Sciences, Oak Ridge National Laboratory, Oak Ridge, TN Sciences of Ukraine, Kyiv, Ukraine Ukraine 88000

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1 Supplementary Material: CuInP S 6 - Room emperature Layered Ferroelectric A. Belianinov, 1 Q. He, 1 A. Dziaugys, P. Maksymovych, 1 E. Eliseev, a A. Borisevich, 1 A. Morozovska, J. Banys, Y. Vysochanskii, 4 and S.V. Kalinin 1 1 he Institute for Functional Imaging of Materials and the Center for Nanophase Materials Sciences, Oak Ridge National Laoratory, Oak Ridge, N 781 Faculty of Physics, Vilnius University, Vilnius, Lithuania L-0151 a Institute for Prolems of Material Sciences, Institute of Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine Institute of Solid State Physics and Chemistry, Uzhgorod University, Uzhgorod, Ukraine Prolem statement and asic equations Within the framework of the LGD theory, equilirium one-dimensional distriution of the out-of-plane polarization component P (z) in a single-domain ferroelectric film can e found from the Euler-Lagrange oundary prolem: Here α =α ( ) C β temperature FE = C + 4γα 5 P αp +βp +γp g P P +λ = 0, z=+ L ϕ =, P P λ = 0. z= L, C is Curie temperature, which is lower that the phase transition ; λ is the extrapolation length [ i ]. Electric potential ϕ should e found self-consistently from the Poisson equation with the open-circuited oundary conditions: (1) 1

2 ε D ϕ 1 P = ε0 ± L ε ε 0 ρ( ϕ) ϕ P ± L, = 0 () Here ε is the ackground permittivity of ferroelectric [1]. For a proper semiconductor the space charge density is ρ ϕ) = ensinh( eϕ k ) (. B In order to derive analytical expressions for the film critical thickness corresponding to the transition into the paraelectric phase, one can use Deye approximation for the charge density in the immediate vicinity of the transition, since the spontaneous polarization and corresponding depolarization field ecome very small here. In accordance with our numerical calculations ased on Eqs.() Deye approximation is valid in the immediate vicinity of the phase transition into the paraelectric phase from either single-domain or poly-domain ferroelectric states, ecause the condition of its validity, ϕ k << 1, holds true here. Below e B we will use Deye approximation ( eϕ kb << 1 ) in the Poisson equation, ϕ ϕ R 1 P = d ε 0ε, where R = kb ε ε e n is Deye radius. For numerical calculations d 0 ( ) we will assume that the temperature dependence of the carries concentration n oeys the activation law, n= n exp( E k ) 0 d B, where d E is the activation energy. he critical thickness of the film transition into a paraelectric phase should e found from the characteristic equation otained from the oundary prolem (1) and () y differentiation, namely from the system of equations listed in the [ ii ]. After cumersome transformations we derived that it can e found as []: κ λ κ = g 1 1 L cr κarctan πκ. () * α λ κ ξ ξ ξ Here the screening length is * κ R d εε0α and the correlation length is ξ = ε ε 0g. * * Coefficient =α ( ) α for the strained ferroelectric film, FE u Q1 4 + s * * β m 4 FE = C + + γα α s11 1 is the FE transition temperature, FE = C β + 4γα, renormalized y the epitaxial misfit strain u m ( a ) 1 = c.he screening length can e rather high, while the correlation length is typically smaller than the lattice constant. he equation () is valid under the condition physical roots at α * < 0. L >> ξ and have

3 As one can see from the Supp. Figure 1(a,) and main text Figure 5, depending on the material parameters and R d values the critical thickness () can vary in a wide range ( nm). Appeared that the thickness () is almost independent on the extrapolation lengths, i.e. the approximation L πκ is rather rigorous for the open-circuited oundary conditions. CuInP S 6 cr (CIPS) material parameters are listed in the Supp. ale 1. he Figure in the main text also can e interpreted as a phase diagram in coordinates "film thickness temperature". he unusual shape of the ferroelectric phase originated from the semiconductor properties contriution, namely from the temperature dependence of R d. LGD potential estimations Bulk LGD potential (for one polarization component) α F = P β + P 4 4 γ + P 6 6 PE Here we suppose that only the first coefficient depends on temperature as follows: α = ( ) ( ) 0 ε0c CW Using experimentally oserved temperature dependence of dielectric permittivity (see Fig. elow) and the value of spontaneous polarization at room temperature (see Ref.[],.55 µc/cm ), one could get the following values: 0 =9 K, C CW = K, β= m 5 /(F C ), γ= m 9 /(F C 4 ) References 1. Eugene A. Eliseev, Sergei V. Kalinin, Anna N. Morozovska. (Accepted to JAP) V. Maisonneuve, V. B. Cajipe, A. Simon, R. Von Der Muhll, and J. Ravez. Physical Review B 56, no. 17: (1997).

4 Screening radius Rd (nm) (a) Concentration n (m - ) Screening radius Rd (nm) () emperature (K) Figure S1. (a) Deye screening radius vs. carriers concentration calculated at 9 K. () he critical thickness of the film transition from paraelectric to a single-domain ferroelectric phase vs. carriers concentration calculated from Eq.(4) at 10, 9, 50 and 00 K. 4

5 coefficient CuInP S 6 ε 7 α(c - mj) at 9 K α (C - mj/k) C (K) 9 FE (K) 16 (calculated) β (C -4 m 5 J) γ(c -6 m 9 J) g (C - m J) ale S1. Material parameters for ulk ferroelectric at room temperature 9 K. 5

6 Figure S. Single Frequency PFM results (Amplitude and Phase respectively) for the surface area shown in the Main ext Figure 1a 6

7 Figure S. 0x0 um Band Excitation PFM results (a) Amplitude and () Phase after DC switching +V in 5x5 um and -V in 10x10um 7

8 Figure S4. 0x0 um Band Excitation PFM results (a) Amplitude and () Phase after DC switching +4V in 5x5 um and -4V in 10x10um 8

9 a Figure S5. 0x0 um Single Frequency PFM results (a) Amplitude and () Phase for the area shown in the manuscript (Figure ) prior to DC switching +5V in 5x5 um and -5 in 10x10um 9

10 Figure S6. Domain area histograms for ulk thickness sample area and thin flakes sample area. One pixel corresponds to 44 nm 10