Many-body effects and possible superconductivity in the 2D metallic surface states of 3D topological insulators
Abstract
We theoretically consider temperature and density-dependent electron-phonon interaction induced many-body effects in the two-dimensional (2D) metallic carriers confined on the surface of the 3D topological insulator (e.g. BiSe). We calculate the temperature and the carrier density dependence of the real and imaginary parts of the electronic self-energy, the interacting spectral function, and the phonon-induced velocity renormalization, enabling us to obtain a simple density and temperature dependent effective dimensionless electron-phonon coupling constant parameter, which increases (decreases) strongly with increasing density (temperature). Our theoretical results can be directly and quantitatively compared with experimental ARPES or STS studies of the 2D spectral function of topological insulator surface carriers. In particular, we predict the possible existence of surface superconductivity in BiSe induced by strong electron-phonon interaction.
pacs:
72.80.Vp, 81.05.ue, 72.10.-d, 73.22.PrElectronic properties of 2D metallic states on 3D topological insulator (e.g. BiSe) surfacesKim et al. (2012, 2012); Li et al. (2012) continue to be one of the most active areas of condensed matter physics and materials science research world-wideHasan and Kane (2010); Fu et al. (2007); Qi and Zhang (2011); Culcer (2012); Bernevig (2013). The reason for this intense interest is partly fundamental and partly technological. The 2D metallicity of the surface carriers here is topologically protected by the time reversal symmetryKane and Mele (2005) as long as the bulk is a gapped insulatorXia et al. (2009), and as such, it is a new kind of a 2D electron system where carrier back-scattering is completely suppressed thus making it possible for a 2D system to be a metal even at in the presence of arbitrary amount of impurity disorderHsieh et al. (2009a). This is in sharp contrast to ordinary 2D electron systems which are generically insulators at in the presence of any finite disorder due to the complete destructive quantum interference arising from the scattering. The 2D metallic surface carriers in topological insulators form helical bands with massless linear Dirac like energy dispersion protected by time reversal invarianceFu et al. (2007); Moore and Balents (2007).
Since the basic phenomenon of 3D topological insulators (TI) with the associated protected 2D gapless metallic surface carriers arises entirely from purely single-particle physics, much of the theoretical work on the subject has focused on the single-particle aspects involving topology and symmetry, including band structureZhang et al. (2009), magnetic impuritiesHe et al. (2011), magneto-electric responseEssin et al. (2009); Qi et al. (2009); Maciejko et al. (2010), integer quantum Hall effectLee (2009), localizationLu et al. (2011), and topological classificationSchnyder et al. (2008); Kitaev (2009). In the current work we focus on many-body TI properties with a theoretical investigation of the interaction-induced renormalization of the single-particle properties of the surface 2D carriers. Such many-body renormalization of the single-particle properties manifest themselves directly in various spectroscopic measurements such as angle resolved photoemission spectroscopy (ARPES)Park et al. (2010); Hatch et al. (2011); Pan et al. (2012) and scanning tunneling spectroscopy (STS)Roushan et al. (2009), and therefore, our theory presented in this work provides direct predictions for ARPES and STS studies of TI surface metallic states. Since our theory provides the renormalized quasiparticle energy dispersion and level broadening, it is of crucial importance in understanding many properties of the TI surface states.
In discussing electronic many-body effects in general, it is useful to distinguish between electron-electron and electron-phonon interaction effects. Both are, of course, always present in any experimental solid state system, and can, in fact, be studied on an equal footing theoretically since the electron-phonon interaction can be eliminated in favor of an effective phonon-mediated electron-electron interaction through a canonical transformation which simply adds to the direct electron-electron Coulomb interaction to produce a complicated effective electron-electron many-body interaction including phonon effects implicitlyJalabert and Das Sarma (1989). Such an effective Hamiltonian is a good starting point when electron-electron and electron-phonon interactions are of comparable strengths. 2D carriers on TI surfaces are, however, very weakly interacting Coulomb systems by virtue of the bulk material (i.e. BiSeXia et al. (2009); Hsieh et al. (2009b), BiTeChen et al. (2009); Hsieh et al. (2009c), BiSbHsieh et al. (2008), etc.) usually having a very large lattice dielectric constant () which strongly suppresses the effective 2D Coulomb-interaction (going as where is the 2D momentum transfer with the background lattice dielectric constant). The Coulomb interaction strength (i.e. the ratio of the Coulomb potential energy to the noninteracting kinetic energy) in the 2D surface carriers, defined by the effective fine-structure-constant where is the velocity (typically cm/s) of the linear, massless Dirac band dispersion of the 2D carriers, is very small () by virtue of the large background lattice dielectric constant. As such all effects of direct electron-electron Coulomb interaction can be safely neglected in discussing TI electronic properties except perhaps at extremely small 2D carrier density of little experimental or technological interestAbergel and Das Sarma (2013).
The starting point of our theory is the calculation of the finite-temperature electron self-energy arising from the electron-phonon interaction, which can be written in the leading-order theory asGrimvall (1981)
(1) |
and,
(2) |
In Eqs. (1), (2), is the wavevector or momentum () and energy or frequency () dependent finite-temperature 2D electron self-energy arising from the interaction of the 2D carriers with the effective TI surface acoustic phonons with the phonon dispersion given by where is the surface phonon velocity. The electron-phonon interaction is considered to be mediated by the deformation potential couplingThalmeier (2011) quantified by the coupling strength given by: with and respectively being the deformation potential coupling strength and the 2D (ionic) mass density of the TI material. In Eqs. (1) and (2), denotes the linear massless 2D energy dispersion of the surface metallic electron-hole bands (with denoting electrons/holes in the conduction/valence band) measured with respect to the finite-temperature chemical potential , and is the finite-temperature Fermi distribution function corresponding to the carrier energy . Finally, is the finite-temperature Bose distribution function for phonons with energy . We mention that the factor is a matrix element effect arising from the topological nature of the surface bands which suppresses backscattering (i.e. for ) due to the chiral nature of the surface 2D bands associated with the intrinsic spin-orbit coupling in the system. We note that although we explicitly consider 2D surface phonons, the same theory should also apply for the interaction of the surface 2D electrons with 3D bulk acoustic phonons where a sum over all the bulk transverse phonon wavevectors have to be carried out, thus reducing the bulk phonon coupling to an effectively 2D coupling problem any way. We do not, however, believe that the coupling to bulk phonons would play a significant role in the physics under consideration since our interest here is restricted entirely to the surface 2D carriers.
The finite-temperature electron-phonon self-energy defined by Eqs. (1) and (2) arises directly from the formal expression for the self-energy where an integration over all internal energy/momentum is implied with being the electron propagator, the phonon propagator, and the electron-phonon interaction matrix element respectively. This leading-order expression for the self-energy is, in fact, essentially exact for the weak-coupling problem by virtue of Migdal’s theorem guaranteeing all vertex correction contributions being parametrically small in powers of ( in the current problem).
Once the electron self-energy is calculated using Eqs. (1) and (2), there are two possible alternative (and, in principle, inequivalent) methods to define an effective dimensionless electron-phonon coupling strength parameter ’’ for the system (which depends on the carrier density and temperature ) given byGrimvall (1981); Allen and Cohen (1970)
(3) |
where , andAllen (1971)
(4) |
where , are respectively the Fermi temperature and the Bloch-Grüneisen temperature with . The effective coupling constant in Eq. (3) directly provides the (density and temperature dependent) 2D carrier velocity renormalization due to the electron-phonon interactionAllen (1971): . The effective coupling constant in Eq. (4) is the high-temperature electron-phonon coupling parameter directly contributing to the phonon-induced carrier resistivity for — in particular, for , where is the phonon-dominated linear-in-temperature electronic resistivity at high temperatures above the Bloch-Grüneisen regimeGiraud and Egger (2012); Kim et al. (2012).
Before presenting our detailed results for the electron-phonon self-energy, we mention that the standard textbook definition of the dimensionless electron-phonon coupling constant in metals (e.g. the coupling strength entering the standard BCS-Eliashberg theory for phonon-induced superconductivity) is based on the electronic density of states (DOS) at the Fermi energy and is defined asAllen and Cohen (2011):
(5) |
where is the electronic density of states of the surface 2D Dirac carriers at the Fermi surface ( or ).
How do , , relate to (or compare with) each other? This is, in fact, a main topic discussed in this work, but it maybe useful to mention here that we find the three definitions to be completely consistent with each other in their respective regimes of validity.
Using the delta function in Eq. (2), and using the high-temperature () expression for the Bose distribution function , it is straightforward to calculate the asymptotic form for in the high-temperature () limit of the equipartition regime for the acoustic phonons, obtaining
(6) |
with denoting electron/hole energy. Eq. (6) combined with the definition for given in Eq. (4) leads to given in Eq. (5). Since also gives the phonon scattering contribution to the high-temperature carrier resistivity, we conclude that the DOS and the transport definition of the electron-phonon coupling parameter agree with each other, i.e. where is the coefficient of the phonon contribution to the temperature dependent electronic resistivity for where the electronic resistivity is linear in Kim et al. (2012). , which is exactly the same as the DOS definition for the coupling strength
The velocity renormalization factor defining the many-body suppression of the carrier Fermi velocityAllen (1971), , is obtained from the real part of the electron-phonon self-energy defined in Eqs. (1) and (3). At , the frequency derivative at the Fermi energy in Eq. (1) can be analytically evaluated to obtain , , and ), we now focus on the electron-phonon self-energy function at finite temperature and carrier density by directly numerically integrating Eqs. (1) and (2). The corresponding electronic spectral function is given by , where is the renormalized propagator for the 2D surface electrons including electron-phonon interaction. , which agrees precisely with the DOS definition of the coupling parameter. Having established the complete equivalence among the definitions of the three distinct electron-phonon coupling strength parameters (
In Figs. 1, 2 and 3, we show our representative numerical results for , and , respectively. For numerical calculations, we use the deformation potential eV, m/s, the mass density of BiSe for a single quintuple layer g/cm, and the surface phonon velocity m/sKim et al. (2012). In addition to the well-defined quasiparticle peak, there is substantial background incoherent contribution to the spectral weight and a sharp (but very small) zero-energy feature associated with the carrier coupling to the acoustic phonon mode.
Our most important results are shown in Fig. 4, where we show our calculated electron-phonon coupling parameter (), calculated on the basis of Eq. (1) through a direct numerical differentiation of the self-energy function. Since the quasiparticle velocity is given by , Figs. 4(a) and 4(b) directly provide, as a function of density and temperature, the experimentally relevant carrier velocity renormalization by the electron-phonon interaction. It is interesting to note the very strong (weak) density (temperature) dependence of the coupling parameter for low (high) temperature (density), respectively. Thus, while at varies in Fig. 4(a) as with carrier density (consistent with the analytical behavior), the -parameter at K in Fig. 1 varies very little with carrier density (being very small, throughout). Similarly, we see in Fig. 4(b) that for cm (relatively high density) starts at a high value (), but quickly falls to around for K.
Although all our calculations are done using the parameters for the BiSe TI system, which is by far the most extensively studied TI system in the literature, we believe that the general trends we find in our work (e.g. the functional dependence of on density and temperature) should be valid for 2D surface states on all TI materials. In this context, it is crucial to emphasize the obvious fact that , and as such all our quantitative results depend on our choice of eV for BiSe system, which we take from the recent detailed transport measurementsKim et al. (2012) of the temperature-dependent resistivity of the BiSe surface 2D carriers, which then lead directly to an estimate of and hence of . We emphasize that the subject matter of the electron-phonon coupling strength in the context of the BiSe surface metallic carriersHatch et al. (2011) has been highly controversial with claims ranging from very weak coupling ()Pan et al. (2012) to very strong coupling ()Zhu et al. (2012). Our work should resolve this controversy because, as our Figs. 4(a) and 4(b) clearly show, the electron-phonon coupling parameter in the BiSe surface 2D states depends very strongly on the temperature and carrier density. Since increases strongly with carrier density (Fig. 4(a)) particularly at lower temperatures ( K), the coupling can be increased arbitrarily by increasing carrier density and could reach as high as for cm. By contrast, decreases sharply with temperature (Fig. 4(b)), and thus even at high carrier density ( cm), the coupling constant will be very small () for K. A direct observation of the strongly temperature and density dependent quasiparticle velocity renormalization, as predicted in our Figs. 4(a) and 4(b), could completely settle the controversy about the precise value of (and hence ) in BiSe. We emphasize that the electron-phonon coupling in the 2D surface states of BiSe is much stronger () quantitatively than the corresponding coupling in graphene ()Tse and Das Sarma (2008) and 2D GaAs ()Kawamura and Das Sarma (2012) systems for comparable temperatures and carrier densities.
Since the original definition of the electron-phonon coupling parameter (as used in our work) arose in the context of the BCS superconductivity (and the associated electron-phonon interaction in 3D metals), where is indeed a constant for each metal, it may appear strange that we are discussing here (Figs. 4(a) and 4(b)) a -parameter which depends strongly with temperature and density. This seeming paradox is resolved by realizing that in regular 3D metals, the carrier density is very high (corresponding to a Fermi energy of eV) which cannot be varied at all except for small changes in going from one metal to another. Thus, the 3D metals are essentially always in the high-density regime. Equally importantly, the relevant temperature scale for 3D metals is the Debye temperature () which is high K. Thus, any temperature dependence of the -parameter in 3D metals can only manifest itself at rather high temperatures , which perhaps does happen as reflected in the so-called “resistivity saturation” phenomenon. In our 2D metallic system on the TI surfaces, however, the electron temperature scale () and the phonon temperature scale () are both relatively small, leading to a very strong temperature dependence of the electron-phonon coupling “constant”. (As an aside we mention that the relevant phonon temperature scale is either or depending on whichever is smaller–in 3D metals K since is a very large energy scale because of the very high value of in metals, on the other hand in 2D systems since is typically small.)
We conclude by mentioning an important consequence of our theoretical findings. Given the large value of (at least for cm) and very weak Coulomb repulsion in the 2D TI surface states on BiSe we predict the possibility of phonon-induced surface BCS superconductivity in BiSe. In fact, using our calculated (and taking the Coulomb repulsion parameter ) we predict K for BiSe surface 2D states for cm. The relevant mean-field formula here is easily obtainable from a BCS-Eliashberg theory, where in our work is related to the Eliashberg function by Grimvall (1981), which gives the following approximate expression for the surface 2D system: . Using K, , and , we get K. We propose that 2D superconductivity (with K) should be looked for in the surface metallic states of BiSe.
Acknowledgements This work is supported by ONR-MURI, LPS-CMTC and Microsoft Q.
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