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We report charge and heat transport studies in copper-intercalated topological insulator Bi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}2Se\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3$$\end{document}3 hybrid devices. Measured conductivity shows impact of quantum corrections, electron–electron and electron–phonon interactions. Our shot noise measurements reveal that heat flux displays a crossover between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^2$$\end{document}T2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^4$$\end{document}T4 with the increase of temperature. The results might be explained by a model of inelastic electron scattering on disorder, increasing the role of transverse acoustic phonons in the electron–phonon coupling process.

We report charge and heat transport studies in copper-intercalated topological insulator Bi 2 Se 3 hybrid devices. Measured conductivity shows impact of quantum corrections, electron-electron and electron-phonon interactions. Our shot noise measurements reveal that heat flux displays a crossover between T 2 and T 4 with the increase of temperature. The results might be explained by a model of inelastic electron scattering on disorder, increasing the role of transverse acoustic phonons in the electron-phonon coupling process.
Charge and heat transport in metals are governed by the interplay between the electron scattering from impurities, quantum corrections, electron-electron (EE) and electron-phonon (EP) interactions. Here we study the temperature dependence of conductivity and heat flux in weakly copper-intercalated topological insulator (TI) Bi 2 Se 3 . Although, the quantum and EE corrections to the conductivity in Bi 2 Se 3 have been extensively studied [1][2][3][4][5][6][7] and the strength of the electron-phonon coupling has been determined e.g. by using ARPES 8 and surface phonons 9 , very little is known about the effect of the interference between EP and electron-impurity scattering on electrical transport, which has been of great interest in disordered metallic conductors 10 and nanowires 11 .
This interference arises via impurities that move along with the phonons, i.e. via vibrating impurities. In disordered materials, for example intercalated crystals, the vibrating impurities due to the EP and electronimpurity interference result in a T 2 contribution to conductivity 12 . Besides intercalated atoms, also interstitial doping can lead to a T 2 contribution in the resistance 13 , whereas in clean Bi 2 Se 3 only very limited T 2 contribution is observed 14 .
In general, the contribution of phonons to the low-temperature conductivity can be analysed by introducing two characteristic temperatures. First is the Bloch-Grüneisen temperature T BG , which is associated with the particles' Fermi surface. Second is the temperature at which the phonon wave length becomes comparable to the mean free path of electrons due to scattering on impurities 12,[15][16][17] . It can be estimated as where v s is the sound velocity, L e the electron mean-free path, and , k B are the Planck and Boltzmann constants, respectively. Note the sound velocities of longitudinal and transverse modes are different, which has to be taken into consideration in an analysis of the EP interaction and determination of characteristic temperatures in the material.
Coupling of electrons to longitudinal and transverse acoustic phonons is a complex problem, which has been widely analysed in clean systems T dis ≪ T 18 . In pure materials, electrons do not interact with transverse phonons in the lowest order in the EP interaction term. Fundamentally, for clean Bi 2 Se 3 , the phonon contribution to the resistance in the bulk is given by the Bloch-Grüneisen term ∝ T 5 at low temperatures T dis << T < T BG 19 , while ∝ T 4 is expected for thin crystals 20 .
In impure materials, however, electron interactions with transverse phonons contribute to the charge transport. In the case of all impurities moving with the phonon displacement field, the resistance at temperatures T dis T T BG is given by 10,12 : where R 0 is the resistance determined by the electron scattering on disorder, E F is the chemical potential, p F is the Fermi momentum, β ℓ and β t are the constants of interaction with longitudinal and transverse phonons with velocities v ℓ and v t , respectively. The interaction constants satisfy β t /β ℓ = (v ℓ /v t ) 2 and can be retrieved from the experimental data. For Bi 2 Se 3 , taking into account v t = 1.7 km/s and v ℓ = 3.5 km/s , β t /β ℓ ≈ 4 emphasises  10 . We note that the crossover between T 2 and T 5 laws in resistivity depends on the mean free path and the coefficients of electron interaction with transverse and longitudinal phonons 16 .
The heat transport by electrons as opposed to relaxation by coupling to phonons can be distinguished in shot noise measurements. In particular, the heat flux between hot electrons and phonons with temperatures T e and T ph , respectively, scales as P(T e , T ph ) ∝ T k e − T k ph ( T e will be defined in Methods section). The power law k is sensitive to the dimensionality of the sample 21 , disorder 12,17,22 , type of phonons 12 , screening 23 and chirality 24 of charge carriers. Here we distinguish the heat flux mechanism, which is determined by the transverse phonons at where v F is the Fermi velocity and V is the volume of the sample. In this paper we present experimental results for the temperature dependence of the conductivity and heat flux in the copper intercalated Bi 2 Se 3 . We focus on the signatures of electron-phonon interaction contribution to the observables. First we comment on a crossover between EE and EP interaction corrections to the conductivity. Then we analyze the contribution of the EP interaction to the heat flux.

Results and discussion
Conductivity. Measurements of magnetoconductivity were performed at temperature T = 100 mK and magnetic field ranging from 0 up to 3.5 T, as shown in Fig. 1. Fitting of the results was based on a modified Hikami-Larkin-Nagaoka approach [25][26][27] . Besides the quantum corrections, we employed both linear ( ∝ B ) and quadratic ( ∝ B 2 ) magnetoresistance components. Quadratic terms arise either from classical magnetoresistance or spin-orbit scattering. The origin of the linear magnetoresistance (LMR) ∝ B is not apparent but, indeed, it has been observed in topological insulators such as Bi 2 Se 3 28,29 and Bi 2 Te 3 30 .
The magnetoconductivity �σ = σ (B, T) − σ (T) for Bi 2 Se 3 can be written in the form 26,27,31 : where α denotes the strength and nature of quantum corrections and B φ = 4e L −2 φ defines a characteristic field related to dephasing length L φ . The term where the first term −σ (T)µ 2 B 2 denotes the classical, mobility-µ-dependent magnetoresistance ( µ 2 B 2 ≪ 1 ), while the latter terms are the quantum corrections specified using characteristics fields B SO,e = 4e L −2 SO,e given by spin-orbit scattering length L SO and the mean free path L e , respectively.
In general, the parameter α is a measure of the relative strengths of the spin-orbit interaction, inter-valley scattering, magnetic scattering, and dephasing, as discussed in Ref. 32 for topological insulators, and their value may vary between −1 · · · + 1 (or even down to − 2 27 ). Using weak localization (WL) α = 1 , our fit displayed in Fig. 1 yields L φ = 293 ± 100 nm for the dephasing length, while the linear and quadratic terms amount to γ = −0.11 mS/T and β = −41 µS/T 2 , respectively. The value obtained for bulk dephasing length L φ is consistent   33 . We note that short dephasing length due to electron-electron scattering, would lead to the so-called hot electron regime, in which the noise power would increase linearly with temperature or bias voltage. This is clearly against our observations and electron-phonon scattering plays a role in the noise generation. Furthermore, the use of weak antilocalization theory with small L φ is not able to reproduce the σ (B) over the measured range of magnetic fields. In other words, the positive trend owing to weak localization corrections dG/dB > 0 is expected at magnetic fields B 30 mT which is not discernible in Fig. 1.
The transport in our sample is nearly fully governed by the bulk contribution. In this situation, the term βB 2 turns out to be dominated by classical magnetoresistance �R ∝ µ 2 B 2 , because L SO is expected to be much shorter than L φ ≃ 300 nm in the bulk 5 and B SO >> B φ . Hence, we may identify β = σ 0 µ 2 and obtain for the mobility of carriers µ = 0.0657 m 2 /Vs. Using g = σ 0 π e 2 , where σ 0 = 9.5 mS, we estimate the mean free path L e = g/k F =35 nm, where the Fermi momentum k F = 3.5 × 10 9 1/m was obtained from the Bloch-Grüneisen temperature (see text related to Fig. 2). The mean free path L e = 35 nm agrees directly with the one obtained from mobility µ = qL e /2m eff v F , using an effective mass m eff /m 0 = 0.11 where m 0 is the free electron mass. This m eff is close to the values obtained in Refs. 34,35 . Three principal remarks on the measured σ (B) are in order. First, neglecting contribution of the surface states, the bulk WL behavior in σ (B) suggests a situation of weakly doped topological insulator with the chemical potential close to the conduction band gap 31,32 . To describe the data, the model of parabolic bulk conduction band may be utilized to describe the properties of copper intercalated Bi 2 Se 3 topological insulator. In this limit, one may expect similar temperature dependencies of conductivity and heat flux between hot electrons and phonons. Second, taking the bulk transverse and longitudinal phonon velocities we can estimate the characteristic temperatures at which thermal phonon wave-length becomes comparable to the electron mean free path. We estimate the T dis = 2.3 K and 4.8 K for transverse and longitudinal phonons, respectively. Third, at low temperature T ≃ 100 mK, the dephasing length L φ ≃ 300 nm is appreciably longer than L e ≃ 35 nm, which implies several scattering events before the phase coherence is lost.
To proceed, the temperature dependence of conductivity was measured and the Bloch-Grüneisen temperature was identified as an inflection point on the R vs. T dependence at T BG = 90 K.
We observe crossover between the logarithmic temperature dependence at low temperatures originating from competing interference and EE interaction corrections as well as T 2 behavior at higher temperatures, which we assign to the EP interaction processes. We note that estimated T dis is slightly lower than the crossover temperature. The conductivity may be described by the formula: where α and d k are parameters and K stands for Kelvin. At T 10 K, the temperature dependence of conductivity is governed by the quantum interference and EE interaction corrections for a system with effective 2D diffusion, which is described by the second term in Eq. (4), having most commonly a positive sign due to the domination by EE interactions 31 . In the interval 10 K T < T BG , we observe signatures of the interference between the EP and electron-impurity scatterings Ref. 12 . This mechanism is described by the third term in Eq. (4), with k = 2 . For surface states, the electron-phonon coupling has been found to display linear temperature dependence in   36 . Under large bias, coupling to optical phonons with exponential activation behavior has also been observed 37 . Equivalently, one can rewrite the last term in Eq. (4) as a correction to the resistance as is done in Eq. (1). There the ratio of v ℓ /v t determines the sign of the correction. Noting 1 2 (v t /v ℓ ) 3 1 − π 2 /16 ≈ 0.02 suggests that the dominant contribution to the EP interaction is due to transverse phonons in the investigated temperature range.
To find the EP interaction constants, one can rewrite Eq. (1) as (R − R 0 )/R 0 = AT 2 . The prefactor A = 59 × 10 −6 K −2 is obtained by fitting the quadratic behavior to the experimental data in Fig. 2. According to theory 10 , the parameter A does not depend on the amount of impurities but on the ratio of dynamic impurities to static ones. Because precise control of Cu intercalation in the TI is hard to achieve at low levels, we decided to study only samples with an average Cu intercalation of 0.02%. Across the studied samples, the obtained values for A remained constant within a factor of 2. The Fermi momentum can be estimated as k F = k B T BG /2 v t ≈ 3.5 nm −1 , which allows us to estimate the chemical potential E F = v F k F /2 ≈ (0.46 ± 0.11) eV for typical Fermi velocity v F = (4 ± 1) × 10 5 m/s in Bi 2 Se 3 . Please note that the chemical potential is measured with respect to the bottom of the conduction band. We find the EP interaction factors β t = 1.07 ± 0.26 and β ℓ = β t (v t /v ℓ ) 2 = 0.27 ± 0.07.

Heat flux between electrons and phonons.
At low temperatures, due to the depopulation of phonon modes and the degeneracy of the electron system, the two subsystems of a normal metal, the electrons and the phonons, become almost isolated from each other. For this reason, they can 'equilibrate' independently, that is, they may reach equilibrium distributions over the quasiparticle states at some effective temperatures that we shall denote by T e (for electrons) and T ph (for phonons). Here we consider the so called "hot electrons" regime T e ≫ T ph . In the hot electron regime, electrons in Bi 2 Se 3 have such a large energy that they can transfer heat by diffusion to the aluminum measurement leads effectively. Consequently, electronic thermal conductance in the Al lead dominates over heat transfer to phonons at bias voltages on the order of 1-2 mV 38 .
The temperature dependence of the heat flow P(T) out from electrons in Bi 2 Se 3 displays both T 2 and T 4 dependencies as seen in Fig. 3, which depicts Joule heating P = IV vs. T e determined from the measured nonequilibrium noise. At temperatures T e < 9 K , electrons are the main heat carriers, which is reflected in the observed dependence P ∝ T 2 e . Assuming homogeneous structure of investigated samples an analysis of the power dissipation at T e > 9 K can be performed taking P ∝ T 4 e (Fig. 3). If we were to fit a higher order polynomial to the data, the terms for T 3 behavior would be on the 1%-2% level. Hence, the decomposition of the heat flow to T 2 and T 4 is accurate, but the uncertainty in the effective volume introduces an error on the order of ± (30-40)% in the electron-phonon coupling per unit volume. In the T 4 case one can relate the energy dissipation with contribution of dynamic scattering potential to the electron interaction with transverse phonons as has been analysed in Refs. 15,16 . A combined fit P(T e ) = c 2 T 2 e + c 4 T 4 e yields a perfect match to the data using parameters given in the caption of Fig. 3.
A T 4 temperature dependence could also arise for a heat flow limited by thermal boundary resistance in 3D systems (Kapitza resistance). It would reflect acoustic impedance mismatch of phonons across the interfaces made of Bi 2 Se 3 and a SiO 2 substrate, as well as Bi 2 Se 3 and metallic leads. This heat flow, P(T ph , T 0 ) = S 4 A K (T k ph − T k 0 ) , typically displays a Kapitza conductance of A K = 300 . . . 500 Wm −2 K −4 between metals and dielectric material [39][40][41] ; here T 0 denotes the substrate phonon temperature. Using the contact area of the whole Bi 2 Se 3 www.nature.com/scientificreports/ crystal to the SiO 2 substrate ( S SiO 2 = 4 µm 2 ) and the Al leads ( S Al = 3 µm 2 ), we find that the Kapitza resistance cannot form a bottle neck in the thermal transport out from the sample. An estimate for the strength of the interference contribution of the transverse phonons and randomscattering potential to the heat flux can be obtained from Eq. (2) in the limit T ph = 0 K . By using β t = 1.07 , v F = 4 × 10 5 m/s , L e = 35 nm , v t = 1.7 km/s , and taking the volume of the sample as V = 0.6 × 10 −13 cm 3 , Eq.
(2) evaluated at T e = 12 K yields for the heat flow P(T e , 0) = 1.7 µW , which is in pretty good agreement with the heat flow given by the T 4 component extracted from the experimental data in Fig. 3. Note that there is no explicit dependence of P(T e , 0) on the Fermi velocity. However, if we identify T BG as 100 K, P(12 K, 0) = 2.2 µW , nearly matching the measured power.

Discussion and conclusions
Typically, charge and heat transport in disordered materials is considered as an effect of electron scattering from the static and vibrating potentials, taking into account that the vibrating potential is completely dragged by phonons. A tacit assumption is made about averaged velocity of the phonons in the investigated system. In many materials, including layered topological insulators, such assumption is invalid. Velocities of transverse and longitudinal phonons differ significantly. Measurements of charge transport at low temperatures provide clear distinction of the effect of scattering on transverse and longitudinal phonons. According to Sergeev and Mitin 16 scattering of electrons on longitudinal phonons dominates in clean materials. Dynamic disorder, however, enhances electron coupling to transverse phonons.
Our studies of the transport properties of copper-intercalated Bi 2 Se 3 indicate that intercalation may provide a tool for modification of the charge and heat transport in a topological insulator. The first influence of intercalation is slight enhancement of linear magnetoresistance at low fields B < 2.5 T. Similar to the work of Ref. 28 , the observed LMR could be related to electronic inhomogeneity, enhanced by intercalation in our work. We believe that the intercalated amount x = 0.02 % is too small to validate the use of the disorder-induced LMR model of Parish and Littlewood in our samples 42 .
At temperatures T 10 K , the quantum and EE interaction corrections dominate the conductivity and the electronic heat flux, yielding σ ∝ ln(T) and P ∝ T 2 , respectively. We emphasize that T dis in principle does not coincide with the crossover temperature between the electron-phonon and electron-electron mechanism contributions to the electric conductivity. Here, we estimate T dis to be lower than the crossover temperature between T 2 and T 4 behaviors. At 10 K T T BG , the electron-transverse phonon interaction results in a T 2 temperature dependence of the conductivity, while the electron-phonon heat flux becomes P ∝ T 4 . The T 4 dependence for heat flux is predicted also for clean 2D conductors 17 , and it has recently been observed e.g. in epigraphene 43 . In our case, however, the combination of T 2 dependent resistivity and T 4 dependent heat flow point strongly towards a disordered system with electron-phonon-impurity interference phenomena, which arise from vibrating impurities.

Methods
Single crystals of Bi 2 Se 3 were grown using the vapour-liquid-solid (VLS) method and intercalated with zerovalent copper atoms using wet chemical intercalation 44 . It was assumed that copper intercalation in Bi 2 Se 3 introduces disorder, which affects charge transport by enhanced scattering in the TI. Disorder in TIs can be introduced as doping (e.g. substitution of Bi atoms, which is called a compositional disorder) or as intercalation of van der Waals gaps (e.g. by zero-valent atoms as Cu, which is a structural disorder). Large intercalation rate takes place not only the van der Waals gaps but also in the interstitial sites of each layer, or it may exist as biatomic or triatomic layers in the van der Waals gaps 45 . This results in structural deformation, particularly in z-direction ( 0.12% < x < 0.2 %) with Cu atoms results in superconductivity of the TI 44 . Low intercalation ( x < 0.1 %) doesn't affect the crystallographic structure of the TI and is a subject of our interest as a tool for modification of band structure of the TI.
The intercalation ratio x = 0.02 % was determined from the energy-dispersive X-ray spectroscopy (EDX). Sound velocity, in plane, was investigated using the 90-degree equal-angle scattering geometry of the Brillouin light spectroscopy (BLS) with a five pass scanning tandem Fabry-Pérot interferometer at = 532 nm. Results of the BLS experiments revealed transverse phonon velocity of 1700 m/s and longitudinal phonon velocity of 3500 m/s (Fig. 4).
We investigated hybrid devices made of Bi 2 Se 3 topological insulator on SiO 2 /Si substrate. We concentrated on representative data obtained for a sample with total dimensions Thickness × Width × Length = 60 nm × 1560 nm × 2500 nm , intercalated with 0.02% of Cu. A central section of length L = 650 nm and width W = 1560 nm was contacted using titanium/aluminium electrodes (5/45 nm). The resistance of the sample was 41 at 4 K.
Our conductivity and current fluctuation experiments were performed on two pulse-tube-based dilution refrigerators operated around 0.015 K and 0.05 K. For experimental details, we refer to Refs. 46,47 . To determine the EP interaction and its effect on heat transport in our hybrid device, we deduce the non-equilibrium electronic temperature from shot noise measurements [48][49][50] . The Fano factor F is defined as the ratio of the measured noise level S I and the Poissonian noise S P , F = S I /S P . For diffusive electron transport the electronic temperature T e is linearly dependent on F and the bias voltage V, T e = FeV /2k B 48 . Equivalently, we can use a calibration constant M to obtain the electronic temperature of the investigated system, T e = MS(I) . The constant was obtained as the ratio of the difference of the shot noise S measured across two temperatures T 1 = 900 mK and T 2 = 20 mK: M = (T 2 − T 1 )/�S . As defined in this manner, T e denotes a spatial average over the temperature distribution in the Joule heated regime where EP scattering is investigated.