Spectroscopic trace of the Lifshitz transition and multivalley activation in thermoelectric SnSe under high pressure

Multivalley systems offer not only exciting physical phenomena but also the possibility of broad utilization. Identifying an important platform and understanding its physics are paramount tasks to improve their capability for application. Here, we investigate a promising candidate, the semiconductor SnSe, by optical spectroscopy and density functional theory calculations. Upon applying pressure to lightly doped SnSe, we directly monitored the phase transition from semiconductor to semimetal. In addition, heavily doped SnSe exhibited a successive Lifshitz transition, activating multivalley physics. Our comprehensive study provides insight into the effects of pressure and doping on this system, leading to promising routes to tune the material properties for advanced device applications, including thermoelectrics and valleytronics. Pressure-sensitive spectroscopy measurements have uncovered promising routes to tune semiconducting materials for energy harvesting and advanced electronic applications. Tin selenide (SnSe) semiconductors have an electronic structure favorable for the transformation of heat into electricity. Tobias Biesner from the University of Stuttgart in Germany and colleagues now demonstrate how electrons in SnSe can be manipulated to access special features of the energy landscape known as multivalleys. The team used infrared light to characterize SnSe crystals subjected to extremely high pressures, similar to forces experienced by minerals under Earth’s crust. These experiments mapped out when multivalleys appeared and how they influence properties such as phase transitions and energy conversion. Identification of accessible multivalley states can also aid device designers looking for additional techniques to store and read out information. The semiconductor SnSe hosts a unique multivalley electronic band structure, offering rich physical phenomena for various applications. Especially its excellent thermoelectric properties are ceaselessly under great attention. By employing pressure-dependent infrared spectroscopy combined with density functional theory calculations, the evolution of the multivalley is mapped under external pressure for different self-dopings. The presented diagram summarizes the findings. In lightly doped SnSe, a phase transition from a semiconductor to a semimetal is observed by increasing the pressure. However, for heavily doped SnSe, altering the Fermi surface, successive Lifshitz transitions take place under pressure, further activating the multivalley physics.


Introduction
Multivalley electronic systems confining charge carriers to a specific momentum have emerged as a key aspect of new technologies 1,2 . These systems have a unique band structure that consists of multiple valence band maxima and conduction band minima with different momenta near the Fermi energy level, providing specific optoelectronic properties 3 . The distinguished momentum dependence in such systems provides an additional degree of freedom for information processing, opening an avenue for valleytronics 4,5 . On the other hand, the multivalley effect improves the efficiency of thermoelectric materials 6,7 . In particular, the Seebeck coefficient can be tuned by increasing the band degeneracy number and the effective mass with an increasing number of valleys [8][9][10] . In addition to identifying accessible multivalley systems, it is highly desirable to develop proper ways to tune them.
The semiconductor SnSe is a promising multivalley system and is known for its outstanding thermoelectric properties [11][12][13][14][15] . In SnSe, the strong electronic anisotropy with multiple valence band maxima and conduction band minima induces two distinct gaps along the Γ − Y and Γ − Z directions of the Brillouin zone, corresponding to the zigzag and armchair anisotropy, respectively (see Fig. 1a for the crystal structure) 3,11,[16][17][18][19][20][21] . To further promote multivalley physics, chemical doping was performed to push the Fermi level deep into the valence band, activating multiple hole pockets (Fig. 1b) 6,16,19,[21][22][23] . Furthermore, samples were unintentionally self-doped by Sn vacancies or SnSe 2 microdomains, resulting in a small shift in the Fermi level that caused the resistivity of heavily doped SnSe to deviate from its semiconducting behavior, generating a metallic temperature dependence, as depicted in Fig. 1c 17,24 .
Pressure is a clean tuning parameter for manipulating material properties 25 . In heavily doped SnSe, a pressureinduced Lifshitz transition occurs, i.e., a change in the Fermi surface topology without a structural transition 26 . At 0.86 GPa, the compound transforms from a two-valley system to a four-valley system, accompanied by improved thermoelectric properties, providing proof that pressure is a powerful tool to tune the electronic multivalley 27,28 . Densityfunctional theory (DFT) calculations and transport studies have reported enhanced thermoelectric properties of SnSe upon application of external pressure [29][30][31][32] . Although transport measurements 33,34 and theoretical approaches 35,36 provide evidence for a semiconductor-semimetal transition, direct investigations of the electronic structure and its pressure dependence are still lacking. In fact, transport measurements are confined to a cumulative probe of the conduction bands without spectroscopic resolution.
In this study, we employ pressure-dependent infrared spectroscopy to directly map the multivalley electronic band structure of SnSe over a wide pressure range up to 12 GPa in lightly doped and heavily doped samples. By combining these experiments with first-principles calculations, we explain the electronic structure evolution of SnSe from semiconductor-to-semimetal under pressure. In addition to the results from transport studies, in heavily doped SnSe, we observe a Lifshitz transition in the highpressure range above 7 GPa, where a new electron pocket emerges (see the sketched Fermi surface in Fig. 1d). Our study provides a comprehensive insight into SnSe under pressure, suggesting strategies to tune the optoelectronic and thermoelectric properties to allow device applications in more efficient ways.

Sample preparation and resistivity measurements
Two <100 > -cleaved SnSe samples (S1 and S2) with different carrier concentrations (lightly doped and heavily doped) were prepared by the Bridgman method 24 . The carrier concentrations (resistivity) of sample S1 and sample S2 were measured by the Quantum Design @ Physical Properties Measurement System (PPMS) to be 3 × 10 15 cm −3 (50.5 Ω cm) and 5 × 10 17 cm −3 (0.086 Ω cm) at 300 K, respectively. With decreasing temperature, sample S1 showed semiconducting-like behavior, whereas sample S2 showed metallic-like behavior. The size of the two samples used for the pressure-dependent optical measurements was approximately 180 (length) × 160 (width) × 40 µm (thickness). Further information about the sample homogeneity can be found in the supplemental material (Fig. S1) Optical measurements under high pressure A screw-driven diamond anvil cell (Almax easyLab) with Type IIa diamonds and an 800 µm culet was used. CuBe gaskets with a thickness of 60 µm and a hole size of 250 µm were prepared for sample space, leaving enough free area for multiple ruby spheres, which were used for pressure calibration 37,38 (cf. Fig. S2). CsI was utilized as a quasi-hydrostatic pressure transmitting medium. Reflectance and transmittance spectra were measured by employing a Fourier transform spectrometer (Bruker Vertex 80 v) under near-normal incidence, E||(b,c) at T = 300 K. For the reflectance measurements, we used the CuBe gasket as a reference and corrected the final spectra by the reflectance of CuBe. The transmittance was measured by referencing the transparent pressure media inside the diamond anvil cell. The optical conductivity was determined by modified Kramers-Kronig analysis of the reflectance, as described in the supplemental material. From the data in the mid-infrared range, we calculated the optical absorption coefficient as follows: , where R and Tr are the reflectance and transmittance, respectively, and d is the sample thickness. Direct optical interband transitions, E g,opt , were determined using the Tauc method 40 by extrapolating the linear regime of (αω) 2 . The spectral weight, SW, was determined by integrating the optical conductivity, σ 1 (ω), over the frequency, SW ¼ R ω max ω min σ 1 ω ð Þ dω, and was directly related to the carrier density and effective mass (see the supplemental material for further information) [40][41][42] . Extended data sets and further details of the experimental analysis can be found in Figs. S3-S8.

DFT calculations
Relativistic density-functional band structure calculations were performed in the Wien2K code 43 with the modified Becke-Johnson (mBJ) exchange-correlation functional 44 parametrized according to Ref. 45 . This approach delivers an accurate bandgap for SnSe at ambient pressure and correctly reproduces the pressure of the semiconductor-to-metal transition. In contrast, standard DFT functionals, such as the widely used Perdew-Burke-Ernzerhof exchange-correlation potential 46 , tend to underestimate the ambient pressure bandgap and, consequently, the critical pressure where SnSe becomes metallic. Atomic positions from Ref. 47 were used. The band structure and density of states were computed on a well-converged k mesh with 1728 points in the irreducible part of the first Brillouin zone.

Results
Optical spectra under pressure The optical conductivity of the lightly doped SnSe in the far-infrared range upon application of pressure is displayed in Fig. 2a. At ambient pressure, SnSe exhibits a phononic feature at low frequencies, which is also visible in our low-pressure spectra (see Fig. S9). At low pressures, we observe an insulating characteristic without pronounced free charge carriers, i.e., no Drude response 40,42 . With increasing pressure, the low-frequency optical conductivity is increasing with the development of a Drude mode when the pressure exceeds 6 GPa. In addition, the interband contribution changes with increasing pressure, as displayed in Fig. 2b. In particular, the direct optical gap is suppressed continuously, resulting in a redshift. However, at all pressures up to the highest pressure available (11.6 GPa), the gap does not close completely, indicating that the metallization of SnSe above 6 GPa is caused by a band shift across the Fermi level rather than a collapse of the optical gap owing to high pressure. The heavily doped sample (cf. Fig. 2c, d) yields similar results; however, in this case, a delayed transition above 7 GPa is observed. This delay is most visible in the SW of the low-frequency range (inset of Fig. 2c), where the SW increases more strongly at an earlier pressure for the lightly doped SnSe.

Optoelectronic properties under pressure
To further scrutinize the electronic evolution under pressure, we compare the experimental results with the results of DFT calculations (Fig. 3a-d). The results along the Z-Γ−Y path of the Brillouin zone are shown for selected pressures. In accordance with the ARPES measurements 17 and previous calculations 31, 36 , we identify multiple valence band maxima in the Γ−Z and Γ−Y directions and a minimum of the conduction band at the Γ point, characteristic of the multivalley electronic structure of SnSe. In Fig. 3a, two distinct direct interband transitions, α and β, are most pronounced. In this case, the transition with lower energy determines the experimental optical gap, E g,opt . Thus, at ambient pressure, in the semiconducting phase, E g,opt is governed by transitions along the Γ−Y path, α.
As the pressure increases, the valence and conduction bands approach the Fermi energy level, E F (Fig. 3b), and eventually reach E F at 6.3 GPa (Fig. 3c), rendering a semimetallic picture of the high-pressure phase. Furthermore, above 7 GPa, the optical transition energy β becomes smaller than α (see Fig. 3d) and determines the value of E g, opt . As the valence band penetrates through E F , the interband transition differs from the band separation, Δ; the Moss-Burstein effect 48,49 , ΔE BM g , occurs. The band separation, Δ, can thus be estimated from the experimental value of E g,opt by correcting for ΔE BM g , Δ ¼ E g;opt À ΔE BM g . In Fig. 3e, we summarize the results of the DFT calculations (α and β) and determine the theoretically expected optical gap. From Fig. 2b, we extract the experimentally observed transition energy. To obtain the experimental band separation, we correct the optical gap for the Moss-Burstein effect by using the SW of the lowfrequency optical conductivity shown in Fig. 2a (see the supplemental materials for additional information). Here, we see good agreement between the experimental E g,opt value of lightly doped SnSe and the theoretically predicted E g,opt value (for comparison of the band separation, Δ, see the supplemental materials, Fig. S10). This demonstrates that the lightly doped sample captures the genuine electronic structure of SnSe.

Effect of self-doping under pressure
The differences between lightly doped and heavily doped SnSe are assumed to originate from a small shift of the Fermi level toward the valence band 17,24 . This shift results in qualitatively and quantitatively different features under pressure as well as distinct resistivity at ambient pressure (see Fig. 1c for the resistivity). In the contour plots in Fig. 4a, b, we compare the evolution of the lowfrequency electronic structure of both samples under pressure. At low pressure, the optical conductivity possesses a phonon mode at 120 cm −1 , which disappears as the Drude response becomes dominant, converting the system into a semimetallic phase (above 6 GPa). Increasing the pressure strengthens the Drude mode as the number of charge carriers increases owing to Fermi surface expansion. At the same time, narrowed interband transitions smear into the low-frequency spectra, increasing their contributions.
Heavily doped SnSe shows a weaker and delayed Drude response (above 7 GPa), as indicated by the SW (Fig. 2c) and the plasma frequencies, ω P , obtained by a Drude-Lorentz fit (see the supplemental materials). These observations confirm that the Fermi level shifts toward lower energy, explaining the wider optical gap in the heavily doped sample that causes a larger Moss-Burstein contribution, ΔE BM g . Furthermore, we note that above 10 GPa, the band structure changes only slightly until 14 GPa (Fig. S11 and Fig. S12). In fact, a shift in the Fermi level for heavily doped samples activates the valence band maximum in the Γ−Z direction at ambient pressure, as shown in Fig. 4c, causing metallic-like resistivity behavior (Fig. 1c). However, we note that the resistivity of the heavily doped sample is still relatively high, whereas the contribution to the optical conductivity is barely observable (σ 0 ≈ 12 Ω −1 cm −1 ). The Γ−Y direction is further involved at lower pressures, generating an additional hole pocket. This low-pressure transition is suggested by our DFT calculations, in accordance with previously mentioned transport studies, suggesting a Lifshitz transition at 0.86 GPa 27 from a two-valley system to a four-valley system. We note that in the present study, the low-energy limitations of high-pressure optical measurements mask the appearance of the low-pressure Lifshitz transition, whose carrier density is expected to be too small to be detected in our experiment. Importantly, a shift in the Fermi energy causes the conduction band minimum at the Γ point to touch E F at higher pressures (above 7 GPa) in heavily doped SnSe, creating an additional electron pocket, which is well pronounced in our experimental spectra. This electron pocket further changes the topology of the Fermi surface; that is, a Lifshitz transition occurs. Thus, heavily doped SnSe undergoes successive Lifshitz transitions below 1 GPa and above 7 GPa, as shown in Fig. 1d.
Pressure-induced metallization of SnSe is observed by transport measurements, but the transition pressure is different 33,34 . This discrepancy may be caused by variations in the definition of the semiconductor-semimetal transition pressure and differences between powder and single-crystalline samples 33 . Furthermore, we demonstrated that self-doping effects strongly influence the pressure evolution.

Discussion
For both samples, the semimetallic high-pressure phase originates from a change in the atomic coordinate z of the Sn atom along the c axis. Although the sample remains in the Pnma space group 31,34,47 , the c axis shows more drastic compression than the other axes 47 . In particular, within the scope of the calculations, only adjusting the lattice parameter does not lead to a metallic phase, even up to 10 GPa. Therefore, we suggest that the previously reported decrease in the Sn z-parameter 47 is a crucial factor driving the system into a metallic phase. As previously mentioned 36 , pressure affects the SnSe bonds along the c axis, leading to hybridization of Sn-5s, Sn-5p, and Se-4p orbitals, thus opening an additional conduction channel. In this system, for heavily doped SnSe, the c-direction (armchair) is conductive, followed by the b-direction (zigzag), ultimately inducing a metallic intralayer state. The shift in the Fermi level of heavily doped samples, however, crucially changes the pressure dependence. In fact, the successive Lifshitz-type transitions in heavily doped SnSe are distinct from semiconductor-semimetal transitions and the mere creation of a Fermi surface in the lightly doped samples. Owing to the lowered Fermi level, the creation of the electron pocket at the Γ point takes place at a higher pressure, expanding the already existing Fermi surface of hole pockets in heavily doped SnSe.
The multivalley physics induced by the application of pressure could also play a role in thermoelectric devices since the effective mass is affected by the number of activated valleys in a material 8 . The high thermoelectric properties reported in Ref. 27 prove the effects of the multivalley SnSe. However, as the thermoelectric efficiency is determined by a combination of several competing parameters, the effects of an additional electron pocket on the thermoelectric properties remain to be clarified in other studies.
The modified contributions of the αand β transitions to the optical gap further offer unique tunability and accessibility to the valley degree of freedom. In particular, with the application of pressure, we can tune the system from a dichroic crystal to a non-dichroic crystal (when α and β become comparable). Such a property opens a promising route to valleytronics [1][2][3] .
In summary, we probed the electronic multivalley band structure of SnSe over a broad pressure range by optical spectroscopy. By combining our spectroscopic investigation with DFT calculations, we clarified the semiconductorsemimetal transition in lightly doped SnSe and observed successive Lifshitz transitions in heavily doped samples. We emphasized that self-doping has a considerable influence on the optical properties of SnSe by shifting the Fermi level, which is more pronounced under applied pressure. Our spectroscopic analysis traced the electronic structure evolution under pressure and allowed us to suggest possible applications in the fields of optoelectronics and thermoelectrics by demonstrating versatile tunability under pressure.