Coulomb drag in graphene – boron nitride heterostructures: the effect of virtual phonon exchange
For a system of two spatially separated monoatomic graphene layers encapsulated in hexagonal boron nitride, we consider the drag effect between charge carriers in the Fermi liquid regime. Commonly, the phenomenon is described in terms of an interlayer Coulomb interaction. We show that if an additional electron – electron interaction via exchange of virtual substrate phonons is included in the model, the predicted drag resistivity is modified considerably at temperatures above 150 K. The anisotropic crystal structure of boron nitride, with strong intralayer and comparatively weak interlayer bonds, is found to play an important role in this effect.
If two systems containing mobile charge carriers are spatially separated such that direct charge transfer is not possible, but close enough to allow interaction between the carriers in different layers, the resulting momentum transfer will equalize the drift velocities in both systems. This frictional effect was experimentally observed between (quasi) two-dimensional electron gases in double quantum well structures Gramila et al. (1991); Sivan et al. (1992). In most of the theoretical work the interlayer interaction was attributed to Coulomb scattering, hence the effect now bears the name ‘Coulomb drag’ (see Refs. Zheng and MacDonald, 1993; Flensberg et al., 1995; Kamenev and Oreg, 1995).
Interest in the subject has been revived recently by the experimental progress which made it possible to prepare two-dimensional electron systems based on monolayer graphene. A considerable number of theoretical works Amorim and Peres (2012); Carrega et al. (2012); Katsnelson (2011); Narozhny et al. (2012); Peres et al. (2011); Hwang et al. (2011); Tse et al. (2007); Badalyan and Peeters (2012); Scharf and Matos-Abiague (2012); Schütt et al. (2012); Song and Levitov (2012) studied Coulomb drag between massless Dirac fermions, which effectively describe the charge carriers in graphene Castro Neto et al. (2009). However, a quantitatively correct explanation of the experimental data is still lackingAmorim and Peres (2012); Kim et al. (2011); Kim and Tutuc (2012); Gorbachev et al. (2012).
In the typical experiment, Coulomb drag is studied by driving a constant current through one of the layers (the active one, labeled by the index in Fig. 1). If no current is allowed to flow in the other (passive, index 1) layer, a potential difference builds up there. In terms of these two quantities, the drag resistivity serves as a measure of the momentum transfer between the two layers, where and are, respectively, the width and the length of the layer. A theoretical expression for in second order in the interlayer interaction can be derived either using Boltzmann’s kinetic equationHwang et al. (2011); Peres et al. (2011); Flensberg et al. (1995); Jauho and Smith (1993) or the Kubo formula Narozhny et al. (2012); Hwang et al. (2011); Kamenev and Oreg (1995).
In the present work, we focus on the interlayer interaction responsible for the drag effect in heterostructures composed of two graphene monolayers and hexagonal boron nitride (hBN), see Fig. 1. The large bandgap insulator hBN has a layered structure composed of stacked hexagonal crystal planes. Recently the material received much attention as it allows the construction of graphene – hBN devices with, in comparison to the much used SiO substrates, favorable high carrier mobilitiesDean et al. (2010); Mayorov et al. (2011); Schiefele et al. (2012); Garcia et al. (2012). In particular, the Manchester group reported the fabrication of devices where a few layer thin hBN crystal, obtained by exfoliation, is sandwiched between two monolayers of graphene Ponomarenko et al. (2011); Britnell et al. (2012a, b). If such a structure is used for a Coulomb drag experiment, the Dirac fermions in the active and passive layer can exchange momentum not only via Coulomb interaction but also by phonon exchange through the spacer medium. The effect of a combined Coulomb-phonon coupling on the drag resistivity has previously only been studied for quasi two dimensional electron gases in semiconductor systems Jalabert and Das Sarma (1989); Tso et al. (1992); Zhang and Takahashi (1993); Gramila et al. (1993); Güven and Tanatar (1997); Bønsager et al. (1998).
In the following, we first investigate the effects of the anisotropy of hBN, where the bonds in between the graphene-like planes are much weaker than the in-plane bonds, on the electron – electron interaction via phonon exchange. We then show that the inclusion of phonon exchange into the description of Coulomb drag can significantly alter the temperature, density and distance dependence of the predicted value for at temperatures above 150 K.
Ii Interlayer interaction
ii.1 Combined Coulomb – phonon mediated interaction
In a two-layer system as shown in Fig. 1, where the regions I, II and III are filled with a homogeneous isotropic dielectric medium, the Fourier transform of the bare (unscreened) Coulomb potential between electrons in layers and has the form
where , denotes the dielectric constant of vacuum and accounts for the high frequency screening properties of the medium. Apart from this Coulomb interaction, the charge carriers in each graphene layer interact via a substrate phonon mediated interaction. The charge carriers from each layer couple to the long range electric fields generated by optically active phonon modes in the surrounding material via Fröhlich couplingFröhlich (1954); Mahan (1981); Marder (2010). This remote interaction between carriers in graphene and optical phonon modes in a substrate medium was found to influence the electrical conductivity of graphene on a dielectric substrateFratini and Guinea (2008); Chen et al. (2008); Schiefele et al. (2012).
In Appendix A, we show that in an isotropic medium, the combined interaction between electrons in layers and via the effects of a static Coulomb potential and virtual substrate phonon exchange is of the form of Eqn. (1), with replaced by the frequency dependent dielectric function of the substrate material (see Eqn. (17)).
In the following, we specialize to the anisotropic spacer material hBN. From its three acoustic and nine optical phonon bands, only those that (via dipole oscillations) create long range electric fields couple to the graphene electrons111See Refs. Geick et al., 1966; Michel and Verberck, 2011; Serrano et al., 2007 for details on the phonon dispersions of hBN, and the classification of the vibrational modes into Raman active, infrared active and optically silent. Figure 3 and eqns. (21) and (24) of Ref. Michel and Verberck, 2011 show how the long range Coulomb potential associated with the infrared active modes leads to the splitting of transverse and longitudinal optical frequencies at the point. . Given the layered uniaxial crystal structure of hBN, these (infrared active) optical modes are described by a dielectric tensor of the formLoudon (1964)
The resonance frequencies and of the two retarded222We are here using the retarded expression (defined as being analytic in the upper half of the complex plane) in order to be consistent with the likewise retarded polarizability of graphene taken from Ref. Wunsch et al., 2006. Not keeping this consistency yields significantly different results. dielectric functions
are the phonon frequencies at the point for transverse intraplane shear modes with displacements parallel and perpendicular to the -axis of the crystal (aligned with the direction in Fig. 1), respectively. We make the usual approximation of dispersionless optical phonon bands.Fröhlich (1954); Sarma and Mason (1985); Jalabert and Das Sarma (1989) The values for the high frequency dielectric constants , the oscillator strengths (related to the static, , and high frequency dielectric constants, ), and the damping factors taken from Ref. Geick et al., 1966 are listed in Table 1.
To obtain the combined Coulomb-phonon interaction in the anisotropic medium, we solve Poisson’s equation
with being the free charge density of a point charge at the origin. With Eqn. (2), Poisson’s equation becomes
and as and we get
A generalization of this result to structures where the regions I,II, and III (see Fig. 1) are filled with different insulating materials (or air) is straightforward; then involves different dielectric functions than .
ii.2 RPA screened interlayer interaction
To take into account the screening properties of the conduction electrons in the graphene layers themselves, we employ the standard procedure of solving the Dyson equation for the two-layer system within the random phase approximation (RPA) (see Ref. Kamenev and Oreg, 1995). This finally yields the dressed interlayer interaction
where denotes the (frequency and momentum dependent) polarizability of the graphene layers333In the numerical calculations, we use for simplicity the zero temperature expression for as calculated in Refs. Wunsch et al., 2006; Hwang and Das Sarma, 2007, which is a good approximation for , with the Fermi temperature..
Fig. 2 shows a density plot of , using dimensionless units and , where is the Fermi momentum. The horizontal dashed green lines mark the transverse and longitudinal frequencies of the infrared active modes in hBN, connected by the Lyddane-Sachs-Teller relationLyddane et al. (1941) . For small damping , the real parts of are close to a pole at and close to zero at , respectively. Near these frequencies, the absolute value of the total screening function likewise shows an abrupt change from high values (light colors) to almost zero (dark colors). In regions where is small, the red lines show the coupled plasmon-phonon dispersion relation of the two-layer system.
Iii Results for the drag resistivity
In the following, we assume for the sake of simplicity the same positive carrier density (corresponding to electron doping) in both layers, such that . In particular, we do not address the recently reported drag at charge neutrality pointGorbachev et al. (2012), which was attributed either to contributions from higher order perturbation theorySchütt et al. (2012) or to correlated density inhomogenities in the graphene layersSong and Levitov (2012); Gorbachev et al. (2012).
The drag resistivity then assumes a negative valueKim et al. (2011), and the first non-vanishing contribution to obtained in perturbation theory is of second order in the dressed interlayer interactionFlensberg et al. (1995); Kamenev and Oreg (1995). In terms of the variables carrier density, layer separation, and temperature, and under the assumptions that both layers are with high electron doping and 444We here use a simplified form of the nonlinear susceptibility of graphene, which is valid for electron doping high enough such that the existence of the valence band can be ignored. The condition is important as we use the zero temperature expressions for the polarizability of graphene. See Refs. Peres et al., 2011; Amorim and Peres, 2012 for a discussion of both approximations. , it reads (refer to Refs. Peres et al., 2011; Amorim and Peres, 2012 for details)
where denotes the effective fine structure constant in graphene and the integral kernel
The function , defined in Eqn. (18), is related to the nonlinear susceptibility of graphene, and restricts the integration range in the -plane to the region .
In order to estimate the contribution of phonon exchange to the drag effect, we note that the drag resistivity resulting from Coulomb interaction only (which is usually taken as a measure for Coulomb drag) is obtained by substituting the static value of the electron-electron interaction into the integral kernel Eqn. (8):
For low temperatures , the resistivity can be approximated by of Eqn. (19) (see Ref. Amorim and Peres, 2012 for a detailed derivation), under the additional condition (large layer spacing), this can be further approximated to yieldAmorim and Peres (2012)
(Note that in the static limit, one only needs to rescale and to take into account the anisotropy of hBN.) The full blue curves in Figures 3-5 show the absolute value of Eqn. (7) for different parameters , , and , while is shown by dashed red curves, by the dashed black lines in Figures 3 and 5, and the dotted green line in Fig. 5 shows .
As Figures 3 and 4 show, the contribution of phonon mediated interaction to the drag resistivity is vanishingly small at low temperatures, but becomes noticeable for K, the effect being more pronounced the larger the layer separation. This temperature dependence is due the factor in the integration kernel Eqn. (8), which suppresses the integrand for values of . Thus at low temperatures, the main contribution to the -integration in Eqn. (7) comes from a frequency range where the dielectric functions in the integrand are still close to their static values. However, the phonon contribution becomes noticeable at lower temperatures than one would expect, taking into account that the energy of the lowest phonon mode . It is also interesting to notice that in the range from 100 to 250 K, the drag resistivity , including the effect of phonons, is closer to the behaviour than the purely Coulomb drag result, . The plot of as a function of in Fig. 6 shows the origin of the phonon contribution to the integral : With rising temperature, peaks near the resonance frequencies and appear in the integrand, which enhance the magnitude of .
Fig. 4 shows that the relative effect of phonon exchange on is larger for high densities. For high values of , the argument of the dielectric functions in Eqn. (8) reaches the resonance frequency already at lower values of . While decreases rapidly with due to increased screening of the Coulomb interaction, the modification of the screening function by phonon interaction is seen to counteract this decrease at high temperatures.
Finally, Fig. 7 illustrates the effect of the anisotropy in hBN that enters through the electron-electron interaction Eqn. (4). We compare the drag resistivity in hBN with that in an isotropic medium with dielectric functions and , respectively (see Eqn. (2)). The difference in magnitude between and is seen to be greatest in the anisotropic case, where both in-plane and out-of-plane phonon modes contribute to the interlayer interaction.
|3.61 meV||0.995 meV|
|170 meV||97.4 meV|
Iv Summary and discussion
We showed that including the electron – electron interaction via phonon exchange into the theory of Coulomb drag significantly changes the magnitude of the predicted drag resistivity in graphene-hBN heterostructures. For large layer separations, the deviations become noticeable at temperatures higher than 150 K.
As the lowest phonon resonance frequency in the spacer material hBN corresponds to a temperature of approximately 1100 K, our result at first sight seems to be at odds with the notion that phonon effects should be proportional to the thermal population factor of the relevant modes. This is indeed the case for other transport phenomena, like the substrate limited electron mobility in graphene, where real momentum transfer from an electronic state (in graphene) to a phonon mode (in a dielectric substrate material) plays a roleSchiefele et al. (2012); Fratini and Guinea (2008). The decay rate of the electronic state is then overall proportional to the thermal population of the phonon mode. Our scenario however involves the exchange of virtual phonons in a process that is of second order in the interlayer interactionKamenev and Oreg (1995), and no decay processes into real phonon states are relevant for . We note that in Ref. Carrega et al., 2012, the effect of substrate phonons on Coulomb drag was considered for the case where a material described by a uniform dielectric function fills what is our region III of Fig. 1, and a deviation from the low-temperature behavior of was predicted for temperatures roughly an order of magnitude lower than the phononic resonance frequency of the substrate material.
Up to date, there remains considerable discrepancy between experimental data on Coulomb drag between graphene layers embedded in SiO/AlOKim et al. (2011); Kim and Tutuc (2012) and hBNGorbachev et al. (2012) and the existing theoretical work. For hBN, the reported drag resistivities in the Fermi liquid regime are roughly a factor of three larger than predicted, and the results of the present paper do not change this situation. The experimentally reported dependence of for nm and nm up to temperatures of 240 K777See Fig. 2a in Ref. Gorbachev et al., 2012. does not disagree with our results presented in Fig. 3. Actually, it appears that in the temperature range of 100 to 250 K the inclusion of phonon mediated interaction brings the behaviour of drag closer to the low temperature behaviour than with static Coulomb interaction only. Nevertheless, an extension of the experimental data shown in Ref. Gorbachev et al., 2012 up to room temperature would be needed to distinguish clearly between and . The use of other substrate materials, such as SiO, should not qualitatively alter the results of this paper. We think that future experiments with devices as considered in the present work will be able to check our predictions.
Acknowledgements.The authors would like to thank N.M.R. Peres for useful discussions. Financial support from Fundação para a Ciência e a Tecnologia (Portugal) through Grant No. SFRH/BD/78987/2011 (B.A.), the Marie Curie ITN NanoCTM (J.S.) and from MICINN (Spain) through Grant No. FIS2010-21372 (F.S.) and FIS2008-00124 (F.G.) is acknowleged.
Appendix A Fröhlich electron–phonon coupling and phonon mediated electron–electron interaction
Throughout the present work, we assume the dielectric properties of hBN layers forming heterostructures as shown in Fig. 1 to be the same as for bulk hBN.
The Fröhlich Hamiltonian describing the coupling of electrons to a bulk polar longitudinal phonon mode (in an isotropic homogeneous dielectric material) is given byFröhlich (1954); Mahan (1981); Marder (2010)
where denotes the electron density operator, () the creation (annihilation) phonon operator with momentum and the matrix element reads
with the longitudinal optical phonon frequency . The phonon mediated interaction between electrons is given by
with the bare phonon propagator
The bare Coulomb interaction is given by
where takes into account the high frequency screening properties of the medium.
with the dielectric function of the medium, see Eqn. 3. Since the Fröhlich coupling is derived in a phenomenological approach based on the dielectric properties of the material, the combined Coulomb and phonon mediated interaction simply reduces to the Coulomb interaction screened by , as it should.
where is a two-dimensional momentum vector. In analogy to the above, we now get for the combined Coulomb–phonon interaction in a homogeneous isotropic medium
Although it is possible to generalize the Fröhlich electron–phonon coupling for the case of anisotropic materialsLoudon (1964); Stroscio and Dutta (2003) and inhomogeneous layered materialsMori and Ando (1989), the easiest way to obtain the effective electron–electron interaction, taking into account the phonon mediated interaction, is by solving Poisson’s equation for the electric potential created by a point charge in the dielectric medium taking into account the frequency dependence of its dielectric tensor.
Appendix B Mathematical details
The dressed interlayer interaction Eqn. (5) is the solution of the coupled set of Dyson equations
where the dashed and wiggled lines denote the bare Coulomb and phonon interaction, respectively, and the full curves electron propagators (see Refs. Zhang and Takahashi, 1993; Kamenev and Oreg, 1995).
For the low-temperature approximation of , the factor in the integration kernel Eqn. (8), which suppresses the integrand for values of , allows one to expand the remaining integrand to the lowest order of . The integration can then be performed, yielding
where (see Ref. Amorim and Peres, 2012 for details).
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