Direct probing of phonon mode specific electron–phonon scatterings in two-dimensional semiconductor transition metal dichalcogenides

Electron–phonon scatterings in solid-state systems are pivotal processes in determining many key physical quantities such as charge carrier mobilities and thermal conductivities. Here, we report direct probing of phonon mode specific electron–phonon scatterings in layered semiconducting transition metal dichalcogenides WSe2, MoSe2, WS2, and MoS2 through inelastic electron tunneling spectroscopy measurements, quantum transport simulations, and density functional calculation. We experimentally and theoretically characterize momentum-conserving single- and two-phonon electron–phonon scatterings involving up to as many as eight individual phonon modes in mono- and bilayer films, among which transverse, longitudinal acoustic and optical, and flexural optical phonons play significant roles in quantum charge flows. Moreover, the layer-number sensitive higher-order inelastic electron–phonon scatterings, which are confirmed to be generic in all four semiconducting layers, can be attributed to differing electronic structures, symmetry, and quantum interference effects during the scattering processes in the ultrathin semiconducting films.

where is the derivative of the effective potential generated from the displacement of atomic arrangements by the phonon mode , and | 〉 is the electronic state with crystal momentum .
We highlight that special attention to + , is required for the systems with spinmomentum locking, since the spin states allocated to the electronic states with momenta and + are different. In a system with the spin-momentum locking, therefore, we need to treat the electronic wave function 〈 | 〉 = ( ) with the momentum-dependent spinor and spatial wave function ( ), such that Note that the pre-factor + † appearing in + , gives rise to the geometric phase, while the pre-factor becomes + † = 1 in systems without spin-momentum locking. The quantal phase of the pre-factor has a close connection to the geometric interpretation, i.e., arg( + † ) = ∫ ⋅ † ( ) ( ), as a line integral of the Berry connection along the geodesic connecting the spin states of + and on the Bloch sphere. Once the combined geodesic lines form a closed geodesic polygon, the total quantal phase factor becomes gauge-independent and physically detectable.

Theoretical evaluation of the interacting two-particle Green function
Here, we theoretically evaluate the two-particle Green function ′ ( , , ) of the conducting electrons that interact with phonons up to the second order of electron-phonon coupling strength ( < 1) . First, we briefly introduce many-body perturbation theory ′ ( , , ) with Feynman diagrams. We combine the parametric terms in the perturbative expansions of ′ ( , , ) according to the dG/dVb peak order in the main text, i.e., ′ ( , , ) ∼ ′ 0 ( , , ) + ′ 1 ( , , ) + ′ 2 ( , , ). Next, we evaluate ′ 1 ( , , ) that is responsible for the first-order dG/dVb peaks. Last, we demonstrate that quantum interference with a consideration of the geometric phase plays an important role in ′ 2 ( , , ), which is attributed to the secondorder peaks in inelastic electron tunnel features dG/dVb.

Perturbative expansion of ′ ( , , ) and Feynman diagrams
The perturbative expansion of the two-particle Green function in the interaction picture is where ̂ is given in the interaction picture. For convenience, we denote the second and third terms in the right-hand side as 1 ( , ) and 2 ( , ), respectively. In Supplementary Figure   10 being responsible for the second-order peaks in dG/dVb.

Evaluation of ′ ( , , ) for the first-order peaks in dG/dVb
We evaluate By using the Feynman diagram, we obtain where electrons in the conduction band emit phonons at ′, and holes in the conduction band ′ absorb the emitted phonons at ′ with momentum conservation = ′ + . We define the retarded bare Green function The level broadening is defined as , which are energy-independent constants with the wide-band approximation.
The transmission probability 1 ( , ), at which incident electrons at can tunnel after the interaction with phonon mode and momentum exchange , is expressed in , where the energy of an incident electron is and that of a transmitted electron is after interaction with a phonon. Notice that the energy is Since the energy scale of phonons, tens of meV, is much smaller than that of the energy gap between band bottoms and the Fermi level, a few eV, the tunneling processes with destructive interference by geometric phase = hardly contribute to the second-order peaks in dG/dVb. Owing to the time-reversal symmetry between and ′ ( 3 and 6 ) in Supplementary Figure 11a and 11b, the solid angle Ω = 2 (blue shade) and thus the geometric phase is = . Two tunneling processes with a reversed order of phonon interactions (red and blue) exhibit quantum interference.

Discussion on the quantum interference in the second-order inelastic scattering processes
We further elaborate on the non-vanishing quantum interference in the second-order inelastic scattering processes when the spin-momentum locking comes into play, which is indeed the central theme of our data interpretation. According to the common wisdom, virtual processes in quantum mechanical scattering events should cancel their redundant phases through an appropriate gauge transformation. Those virtual processes, evidently distinct from the Aharonov-Bohm effect, should occur in some electron-phonon scattering processes as well. This argument is so powerful and fundamental that there seems no room for any further alternatives. Against all these odds, however, a non-trivial phase can be associated with IETS once the virtual inelastic scattering processes are effectively quenched to a few limited channels and the scattering routes are involved with abrupt turnarounds of spin degrees of freedom, as in the monolayer SC-TMDs.
With these restrictions, the non-trivial phase should be emerged because the quenched virtual scattering phase space effectively introduces a specific close path for a certain two-phonon process and the electron spin along the path should rotate following the unique spin-momentum locking in the monolayer SC-TMDs.
We try to elaborate the above statement with the second-order perturbation theory and the Fermi's Golden rule. At first, we should point out that the electron-phonon scatterings and their strengths are particularly strong at high symmetric points in the SC-TMDs, as discussed in the main text. Accordingly, the electron-phonon scattering phase space should be effectively quenched to the selected scattering processes that involve the phonons at the high symmetric points of Q, M and K. Next, we introduce a simple harmonic perturbation with two distinct phonon modes to capture the essential aspect of the inelastic two-phonon electron scattering processes, Here, = , denotes the electron-phonon coupling matrix of the phonon modes . The harmonic perturbation describes the energy exchanges between the phonons with momenta of and and conducting electrons. We now consider the transition rate W to describe the electron tunneling with an incident momentum and a transmitted momentum . In particular, we focus on the second-order W (2) , which describes two-phonon inelastic electron scattering With the Fermi golden rule, the two-phonon IETS processes with the conducting electrons around the conduction-band minimum at Q can be described as The momentum conservation law allows the two independent inelastic scattering paths; electrons are scattered along the path-A through a virtual state |κ A 〉, and the path-B through a virtual state the different phase factors and from the virtual states | 〉 and | 〉 should be vanished and no quantum interference effects are expected in inelastic electron-phonon scattering processes.
However, when the scattering routes involve with the abrupt spin state ( η k ) changes introduced by the spin-momentum locking and the inversion symmetry breaking in monolayer SC-TMDs, a non-trivial quantum phase can emerge and consequently influence the two-phonon electron scattering processes, as explicitly observed in our measurements. In monolayer SC-TMDs, the second-order transition rate W should then be modified with the spin states : (2) ∝ | The differing spin states at the high symmetry points demand that the inverted brackets should be expressed as | 〉 † 〈 |, and † is far from the identity matrix for any spinor . Therefore, the non-trivial phase difference γ is introduced in (2) as where Arg[ ] denotes the phase of a complex number . Since the spinors appear with their Hermitian conjugates in the closed path, it is straightforward to show that the phase γ is gauge invariant and cannot be eliminated by a gauge transformation. The gauge invariant phase in twolevel systems becomes γ = Ω/2 and Ω is the solid angle of geodesic polygon enclosed by the spinors on the Bloch sphere (Supplementary Figure 12).
The observable physical content of the solid angle Ω was firstly recognized by Prof. M.
Berry in the following paper [Berry, M. V. The adiabatic phase and Pancharatnam's phase for polarized light. J. Mod. Opt. 34, 1401-1407(1987]. In particular, when ( ) is a timereversal partner of ( ) , it is straightforward to show γ = π (equivalently Ω = 2π), resulting in destructive quantum interference. Note that some of two-phonon inelastic scattering processes in the SC-TMDs, particularly with the Q and M phonons involve such time-reversal spin-state partners. In comparison, inelastic electron scatterings with K and M phonons do not involve with those spin states. Based on these, IETS spectra relating to the Q and M phonon modes become absent in the monolayer SC-TMDs thanks to the destructive quantum interference, but become prevalent in the bilayer SC-TMDs where the additional quantum phase plays no roles.
Supplementary Table   Two Table 1. Two-phonon inelastic electron tunneling processes considered in the quantum transport simulations. List of all the two-phonon electron-phonon scattering processes that are considered in the theoretical calculations in Figure 3c and 3d of the main text.