Supporting Information GeP3: A small indirect band gap 2D crystal with high carrier mobility and strong interlayer quantum confinement Yu Jing 1,3, Yandong Ma 1, Yafei Li 2, *, Thomas Heine 1,3 * 1 Wilhelm-Ostwald-Institute for Physics und Theoretical Chemistry, Leipzig University, Linnéstr. 2, 04103 Leipzig, Germany; 2 College of Chemistry and Materials Science, Jiangsu Key Laboratory of Biofunctional Materials, Nanjing Normal University, Nanjing, Jiangsu 210023, China; 3 Department of Physics and Earth Sciences, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany. *Correspondence and requests for materials should be addressed to Y. L. (email: liyafei @njnu.edu.cn) or to T. H. (email: thomas.heine@uni-leipzig.de) Computational details The cleavage energy (Ecl) of GeP3 thin layer is defined as: Ecl = EnL + Ebulk-nL Ebulk where EnL, Ebulk-nL and Ebulk are the total energies of 1L or 2L GeP3, the total energy of the bulk after exfoliation, and the total energy of the bulk before exfoliation, respectively. Here, a 5L GeP3 slab is used as a model of the bulk. Larger value of Ecl indicates more difficulties to cleave GeP3 layers from the bulk. In order to avoid the interactions between neighboring images, the vacuum layer of 1L GeP3 was set to be 20 Å along the z direction. The k-point mesh of the Brillouin zone was set to be 6 6 1 for geometry optimization of the thin layers, and HSE06 functional with a mixing parameter alpha value of 0.25 was employed to calculate their band structure and optical properties. 1 Note that this default alpha value may be not sufficient to predict the experimental value very accurately. The band structures of bulk GeP3 and the carrier mobilities of GeP3 thin layers were calculated using the PBE method, as the rigid band shift between DFT-PBE and
HSE06 will not affect the prediction of carrier mobility in 1L and 2L GeP3. Phonon spectra were computed by means of finited displacement method as implemented in the CASTEP code. PBE functional is used for the exchange-correlation energy functional and ultra-soft pseudo-potentials were adopted to describe the valence-ion interaction. A cutoff energy of 500 ev is used with k-point mesh of 6 6 1. 2 The thermal stability of GeP3 monolayer was accessed by performing ab initio molecular dynamics (AIMD) simulations at 500 K. The PAW pseudo-potential and PBE functional as implemented in VASP software were used to examine the stability of the monolayer in a 2 2 supercell. The AIMD simulation at the temperature of 500 K lasts for 10 ps with a time step of 1.0 fs, which was controlled by using the Nosé-Hoover thermostat. 3 The light absorption abilities of 1L and 2L GeP3 were evaluated by calculating the imaginary part ε2(ω) of the dielectric function, which is expressed by: 4 2 2 2 4 e 1 ( ) lim w ( ) 2 q c,v,k k ck vk ck e q vk ck e 0 2 q vk q * (1) Here, c and v refer to the conduction and valence band states, respectively, and ck is the cell periodic part of the orbitals at the k-point k.
Structural details for bulk and single-layered GeP3 Here is the POSCAR information for optimized GeP3 monolayer with lattice geometry and atom positions. ----------------------------POSCAR file begins---------------------------- 1L GeP 3 1.0 6.9600000381 0.0000000000 0.0000000000-3.4800000191 6.0275368434 0.0000000000 0.0000000000 0.0000000000 20.0000000000 Ge P 2 6 Direct 0.842199981 0.179700002 0.465160012 0.175290003 0.847050011 0.585240006 0.338360012 0.683790028 0.507439971 0.338310003 0.172720000 0.507179976 0.849389970 0.683830023 0.507229984 0.679099977 0.343019992 0.542940021 0.168029994 0.342960000 0.543150008 0.679099977 0.854020000 0.543240011 Table S1 The optimized lattice constants (C lattice) and bond length for P-P and Ge-P in GeP 3 monolayer (1L), bilayer (2L), trilayer (3L) and bulk, respectively, in comparison to the experimentally examined values (EXP) for the bulk. Structure C lattice /Å L P-P / Å L Ge-P / Å angle P-P-P / ref 1L GeP 3 6.96 2.18 2.51 110-2L GeP 3 6.95 2.22 2.45 99.3-3L GeP 3 6.99 2.23 2.45 97.6 - GeP 3bulk 7.09 2.23 2.47 97.9 - GeP 3 bulk EXP 7.05 - - - 5
Stability Figure S1. Top and side views of an AIMD snapshot of 1L GeP 3 at T=500 K (P and Ge atoms are Electronic properties denoted by the purple and green balls). Figure S2. Wavefunctions of CBM (a) and VBM (b) from the top and side views for GeP 3 monolayer. The isosurface is set to be 0.003 eå 3. Figure S3. (a) Partial density of states (PDOS) of 1L GeP 3; (b) band structure of 4L GeP 3.
Figure S4. Band gap variation of GeP 3 with respect to the layer number. Carrier mobilities An orthogonal lattice (shown in Figure S5a) was used for the convenience of demonstrating the carrier mobilities along the armchair and zigzag directions. The band structures of 1L and 2L GeP3 calculated by PBE in an orthogonal supercell are presented in Figure S5 b and c. Figure S5. (a) 2L GeP 3 in an orthogonal super cell and the Brillouin zone, and band structures for 1L (b) and 2L (c) GeP 3 in the orthogonal lattice. Since the VBM of 1L and 2L GeP3 are located in the range between the X and points instead at one of the high-symmetry points, we further calculated the band structures of these structures in a selected Brillouin zone (an example is shown in Figure S6a) to access their effective masses along the zigzag and armchair directions.
Figure S6. (a) Band structure of 1L GeP 3 in an orthogonal supercell and the selected Brillouin zone for the orthogonal lattice is plotted inside (b) Energy difference between the total energy of unstrained and strained 1L GeP 3 along the zigzag and armchair directions; (c) and (d) energy shift of VBM and CBM for 1L GeP 3 with respect to the lattice dilation and compression along the zigzag and armchair directions, respectively. According to m*=ħ 2 [ 2 ε(k)/ k 2 ] 1, the effective masses of the electrons and holes along the armchair and zigzag directions are calculated. As shown in Table S2, in the zigzag and armchair directions, GeP3 monolayer shows light and heavy effective masses for holes. Along the armchair direction, the m* for electrons ( m * e ) and light holes ( m * h ) are 0.80 and 0.55 me, respectively. In the zigzag direction, and the m * e and light m * h are 0.59 and 0.72 me, respectively. By simulating the total energy (E) change of the 2D material motivated by the applied strain (δ), the in-plane stiffness (C2D) can be obtained as shown in Figure S6b. According to C2D = [ 2 E/ δ 2 ]/S0, we calculated the C2D along the armchair and zigzag directions, where S0 is the surface area of the optimized supercell. As illustrated in Table 2, the C2D of GeP3 monolayer along the zigzag and armchair directions are 22.35 N/m and 22.10 N/m, respectively. The similar in-plane stiffness from the zigzag and armchair directions indicates the isotropic mechanical stress response of 1L GeP3.
Table S2 Calculated effective mass m*, DP constant E 1, in-plane stiffness C 2D, carrier mobility μ for GeP 3 monolayer and bilayer along the zigzag (x) direction and armchair (y) directions. N L represents the layer number of GeP 3. Carrier type NL m * zig /m0 m * arm /m0 E1zig E1arm C2Dzig C2Darm zig arm G-X G-Y ev N/m 10 3 cm 2 V -1 s -1 electron hole 1 0.80 0.59 3.44 3.48 22.35 22.10 0.04 0.07 2 0.21 0.25 3.70 1.24 53.32 54.70 1.25 8.84 1 2.75(0.55) 1.00(0.72) 1.74 1.29 22.35 22.10 0.014(0.35) 0.19(0.36) 2 0.36 0.31 1.12 0.97 53.32 54.70 4.63 8.48 As presented in Figures S6c and d, the DP constant (E1) can be obtained by fitting the liner relationship of the band edge (Eedge) for VBM and CBM with the strain exertion (δ) along the zigzag and armchair directions, respectively. The calculated E1 along the zigzag (E1zig) and armchair (E1arm) directions are listed in Table S2. It can be found that 1L GeP3 possesses small DP constants for both electrons and holes. Specifically, the DP constants along the zigzag directions are 3.44 ev for electrons and 1.74 ev for holes, while the DP constants along the armchair direction are 3.48 ev and 1.29 ev for electrons and holes, respectively. According to the DP theory, the carrier mobility of 2D materials can be estimated by the following expression, 6 3 2e C2D 2D (1) 2 3k T m * E B 2 1 where ħ is the reduced Planck constant, kb is Boltzmann constant, and T is the temperature (set to be 300 K). Since C2D, m* as well as E1 have already been estimated, the carrier mobilities of GeP3 monolayer along different directions can be calculated by a simple operation of expression (1). The calculated carrier mobilities for GeP3 monolayer are shown in Table S2. Following the same calculation protocol, the carrier mobilities of 2L GeP3 are also calculated (see Table S2).
Band structure engineering of 1L GeP3 Figure S7. (a) Band gap tuning of 1L GeP 3 by applying compressive/ tensile biaxial strain, calculated at the HSE06 level. The applied biaxial strain is defined as (l l 0)/l 0, here l is strained lattice and l 0 is the equilibrium lattice constant. (b) Band structure variations of GeP 3 monolayer after being compressed or streched.the Fermi level is set to be the VBM. As illustrated in Figure S7b, the VBM of GeP3 monolayer moves left under the compressive strain whereas it moves right under the tensile strain. References 1 Heyd, J.; Peralta, J. E.; Scuseria, G. E.; Martin, R. L. J. Chem. Phys. 2005, 123, 174101. 2 Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter 2002, 14, 2717 2744. 3 Martyna, G. J.; Klein, M. L.; Tuckerman, M. E. J. Chem. Phys. 1992, 97, 2635 2643. 4 Gajdoš, M.; Hummer, K.; Kresse, G.; Furthmüller, J.; Bechstedt, F. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 045112. 5 Donohue, P. C.; Young, H. S. J. Solid State Chem. 1970, 1, 143-149. 6 Bardeen, J.; Shockley, W. Phys. Rev. 1950, 80, 72 80.