The magnetic properties and Kondo effect in Ce_(3)TiBi_(5) with a quasi-one-dimensional structure were investigated using in situ high-pressure resistivity measurements up to 48 GPa.At ambient pressure,Ce_(3)TiBi_(5) ...The magnetic properties and Kondo effect in Ce_(3)TiBi_(5) with a quasi-one-dimensional structure were investigated using in situ high-pressure resistivity measurements up to 48 GPa.At ambient pressure,Ce_(3)TiBi_(5) undergoes an antiferromagnetic(AFM)transition at T_(N)∼5 K.Under high pressures within 8.9 GPa,we find that Kondo scattering contributes differently to the high-temperature resistance,R(T),depending on the applied current direction,demonstrating a significantly anisotropic Kondo effect.The complete P–T phase diagram has been constructed,in which the pressure dependence of T_(N) exhibits a dome-like shape.The AFM order remains robust under pressure,even when the coherence temperature T^(*) far exceeds 300 K.We attribute the observed anisotropic Kondo effect and the robust AFM to the underlying anisotropy in electronic hybridization under high pressure.展开更多
We have analyzed magnetic order in the one-dimensional Kondo lattice with classical localized spins.To identify relevant low-energy configurations,we combine the exact diagonalization of the electronic system with a d...We have analyzed magnetic order in the one-dimensional Kondo lattice with classical localized spins.To identify relevant low-energy configurations,we combine the exact diagonalization of the electronic system with a dissipative evolution,described by the Landau–Lifshitz-Gilbert equation.We find that spiral states always relax into a more complex form of noncollinear order,characterized by a periodic modulation of the relative angles between neighboring spins.A finite-size scaling analysis shows that the amplitude of the modulation and the gain in free energy remain finite in the thermodynamic limit.Importantly,the wavelength of the modulation is determined by the Fermi wavevector of the unperturbed spiral.This suggests that complex noncollinear order originates from an instability of the unperturbed spirals,which,in the presence of a weak pairing term,may hinder topological superconductivity.Our final phase diagram is obtained by comparing the modulated spiral states with various complex collinear configurations proposed in the literature.展开更多
Dirac node-line(DNL) materials constitute a distinct category of topological semimetals, defined by the linear crossing of valence and conduction bands along one-dimensional lines within the Brillouin zone(BZ), resemb...Dirac node-line(DNL) materials constitute a distinct category of topological semimetals, defined by the linear crossing of valence and conduction bands along one-dimensional lines within the Brillouin zone(BZ), resembling the behavior of Dirac fermions. However, spin–orbit coupling(SOC) and electronic interactions can typically alter these intersections and break the DNLs. In mostly reported cases, DNLs are classified as non-interacting types, which highlights the significant research value in searching for robust interacting DNLs in practical materials. Here, by employing first-principles calculations that combine density functional theory(DFT) with dynamical mean-field theory(DMFT), and leveraging symmetry-based indicator theory, we identify CeAgSb_(2) as a Dirac semimetal. Our investigation reveals that robust Dirac nodal lines(DNLs)in this Kondo system are driven by Kondo interactions and nonsymmorphic lattice symmetries. Furthermore, our results demonstrate that the properties of these DNLs are substantially modulated by Kondo behavior across varying temperature regimes. The interacting DNLs in CeAgSb_(2) represents a rare example of Dirac semimetal under electronic correlations, and the peculiar variation of Dirac fermions with temperature provides theoretical reference for future experimental explorations of novel electronic-correlation effects in topological materials.展开更多
We study the thermoelectric transport of a series-coupled double quantum dots(SDQDs)system,based on the hierarchical equations of motion approach.The thermocurrent as a function of the energy level of QDs gives rise t...We study the thermoelectric transport of a series-coupled double quantum dots(SDQDs)system,based on the hierarchical equations of motion approach.The thermocurrent as a function of the energy level of QDs gives rise to a sign-changing phenomenon.The temperature difference between the two leads can enhance the thermocurrent.Moreover,the sign changing also generates in thermocurrent as a function of temperature due to the transition from the many-body Kondo resonant tunneling process to the single electron process of the SDQDs system.The inter-dot coupling strength between two QDs not only affects the value of the thermocurrent but also influences the characteristic temperature at which the sign changing of thermocurrent emerges.In a weak coupling regime,the thermocurrent firstly is enhanced by inter-dot coupling strength due to the‘t-enhanced Kondo effect'and then decreases with inter-dot coupling strength due to the effective antiferromagnetic interaction between the two QDs.In the middle coupling regime,the forming coherence bonding and antibonding orbitals channels and the residual Kondo effect co-dominate the transport process.The thermocurrent firstly decreases,then increases,and finally decreases with temperature.However,the thermocurrent shows a transition from increasing to decreasing behavior with temperature in the strong coupling regime.Although the inter-dot coupling strength t has a complex impact on the SDQDs system,the characteristic temperature k_BT_c,at which a sign changing appears,indicates a quantitative relationship with the value of the inter-dot coupling strength t by an identical amount of the Kondo correlation being partially destroyed.展开更多
Understanding the quantum critical phenomena is one of the most important and challenging tasks in condensed matter physics and the two-impurity Anderson model(TIAM) is a good starting point for this exploration. To t...Understanding the quantum critical phenomena is one of the most important and challenging tasks in condensed matter physics and the two-impurity Anderson model(TIAM) is a good starting point for this exploration. To this end,we employ the algebraic equation of motion approach to calculate the TIAM and analytically obtain the explicit singleparticle impurity Green function under the soft cut-off approximation(SCA). This approach effectively incorporates the impurity spacing as an intrinsic parameter. By solving the pole equations of the Green function, we have, for the first time, qualitatively calculated the spectral weight functions of the corresponding low-energy excitations. We find that when the impurity spacing is less than one lattice distance, the dynamic Rudermann–Kittel–Kasuya–Yosida(RKKY) interaction effectively enters, resulting in a rapid increase in the spectral weights of the RKKY phase, which ultimately surpass those of the Kondo phase;while the spectral weights of the Kondo phase are strongly suppressed. From the perspective of spectral weights, we further confirm the existence of a crossover from the Kondo phase to the RKKY phase in the TIAM. Based on these results, the reasons for the phenomenon of the Kondo resonance splitting are also discussed.展开更多
Vanadium-based transition metal chalcogenides VmXn(X=S,Se,Te)with their distinctive quantum effects,tunable magnetism,spin-orbit coupling,and high carrier mobility are a valuable platform to explore the interplay betw...Vanadium-based transition metal chalcogenides VmXn(X=S,Se,Te)with their distinctive quantum effects,tunable magnetism,spin-orbit coupling,and high carrier mobility are a valuable platform to explore the interplay between magnetism and electronic correlations,especially with tunable structural phases and magnetic properties through stoichiometric variations,making them ideal candidates for advanced device applications.Here,we report the synthesis of high-quality V_(5+x)S_(8)single crystals with different concentrations of self-intercalated vanadium.V_(5+x)S_(8)crystals show an antiferromagnetic behavior and a spin-flop-like transition below TN of 30.6 K.The high-quality V_(5+x)S_(8)single crystals exhibit a large negative magnetoresistance of 12.3%at 2 K.Interestingly,V_(5+x)S_(8)crystals show an obvious low-temperature resistance upturn that gradually levels off with the increasing magnetic field,attributed to the Kondo effect arising from the interaction between conduction electrons and embedded vanadium magnetic impurities.With increasing V doping,the antiferromagnetic interactions intensify,weakening the coupling between the local moments and conduction electrons,which in turn lowers the Kondo temperature(TK).Furthermore,the anomalous Hall effect is observed in V5.73S8,with an anomalous Hall conductivity(AHC)of 50.46 W^(-1)·cm^(-1)and anomalous Hall angle of 0.73%at 2 K.Our findings offer valuable insights into the mechanisms of the Kondo effect and anomalous Hall effect in self-intercalated transition metal chalcogenides with complex magnetism and electronic correlation effects.展开更多
The exploration and synthesis of new materials are important for materials science and condensed matter physics.Here, we report the crystal structure, magnetic properties, and electrical transport properties of the si...The exploration and synthesis of new materials are important for materials science and condensed matter physics.Here, we report the crystal structure, magnetic properties, and electrical transport properties of the single crystals of Nd_(5)ScSb_(12), which is a quasi-one-dimensional new compound. Nd_(5)ScSb_(12) exhibits antiferromagnetic transition in both directions perpendicular and parallel to the long axis. Moreover, the magnetic field-dependent magnetization reveals two metamagnetic transitions. The electrical transport properties have been measured on the same sample but with different measurement lengths between the electrodes of the voltage. The resistivity exhibits the metallic behavior. At low temperatures, the Kondo effect and negative transverse magnetoresistance(MR)(B⊥I) have been observed. Interestingly, the measurement length has a significant impact on the Kondo effect and negative MR, providing an intuitive new approach to regulate the Kondo effect. As the measurement length increases, the Kondo effect and negative MR become more pronounced. This not only indicates that the interaction between magnetic impurities and conduction electrons dominates the electrical transport of Nd_(5)ScSb_(12) at low temperatures, but also confirms that the negative MR originates from the suppression of the Kondo effect.展开更多
In the present paper,by applying the Lang-Firsov canonical transformation and the so-called non-crossingapproximation technique,we investigate the joint effects of the electron-phonon interaction and an external alter...In the present paper,by applying the Lang-Firsov canonical transformation and the so-called non-crossingapproximation technique,we investigate the joint effects of the electron-phonon interaction and an external alternatinggate voltage on the transport of a quantum dot system in the Kondo regime.We find that,while the satellite Kondoresonant peaks appear in both the averaged local density of states and the differential conductance,the main Kondopeak at the Fermi energy is greatly suppressed.These results confirm the previous ones derived by other methods,suchas the equation of motion solution.Furthermore,based on the picture of virtual transition between quasi-eigenstates inthe system,we also give a slightly different explanation on these phenomena.展开更多
We have calculated the transport properties of electron through an artificial quantum dot by using the numerical renormalization group technique in this paper. We obtain the conductance for the system of a quantum dot...We have calculated the transport properties of electron through an artificial quantum dot by using the numerical renormalization group technique in this paper. We obtain the conductance for the system of a quantum dot which is embedded in a one-dimensional chain in zero and finite temperature cases. The external magnetic field gives rise to a negative magnetoconductance in the zero temperature case. It increases as the external magnetic field increases, We obtain the relation between the coupling coefficient and conductance. If the interaction is big enough to prevent conduction electrons from tunnelling through the dot, the dispersion effect is dominant in this case. In the Kondo temperature regime, we obtain the conductivity of a quantum dot system with Kondo correlation.展开更多
We report capacitive coupling induced Kondo–Fano(K–F) interference in a double quantum dot(DQD) by systematically investigating its low-temperature properties on the basis of hierarchical equations of motion evaluat...We report capacitive coupling induced Kondo–Fano(K–F) interference in a double quantum dot(DQD) by systematically investigating its low-temperature properties on the basis of hierarchical equations of motion evaluations. We show that the interdot capacitive coupling U12 splits the singly-occupied(S-O) state in quantum dot 1(QD1) into three quasi-particle substates: the unshifted S-O0 substate, and elevated S-O1 and S-O2. As U12 increases, S-O2 and S-O1 successively cross through the Kondo resonance state at the Fermi level(ω = 0), resulting in the so-called Kondo-I(KI), K–F, and Kondo-II(KII) regimes. While both the KI and KII regimes have the conventional Kondo resonance properties, remarkable Kondo–Fano interference features are shown in the K–F regime. In the view of scattering, we propose that the phase shift η(ω)is suitable for analysis of the Kondo–Fano interference. We present a general approach for calculating η(ω) and applying it to the DQD in the K–F regime where the two maxima of η(ω = 0) characterize the interferences between the Kondo resonance state and S-O2 and S-O1 substates, respectively.展开更多
Using the nonequilibrium Green's function technique,we investigate the current induced heat generationin Kondo regime.The Kondo effect influences the heat generation significantly.In the curve of heat generation v...Using the nonequilibrium Green's function technique,we investigate the current induced heat generationin Kondo regime.The Kondo effect influences the heat generation significantly.In the curve of heat generation versusthe bias,a negative differential of the heat generation is exhibited.The symmetry of the heat generation is destroyed bythe strong electron-electron interaction and the electron-phonon interaction.展开更多
基金supported by the National Key Research and Development Program of Chinathe National Natural Science Foundation of China (Grant Nos.2024YFA1408000,12474097,and2023YFA1406001)+2 种基金the Guangdong Provincial Quantum Science Strategic Initiative (Grant No.GDZX2201001)the Center for Computational Science and Engineering at Southern University of Science and Technology,the Major Science and Technology Infrastructure Project of Material Genome Big-science Facilities Platform supported by Municipal Development and Reform Commission of Shenzhen(for J.L.Z.and Y.L.)the Chinese funding sources applied via HPSTAR。
文摘The magnetic properties and Kondo effect in Ce_(3)TiBi_(5) with a quasi-one-dimensional structure were investigated using in situ high-pressure resistivity measurements up to 48 GPa.At ambient pressure,Ce_(3)TiBi_(5) undergoes an antiferromagnetic(AFM)transition at T_(N)∼5 K.Under high pressures within 8.9 GPa,we find that Kondo scattering contributes differently to the high-temperature resistance,R(T),depending on the applied current direction,demonstrating a significantly anisotropic Kondo effect.The complete P–T phase diagram has been constructed,in which the pressure dependence of T_(N) exhibits a dome-like shape.The AFM order remains robust under pressure,even when the coherence temperature T^(*) far exceeds 300 K.We attribute the observed anisotropic Kondo effect and the robust AFM to the underlying anisotropy in electronic hybridization under high pressure.
基金support from the Innovation Program for Quantum Science and Technology under Grant No.2021ZD0301602the National Science Association Funds under Grant No.U2230402+2 种基金supported by FCT-Portugal(PR)the Quant ERA II project‘DQUANT:A Dissipative Quantum Chaos perspective on Near-Term Quantum Computing’via Grant Agreement No.101017733support from FCTPortugal through Grant No.UID/CTM/04540/2020。
文摘We have analyzed magnetic order in the one-dimensional Kondo lattice with classical localized spins.To identify relevant low-energy configurations,we combine the exact diagonalization of the electronic system with a dissipative evolution,described by the Landau–Lifshitz-Gilbert equation.We find that spiral states always relax into a more complex form of noncollinear order,characterized by a periodic modulation of the relative angles between neighboring spins.A finite-size scaling analysis shows that the amplitude of the modulation and the gain in free energy remain finite in the thermodynamic limit.Importantly,the wavelength of the modulation is determined by the Fermi wavevector of the unperturbed spiral.This suggests that complex noncollinear order originates from an instability of the unperturbed spirals,which,in the presence of a weak pairing term,may hinder topological superconductivity.Our final phase diagram is obtained by comparing the modulated spiral states with various complex collinear configurations proposed in the literature.
基金Project supported by the National Natural Science Foundation of China (Grant No. 12364023)the Natural Science Foundation of Guangxi Zhuang Autonomous Regin, China (Grant No. 2024GXNSFAA010273)。
文摘Dirac node-line(DNL) materials constitute a distinct category of topological semimetals, defined by the linear crossing of valence and conduction bands along one-dimensional lines within the Brillouin zone(BZ), resembling the behavior of Dirac fermions. However, spin–orbit coupling(SOC) and electronic interactions can typically alter these intersections and break the DNLs. In mostly reported cases, DNLs are classified as non-interacting types, which highlights the significant research value in searching for robust interacting DNLs in practical materials. Here, by employing first-principles calculations that combine density functional theory(DFT) with dynamical mean-field theory(DMFT), and leveraging symmetry-based indicator theory, we identify CeAgSb_(2) as a Dirac semimetal. Our investigation reveals that robust Dirac nodal lines(DNLs)in this Kondo system are driven by Kondo interactions and nonsymmorphic lattice symmetries. Furthermore, our results demonstrate that the properties of these DNLs are substantially modulated by Kondo behavior across varying temperature regimes. The interacting DNLs in CeAgSb_(2) represents a rare example of Dirac semimetal under electronic correlations, and the peculiar variation of Dirac fermions with temperature provides theoretical reference for future experimental explorations of novel electronic-correlation effects in topological materials.
基金the support of NSFC(Grants No.11804245,No.11747098,No.12247101,&No.12047501)the Fund from the Ministry of Science and Technology of China(Grant No.2022YFA1402704)。
文摘We study the thermoelectric transport of a series-coupled double quantum dots(SDQDs)system,based on the hierarchical equations of motion approach.The thermocurrent as a function of the energy level of QDs gives rise to a sign-changing phenomenon.The temperature difference between the two leads can enhance the thermocurrent.Moreover,the sign changing also generates in thermocurrent as a function of temperature due to the transition from the many-body Kondo resonant tunneling process to the single electron process of the SDQDs system.The inter-dot coupling strength between two QDs not only affects the value of the thermocurrent but also influences the characteristic temperature at which the sign changing of thermocurrent emerges.In a weak coupling regime,the thermocurrent firstly is enhanced by inter-dot coupling strength due to the‘t-enhanced Kondo effect'and then decreases with inter-dot coupling strength due to the effective antiferromagnetic interaction between the two QDs.In the middle coupling regime,the forming coherence bonding and antibonding orbitals channels and the residual Kondo effect co-dominate the transport process.The thermocurrent firstly decreases,then increases,and finally decreases with temperature.However,the thermocurrent shows a transition from increasing to decreasing behavior with temperature in the strong coupling regime.Although the inter-dot coupling strength t has a complex impact on the SDQDs system,the characteristic temperature k_BT_c,at which a sign changing appears,indicates a quantitative relationship with the value of the inter-dot coupling strength t by an identical amount of the Kondo correlation being partially destroyed.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11974420)。
文摘Understanding the quantum critical phenomena is one of the most important and challenging tasks in condensed matter physics and the two-impurity Anderson model(TIAM) is a good starting point for this exploration. To this end,we employ the algebraic equation of motion approach to calculate the TIAM and analytically obtain the explicit singleparticle impurity Green function under the soft cut-off approximation(SCA). This approach effectively incorporates the impurity spacing as an intrinsic parameter. By solving the pole equations of the Green function, we have, for the first time, qualitatively calculated the spectral weight functions of the corresponding low-energy excitations. We find that when the impurity spacing is less than one lattice distance, the dynamic Rudermann–Kittel–Kasuya–Yosida(RKKY) interaction effectively enters, resulting in a rapid increase in the spectral weights of the RKKY phase, which ultimately surpass those of the Kondo phase;while the spectral weights of the Kondo phase are strongly suppressed. From the perspective of spectral weights, we further confirm the existence of a crossover from the Kondo phase to the RKKY phase in the TIAM. Based on these results, the reasons for the phenomenon of the Kondo resonance splitting are also discussed.
基金supported by the National Key R&D Program of China(Grant No.2022YFA1204100)the National Natural Science Foundation of China(Grant Nos.62488201 and 1240041502)+2 种基金the CAS Project for Young Scientists in Basic Research(Grant No.YSBR-003)the Chinese Academy of Sciences(Grant No.XDB33030100)the Innovation Program of Quantum Science and Technology(Grant No.2021ZD0302700).
文摘Vanadium-based transition metal chalcogenides VmXn(X=S,Se,Te)with their distinctive quantum effects,tunable magnetism,spin-orbit coupling,and high carrier mobility are a valuable platform to explore the interplay between magnetism and electronic correlations,especially with tunable structural phases and magnetic properties through stoichiometric variations,making them ideal candidates for advanced device applications.Here,we report the synthesis of high-quality V_(5+x)S_(8)single crystals with different concentrations of self-intercalated vanadium.V_(5+x)S_(8)crystals show an antiferromagnetic behavior and a spin-flop-like transition below TN of 30.6 K.The high-quality V_(5+x)S_(8)single crystals exhibit a large negative magnetoresistance of 12.3%at 2 K.Interestingly,V_(5+x)S_(8)crystals show an obvious low-temperature resistance upturn that gradually levels off with the increasing magnetic field,attributed to the Kondo effect arising from the interaction between conduction electrons and embedded vanadium magnetic impurities.With increasing V doping,the antiferromagnetic interactions intensify,weakening the coupling between the local moments and conduction electrons,which in turn lowers the Kondo temperature(TK).Furthermore,the anomalous Hall effect is observed in V5.73S8,with an anomalous Hall conductivity(AHC)of 50.46 W^(-1)·cm^(-1)and anomalous Hall angle of 0.73%at 2 K.Our findings offer valuable insights into the mechanisms of the Kondo effect and anomalous Hall effect in self-intercalated transition metal chalcogenides with complex magnetism and electronic correlation effects.
基金Project supported by the National Key Research and Development Program of China (Grant No. 2023YFA1406500)the National Natural Science Foundation of China (Grant Nos. 12474098, 12274388, 12174361, 12404043, and 12204004)the Natural Science Foundation of Anhui Province, China (Grant No. 2408085QA024)。
文摘The exploration and synthesis of new materials are important for materials science and condensed matter physics.Here, we report the crystal structure, magnetic properties, and electrical transport properties of the single crystals of Nd_(5)ScSb_(12), which is a quasi-one-dimensional new compound. Nd_(5)ScSb_(12) exhibits antiferromagnetic transition in both directions perpendicular and parallel to the long axis. Moreover, the magnetic field-dependent magnetization reveals two metamagnetic transitions. The electrical transport properties have been measured on the same sample but with different measurement lengths between the electrodes of the voltage. The resistivity exhibits the metallic behavior. At low temperatures, the Kondo effect and negative transverse magnetoresistance(MR)(B⊥I) have been observed. Interestingly, the measurement length has a significant impact on the Kondo effect and negative MR, providing an intuitive new approach to regulate the Kondo effect. As the measurement length increases, the Kondo effect and negative MR become more pronounced. This not only indicates that the interaction between magnetic impurities and conduction electrons dominates the electrical transport of Nd_(5)ScSb_(12) at low temperatures, but also confirms that the negative MR originates from the suppression of the Kondo effect.
文摘In the present paper,by applying the Lang-Firsov canonical transformation and the so-called non-crossingapproximation technique,we investigate the joint effects of the electron-phonon interaction and an external alternatinggate voltage on the transport of a quantum dot system in the Kondo regime.We find that,while the satellite Kondoresonant peaks appear in both the averaged local density of states and the differential conductance,the main Kondopeak at the Fermi energy is greatly suppressed.These results confirm the previous ones derived by other methods,suchas the equation of motion solution.Furthermore,based on the picture of virtual transition between quasi-eigenstates inthe system,we also give a slightly different explanation on these phenomena.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10074029 and 60276005）, and by the State Key Development Program for Basic Research of China （Grant No G1999064509）.
文摘We have calculated the transport properties of electron through an artificial quantum dot by using the numerical renormalization group technique in this paper. We obtain the conductance for the system of a quantum dot which is embedded in a one-dimensional chain in zero and finite temperature cases. The external magnetic field gives rise to a negative magnetoconductance in the zero temperature case. It increases as the external magnetic field increases, We obtain the relation between the coupling coefficient and conductance. If the interaction is big enough to prevent conduction electrons from tunnelling through the dot, the dispersion effect is dominant in this case. In the Kondo temperature regime, we obtain the conductivity of a quantum dot system with Kondo correlation.
基金National Natural Science Foundation of China(Grant Nos.11774418,11374363,and 21373191).
文摘We report capacitive coupling induced Kondo–Fano(K–F) interference in a double quantum dot(DQD) by systematically investigating its low-temperature properties on the basis of hierarchical equations of motion evaluations. We show that the interdot capacitive coupling U12 splits the singly-occupied(S-O) state in quantum dot 1(QD1) into three quasi-particle substates: the unshifted S-O0 substate, and elevated S-O1 and S-O2. As U12 increases, S-O2 and S-O1 successively cross through the Kondo resonance state at the Fermi level(ω = 0), resulting in the so-called Kondo-I(KI), K–F, and Kondo-II(KII) regimes. While both the KI and KII regimes have the conventional Kondo resonance properties, remarkable Kondo–Fano interference features are shown in the K–F regime. In the view of scattering, we propose that the phase shift η(ω)is suitable for analysis of the Kondo–Fano interference. We present a general approach for calculating η(ω) and applying it to the DQD in the K–F regime where the two maxima of η(ω = 0) characterize the interferences between the Kondo resonance state and S-O2 and S-O1 substates, respectively.
基金Supported by the Scientific Research Fund of Hunan Provincial Education Department,China under Grant Nos.08B016 and 09B021
文摘Using the nonequilibrium Green's function technique,we investigate the current induced heat generationin Kondo regime.The Kondo effect influences the heat generation significantly.In the curve of heat generation versusthe bias,a negative differential of the heat generation is exhibited.The symmetry of the heat generation is destroyed bythe strong electron-electron interaction and the electron-phonon interaction.
