by taking into account the actual measuring process. The thermal conductance of the system can be calculated by the NEGF method. As we shall see later, our model systems can produce features similar to experiments, even though our systems are at the nanometer scale while the experimental systems are of millimeter scale and are in the diffusive regime.

We consider the same Hamiltonian of SPI as in [10, 11], which can be expressed in the form . Here, and are the vectors of displacement and momentum of the nth lattice site. In the presence of a magnetic field , each lattice site has a magnetization . For isotropic SPI, the isospin is parallel to , and the ensemble average of the isospin is proportional to the magnetization, that is . Therefore, under the mean-field approximation, the SPI can be represented as

where has units of frequency. According to Sheng et al  [10], Λ is estimated to be 0.1 cm^–1≈3×10⁹ Hz at B = 1 T and T = 5.45 K, which is within the possible range of the coupling strength in ionic insulators [17, 18]. In our calculation, we will use this relation to map Λ to the magnetic field.

Now we turn to the derivation of the general formulae for the temperature difference in the system as illustrated in figure 1, where a two-dimensional (2D) lattice sample, which can be honeycomb lattices or square lattices, is connected with four ideal semi-infinite leads. We denote the lattice as N_R ×N_C where N_R, N_C correspond to the number of rows and columns. We adjust temperatures of the upper and lower probes T₃ and T₄ such that the heat currents from these two leads vanish, namely, I₃ = I₄ = 0. Then we get the relative Hall temperature difference as R = (T₃–T₄)/(T₁–T₂).

The total Hamiltonian is assumed to be

where , and A is an antisymmetric, block diagonal matrix with the diagonal elements . Here, the integers 0 to 4 are associated with the center region, left, right, upper and lower leads, respectively. U_α (P_α) are column vectors consisting of all the displacement (momentum) variables in region α. K_α is the spring constant matrix and V_β,0 = (V_0,β)^T is the coupling matrix between the β lead and the central region. The dynamic matrix of the full linear system without SPI is

We obtain the equation for U₀ and P₀ as

The energy flux to the central region from the lead α is,

We define the contour-ordered Green's function as where α and β refer to the region that the coordinates belong to and T_c is the contour-ordering operator. Then the equations of motion of the contour ordered Green's function can be derived. In particular, the retarded Green's function for the central region in frequency domain is G^r [ω] = [(ω + iη)²–K₀–Σ^r [ω]–A² + 2iωA]^–1. Here, , and Σ_α = V_0,αg_αV_α,0 is the self-energy due to interaction with the heat bath, g_α^r = [(ω + iη)²–K_α ]^–1. The lesser Green's function is obtained through G ^<  = G^rΣ ^< G^a in the usual way. We thus can calculate the heat flux by the following formula,

If the temperature differences among the leads are very small, we can treat the system in the linear response regime, T_α = T + Δ_α. The linearized heat flux from each heat bath can be written as

The conductance from heat bath α to β is defined as

where T_βα[ω] = Tr(G^r Γ_βG^aΓ_α ), f = (e ^ω/k_BT–1)^–1, and Γ_α = i(Σ_α^r [ω]–Σ_α^a [ω]). Therefore, if we set the heat flux of lead 3 and lead 4 to zero, we can get the relative Hall temperature difference as

Equations (8) and (9) are the Landauer–Büttiker theory [19]–[21] applied to the multiple-lead thermal transport.

3. Results

In the following calculation, we assume a lattice constant a = 2.465 Å, and the force constant K_L = 0.02394 eV (amu Å²)^–1, K_T = K_L/4. The ratio of the longitudinal and transverse sound speed to be . Then the speed of sound for longitudinal acoustic phonons is about 4000 m s^–1. As mentioned above, Λ is estimated to be about 3×10⁹  Hz ≈2.0×10^–6  eV at B = 1 T. We set al l the couplings between the leads and central region the same, and all the leads and central region have the same spring constants for simplicity.

From equation (10), we find that the relative Hall temperature difference R is an odd function of magnetic field. The Onsager relation, σ_αβ (Λ) = σ_βα (–Λ), always holds due to the definition of the conductance. Furthermore, if there is a symmetry operation S such that

then σ_αβ (Λ) = σ_αβ (–Λ). If this relation is true, then there is no PHE in the system. These symmetry considerations are consistent with a different treatment for bulk systems based on the Green–Kubo formula [22].

We discuss numerical results in the following. Figure 2 shows the temperature difference changing with magnetic field at temperature T = 5.45 K for the honeycomb and square lattices with nearest-neighbor couplings. For the honeycomb case, the Hall temperature is odd and linear in the magnetic field between 0 and 40 T, in that range the slope of the curve is 3×10^–5 K T^–1, comparable to the experimental data in [8]. When the magnetic field is extremely large, it will decrease. From our calculation, we find that the triangular lattice has a similar behavior. However, for a square lattice with the nearest-neighbor coupling, there is no PHE. The spring constant matrix between every nearest coupling site is diagonal for the square lattice. This matrix and also the full matrix K are invariant with respect to a reflection in the x- or y-direction, thus satisfying equation (11). If we consider next-neighbor couplings of the lattice, the dynamic matrix K will not have mirror reflection symmetry, and the PHE can emerge.

We show the conductances among different leads in figure 3. Because of the symmetry of the system, we have additional relations, σ₁₃ = σ₃₂ = σ₂₄ = σ₄₁, and σ₁₄ = σ₄₂ = σ₂₃ = σ₃₁. We find that the conductances between two longitudinal leads or two transverse probe-leads are even in the magnetic field, which can be seen in figure 3(a); σ₃₄ has the same property. However, for a honeycomb lattice the conductance between one longitudinal lead and one transverse probe-lead is not an even function of magnetic field (figure 3(b)), which gives a contribution to the Hall temperature difference. Therefore, for a honeycomb lattice, the temperature difference is not zero. But for square lattices, σ₁₃ is an even function of magnetic field, and the same is true for other components; no PHE exists in such systems.

We show the numerical results for the ratio R at T = 5.45 K for honeycomb lattices in figure 4(a). The temperature difference will not be linear when the magnetic field is larger than 40 T, after about 110 T it will decrease, and about B_c = 230 T, R changes sign to negative. It is the same critical point for the difference of conductances σ₁₃–σ₁₄, which is consistent with equation (10). Here, the magnetic field is very high, which is just a numerical analysis at the parameter of SPI: Λ = 2.0×10^–6  eV; if we can find a material with SPI ten times larger (at the nanoscale it may be possible), the magnetic field can be ten times smaller. In figure 4(b), we show R versus temperature at B = 1 T. When the temperature increases, R will increase almost linearly. After some value, it decreases, and then increases again. Finally, it tends to a constant. This behavior is due to the competition of the numerator and denominator in equation (10). When the temperature is very high, all conductances tend to constants due to the ballistic thermal transport.

In [5], it is shown that R decreases with increasing ratio of the longitudinal and transverse sound speed δ = v_L/v_T and changes sign when δ becomes larger than 5. However, we find that when the ratio (δ > 1) becomes large, R increases, see figure 5(a). At exactly δ = 1, when the longitudinal speed equals the transverse speed, there is no PHE, which verifies our condition, equation (11), for the absence of the PHE. All the spring constant matrices between the nearest-neighbors become diagonal at δ = 1; the condition (11) holds for a mirror reflection operation. If δ < 1, R increases again with the decreasing of δ. Although the ratio R does not change sign with δ, due to the ballistic nature of a small system, it is sensitive to the geometric details, which is shown in figure 5(b): the magnitude and the sign of R change as the size increases.

4. Conclusion

In conclusion, a theory for the PHE in nanoscale paramagnetic dielectrics by the NEGF approach is developed. The results are consistent with the essential experimental features of the PHE, such as the magnitude and linear magnetic field dependence of the observed transverse temperature difference. We find that there is no PHE if the lattice satisfies a certain symmetry. The symmetry of the dynamic matrix K is the key point for the existence of the PHE. The Hall temperature difference changes with the equilibrium temperature and tends to a constant at high temperatures. It does not change sign with the ratio of the longitudinal and transverse sound speed in the range of δ (0.1,10), but it does change sign as the system size increases.

Most of our results should be verified by experiments on nanoscale paramagnetic dielectrics, which can have potential applications in controlling nanoscale phonon transport. For most paramagnetic dielectric materials, because of the complexity of the coupling (such as next or next–next neighboring interaction), the dynamic matrix does not have the mirror reflection symmetry, and the PHE can be present. The Hall temperature difference behavior with magnetic field can be measured in a strong magnetic field. From our study, the PHE can be measured in nanoscale systems at a relatively high temperature (~100 K).

Acknowledgments

We thank Jinwu Jiang, Yonghong Yan, Jie Chen and Jie Ren for fruitful discussions. LZ and BL are supported by the grant R-144-000-203-112 from the Ministry of Education of the Republic of Singapore. J-S W acknowledges support from faculty research grants R-144-000-173-112/101 and R-144-000-257-112 of NUS.

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