Reversible and nonvolatile manipulation of the electronic transport properties of topological insulators by ferroelectric polarization switching

Reversible and nonvolatile electric-field control of the physical properties of topological insulators is essential for fundamental research and development of practical electronic devices. Here, we report the integration of topological insulator films with ferroelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT) single crystals in the form of ferroelectric field-effect devices that allow us to tune the electronic properties of topological insulator films in a reversible and nonvolatile manner. Specifically, gating of Cr-doped Bi2Se3 films with the PMN-PT layer is shown to provide a means to reversibly tune and modulate the carrier density and carrier type, as well as its other properties, such as the conductance, magnetoconductance, Fermi level, phase coherence length, and screening factor of electron–electron interaction by polarization switching at room temperature. These findings provide a simple and direct approach for probing the quantum transport properties of topological insulator films through ferroelectric gating by using PMN-PT. The combination of topological insulators with both ferroelectrically and piezoelectrically active PMN-PT thus offers a promising step toward exploring topological insulator/ferroelectric(piezoelectric) hybrid devices that could utilize not only the ferroelectric field-effect of topological insulator/PMN-PT structures but also the unique properties of respective materials. Electric gating can tune the properties of topological insulators, modifying the carriers’ density and type and Fermi level. Several gating materials can be used, but finding materials that allow the manipulation of these properties in a reversible and nonvolatile way at room temperature while being compatible with solid-state electronics is difficult. Now, an international team of researchers led by Ren-Kui Zheng, Yu Wang and Yu-Kuai Liu demonstrated the ferroelectric gating of the topological insulator Bi2Se3 doped with Cr, using single crystals of the perovskite material Pb(Mg1/3Nb2/3)O3-PbTiO3. They fabricated field-effect transistors in which they achieved electric-field control of the carrier density and type and of the Fermi level of the topological insulator through polarization switching of the ferroelectric layer. These results represent a step forward in the development of practical electronic devices based on topological insulators.


INTRODUCTION
Topological insulators (TIs) are a new class of quantum materials that continue to attract worldwide attention due to a variety of novel phenomena that they support, such as the topological magnetoelectric effect, Majorana fermions, quantized anomalous Hall effect, topological superconductor, and their potential applications in quantum computation and dissipationless spintronic devices. 1-10 A number of experimental techniques, including chemical doping, [3][4][5] thickness variation, 6,7 mechanical and epitaxial strain, [8][9][10][11][12] hydrostatic pressure, 13 magnetic proximity, 14,15 and electric gating, 7, [16][17][18][19][20][21][22][23][24][25] have been employed in studies of quantum transport properties of TIs. Among these techniques, electric gating offers an effective approach for tailoring the electronic properties of TIs since it enables in situ modification of carrier density and carrier type and thus circumvents a number of problems inherent to TI films, e.g., disorders, 12,26 defects, 12,27 and lattice strains. [8][9][10][11][12] Several dielectric insulators such as the SiO 2 , 3,28 Al 2 O 3 , 16 HfO 2 , 17 h-BN, 18 SiNx, 19 and SrTiO 3 , [20][21][22] have been used as gate materials to tune the carrier density and carrier type, Fermi level, and electronic transport properties of TI films through the application of gate voltages at very low temperatures (millikelvin to several kelvin). It is noted that the electric-field-induced polarization charges achieved by gating with these dielectric insulators used so far for this purpose are itself volatile and involve relatively weak areal charge density (∼10 12 -10 13 /cm 2 ), 3,[16][17][18][19][20][21][22]28 making it impossible to manipulate the carrier density and carrier type of TI films in a nonvolatile manner at ambient temperatures. Although ionic liquid or gel-like electrolyte gate materials have been successfully used to tune the carrier density in a variety of thin-film samples, including topological insulators, 25 iron-based superconductors, 29 two-dimensional (2D) chalcogenides, 30 tungsten oxides, 31 wideband-gap oxide semiconductors, 32 and strongly correlated oxide magnetics, 33 these electrolytes are not compatible with modern solid-state electronics and may react with thin-film samples electrochemically when gate voltages are applied. 25,[29][30][31] To make progress in both basic research and practical applications, it is desirable to realize control of both carrier density and carrier type and related electronic properties of TI films in a reversible and nonvolatile manner at room temperature. Electrostatic manipulation of carrier density via ferroelectric gating provides a promising approach for realizing such goals. Among a number of ferroelectric materials, perovskite Pb(Mg 1/3 Nb 2/3 ) O 3 -PbTiO 3 (PMN-PT) single crystals possess ultra-high piezoelectric coefficients (d 33 > 2000 pC/N) and large remnant polarization of 2Pr ∼ 60-80 μC/cm 2 at room temperature, which corresponds to an areal charge density of ∼3.7-4.9 × 10 14 /cm 2 . This value is one order of magnitude higher than that obtainable with the dielectric gate materials mentioned above (Supplementary SI). For bismuth chalcogenide TI films, the areal carrier density (n 2D ) is generally on the order of ∼10 13 /cm 23,6,7,16,22,25,34 , which is approximately one order of magnitude lower than the remnant polarization charges of PMN-PT. It is therefore anticipated that the Fermi level, carrier density and carrier type, and electronic properties of TI films could be significantly modulated by the nonvolatile polarization charges of PMN-PT.
For this purpose, we have fabricated TI-based ferroelectric fieldeffect transistors (FeFETs) by growing Cr-doped Bi 2 Se 3 (CBS) thin films with different thicknesses on (111)-oriented PMN-PT singlecrystal substrates. Using these TI/PMN-PT structures, we are able to successfully realize reversible and nonvolatile electric-fieldcontrol of the Fermi level, carrier density, and corresponding electronic properties of CBS films through polarization switching of PMN-PT at room temperature, thus leading to a better understanding of carrier-density-related quantum transport properties of TI films on the one hand, and providing a straightforward approach to realize reversible and nonvolatile tuning of the Fermi level and electronic properties of TI films on the other, as discussed in the sections that follow. Figure 1f shows the relative resistance changes (ΔR/R) of the 6-nm CBS film (Fig. 1a) obtained by scanning electric fields from zero to positive (red) and zero to negative (blue), respectively. Note that the PMN-PT substrates were in the unpoled state before scanning the electric fields. The results reveal that both the magnitude and the sign of ΔR/R strongly depend on the polarization direction of the PMN-PT. ΔR/R totally decreases by ∼22% upon positive polarization. In contrast, ΔR/R increases by ∼13% upon negative polarization. We further measured the resistance of the 6-nm CBS film by scanning bipolar electric fields and observed a resistanceelectric-field hysteresis loop (Supplementary Figure S3), which implies that the interfacial charge effect plays a dominant role in controlling the resistance changes. This feature, together with the opposite changes in the sign of ΔR/R upon positive and negative polarization, demonstrates that the lattice strain effect due to the electrostriction is very weak. Further, we found that with increasing film thickness from 3 to 80 nm, ΔR/R decreases significantly (Supplementary Figure S4A). Particularly, for the thickest 80-nm film, the polarization switching has a minor effect on the resistance (ΔR/R = 5% at 300 K). Taking these polarization direction and thickness-dependent behaviors of ΔR/R and the ntype nature of charge carriers of the CBS films (Supplementary Figure S4D) into account, we conclude that the electric-fieldinduced positive or negative polarization charges at the interface contribute significantly to the resistance changes. We thus measured the carrier density of the CBS films by Hall measurements using the Van der Pauw configuration (Supplementary Figure S2A). The areal carrier density n 2D for the positively polarized P þ r and negatively polarized P À r states is plotted as a function of temperature in the inset of Fig. 1g. Indeed, n was modified by ∼46% at 2 K and by ∼23% at 300 K as the polarization is switched from P þ r to P À r , while the electron mobility μ is only weakly modified by the polarization switching (Supplementary Figure S4B). For the present PMN-PT with a rhombohedral crystal structure the polarization switching from P þ r (i.e., the [111] direction) to P À r (i.e., the [-1-1-1] direction), as schematically illustrated in Figure S5 (Supplementary Materials), should not induce any lattice strain, 35,36 which is confirmed by the X-ray diffraction results shown in Figure S6 (Supplementary Materials), where the PMN-PT(111) and CBS(0003), (0006), (00015) diffraction peaks remain unchanged as the polarization state was switched from P þ r to P À r . Further, PFM measurements show that the polarization direction had been fully and reversibly switched between the [111] and the [-1-1-1] directions by electric voltages, as evidenced by the uniform and sharp contrast between the two domain states within the dotted blue boxes (Fig. 1c, d). These XRD and PFM results demonstrate that any possible lattice strain effects associated with the polarization switching can not account for the resistance and carrier density differences between the two polarization states of PMN-PT. Our findings thus strongly suggest that the polarization switching-induced interfacial charge effects play a crucial role in estalishing the electronic changes of the CBS/ PMN-PT structure.

RESULTS AND DISCUSSION
Due to the nonvolatile nature of polarization charges at the interface, 35,36 the resistance for the P þ r and P À r states exhibits distinct differences down to the lowest temperature for all CBS films with different thicknesses (Fig. 1g and Supplementary Figure  S4A). As can be seen in Fig. 1g, for the P þ r state the resistance shows metallic conducitivity (dR/dT > 0) for T > 20 K and a weak increase for T ≤ 20 K. Upon polarization switching from P þ r to P À r , the metallic profile of the resistance is converted to a semiconducting one (dR/dT < 0), with a stronger resistance upturn for T ≤ 20 K, which can be understood by the significant lowering of the areal carrier density (n 2D ) by amount of Δn 2D ∼7.7 × 10 13 / cm 2 at T = 2 K and Δn 2D ∼3.7 × 10 13 /cm 2 at T = 300 K (inset, Fig.  1g). Such a large reduction in n 2D would result in lowering of the Fermi level relative to the valence band, 1,37 thus leading to a metal-to-semiconductor transition. Moreover, we found that the polarization switching between P þ r and P À r results not only in the changes in the carrier density (Fig. 2c) and resistance (Fig. 2d), but also in a reversible and nonvolatile switching of the carrier type between n-type and p-type in the thinnest CBS film (∼3 nm), as evidenced by the reversible changes in the sign of the slope of the Hall resistance R xy versus the magnetic field B curves at T = 300 and 200 K (Fig. 2a, b). For such a thin CBS film, the top surface state and the bottom surface state would hybridize, resulting in a hybridized gap between the conduction band and the valence band. Thus, the Hall results imply that the Fermi level could be reversibly tuned between the conduction band and the valence band by the polarization switching of the PMN-PT layer at room temperature, as schematically illustrated by Fig. 2e, f. Although such reversible switching of carrier type between n and p has been achieved earlier, in earlier works this has only been successful at extremely low temperatures and only in a volatile manner [16][17][18][19][20][21][22] or has been achieved in a irreversible manner via chemical doping. 4,5,28,38,39 Our approach of PMN-PT-based FeFET structure has the advantage of enabling in situ reversible and nonvolatile tuning of carrier density and carrier type, achievable at room temperature. As seen in Fig. 1a, the quality of our 6-nm CBS film prepared by the pulse laser deposition is indeed lower than those prepared by the molecular beam expitaxy (MBE). 6,10,14,22,26,34,37 As a result, the areal carrier density of the 6-nm CBS film (n 2D ∼ 1.8 × 10 14 /cm 2 ) is approximately one order of magnitude higher that those prepared by MBE. 6,22,34,37 Consequently, the polarization direction switching of the PMN-PT layer is unable to convert the carrier type of the 6-nm film from n-type to Fig. 2 Hall resistance, carrier density, and resistance of the ∼3 nm CBS film. a, b Hall resistance as a function of the magnetic field at T = 300 and 200 K for the P þ r and P À r states of the PMN-PT substrate. c, d Temperature dependence of the charge-carrier density and resistance of the CBS film for the P þ r and P À r states of the PMN-PT substrate. e, f Schematic band diagrams of the Fermi level shift induced by polarization switching p-type. Nevertheless, it is anticipated that if high-quality TI films with lower carrier density is prepared on PMN-PT using MBE, the carrier type of thicker CBS films (t > 3 nm) would also be reversibly modulated between n and p in a nonvolatile manner by polarization switching. In any case, the present realization of reversible modulation of carrier type in a nonvolatile manner by polarization switching at room temperature is an encouraging outcome for both basic research and practical applications and demonstrate the feasibility of carrier type modulation using ferroelectric gating at ambient temperature. Finally, it is noted that regardless of the polarization states of PMN-PT, the Ioffe-Regel parameters k F ' 40 for all films are larger than unity (Supplementary Figure S4C), indicating that the electronic transport in this system is in the weakly disordered regime. 22,26,41 We note that the resistance of all CBS films shows upturn at low temperatures with decreasing temperature (dR/dT < 0) and are enhanced by the application of magnetic fields for both P þ r and P À r states (Supplementary Figure S7). Magnetic measurements indicate that the doped Cr ions did not form long-range ferromagnetic order (Supplementary Figure S8), which is similar to that observed in MBE-grown CBS films with high-Cr doping levels (≥ 9.9%). 42,43 Spin-dependent magnetic scattering is clearly unable to account for the enhancement of the resistance in magnetic fields. Nevertheless, these temperature-and magnetic field-dependent resistance behaviors may be reasonably understood by taking both the electron-electron interaction (EEI) and quantum interference into account. 22,41,44,45 To get insight into this, we analyzed the low-temperature resistance behaviors in terms of the 2D Dirac model, which takes both EEI and quantum interference into account. 41 To perform the fitting, we present the resistance data (Supplementary Figure S7) in the form of quantized conductance G(e 2 /h), where e and h are the unit charge and Planck constant, respectively. Figure 3a, b, c, d show the logarithmic temperature dependence of G(e 2 /h) for the CBS films with different thicknesses (6 nm ≤ t ≤ 40 nm) in magnetic fields up to 9 T when the adjacent PMN-PT layers were in the P þ r and P À r states, respectively. In all cases, G(e 2 /h) decreases linearly with decreasing temperature from 10 K down, and can be satisfactorily fitted in the temperature range from 2 to 10 K by the equation (41), where K is the slope of the temperature-dependent conductance, which reflects the contribution of the electron-electron interaction and quantum interference to the surface state conductivity at low temperatures. G 0 is the residual conductance. T 0 is a reference temperature from which one measures the deviation ΔG. The coefficient K for the P þ r and P À r states are plotted against the magnetic field B in the lower panels of Fig. 3a, b, c, d, respectively. For both polarization states, K increases sharply with increasing field within a narrow-field window below B = 1 T, and saturates for B > 1 T. Such a variation of K with B is consistent with the theoretically calculated results, which are obtained based on the corrections to conductivity from EEI and quantum interference for disordered massless Dirac fermions. This, together with the good fits of G to Eq. (1) suggests the existence of surface states at low temperatures for the 6-to 40-nm CBS film. 41 Upon polarization switching from P þ r to P À r , K is seen to decrease appreciably, particularly for the thinner 6-and 10-nm CBS films. It has been shown that K can be written as K/π = K ee + K qi , where K ee ≈ 1 − (η Λ + η Γ )F is a contribution from EEI, and K qi = αp is a contribution from quantum interference. 41 Here, η Λ ∈ [1/2, 1] and η Γ ∈ [1/4, 0] are weighting factors, and F ∈ [0, 1] is the screening factor of electron-electron interaction. In K qi = αp, α is a prefactor that describes the number of conductance channels and the parameter p can be obtained from fitting of the phase coherence length ' ϕ to ' ϕ~T −p/2 (as discussed in sections that follow). In highmagnetic fields (B »B ϕ , where B ϕ is the coherence field B ϕ ¼ h=4e' 2 ϕ ) destructive quantum interference is greatly suppressed, resulting in K qi → 0. The constant K is then solely determined by K ee , i.e., For the 6-nm CBS film, upon switching polarization from P þ r to P À r , the term η Λ + η Γ decreases slightly, from 0.70 to 0.68 (see Supplementary Figure S9 for details). This, together with Eq. (2) indicates that the screening factor F increases as a result of the polarization switching-induced reduction in the areal carrier density n 2D . In the limit B << B ϕ , on the other hand, both K ee and K qi contribute to K. One then has K/π = K ee + K qi = 1 + αp − (η Λ + η Γ )F. 41 Since, as seen in Fig. 5 and Supplementary Figure  S9A, α increases as the polarization is switched from P þ r to P À r , thus the decrease in K implies that the screening factor F increases upon the polarization switching from P þ r to P À r . Namely, for B << B ϕ F also increases as a result of the reduction in n 2D . Therefore, the results experimentally demonstrate that the polarization switching-induced reduction in the areal carrier density enhances the screening factor of the electron-electron interaction, thereby modifying the EEI contribution, which is consistent with the theoretically calculated results from the Bi 2 Se 3 and Pb-doped Bi 2 Se 3 films (ref. 45 ). Now, we turn to the effects of polarization switching on the magnetoconductance of CBS films with different thicknesses and at different fixed temperatures. It is known that topological surface states can be identified by observing quantum corrections to the conductance in the form of WAL at low temperatures, which manifests itself as a sharp cusp in magnetoconductance. 41,46,47 Figure 4a shows the magnetoconductance (ΔG) as a function of B for CBS films with different thicknesses (t = 6, 10, 20, 40, 80 nm). Negative ΔG with a sharp cusp at B = 0 T is observed for all films at T = 2 K, which is a signature of WAL of surface states. For the 6-nm film, the polarization switching affects the magnitude of ΔG and sharpness of the cusp appreciably, implying fairly effective of the polarization switching on the surface states. Note that as the film thickness increases the effects of electric-field-induced polarization charges on ΔG become less pronounced (Fig. 4a), indicating increasing dominance of bulk contributions. With increasing temperature from T = 2 K, the sharp cusp in the conductance is gradually suppressed and almost disappears at T = 50 K, regardless of the polarization states of PMN-PT (Supplementary Figure S10). It has been shown that 2D magnetoconductance associated with the WAL effect can be described by the Hikami-Larkin-Nagaoka (HLN) theory. 47,48 In the low-field regime, quantum corrections to the conductance are given by Fig. 4 Magnetoconductance ΔG versus magnetic field B for CBS films for PMN-PT substrates in P þ r and P À r states. a ΔG for CBS films with different thicknesses measured at a fixed temperature (T = 2 K). b ΔG for the 6 nm CBS film, as measured at different temperatures of T = 2, 5, 10, 20, 30, and 50 K, respectively. The blue and red solid curves represent the fitting results obtained using Eq. (3) Here, ψ is the digamma function, B is the magnetic field perpendicular to the film plane, ' ϕ is the phase coherence length with ' ϕ ¼ ffiffiffiffiffiffiffiffi Dτ ϕ p , where D and τ ϕ are the electron diffuse constant and phase coherence time, respectively. ΔG of the 6-nm film at different fixed temperatures for both the P þ r and P À r states (Fig. 4b) can be well fitted by Eq. (3) with the fitting parameters α and ' ϕ shown in Fig. 5, where α at T = 2 K increases by some 16.7% (from −0.36 to −0.30) as the polarization is switched from P þ r to P À r , indicating a weakening of the conductance channel of the surface states due to the reduction in n 2D . With increasing temperature from 2 to 50 K, not only the absolute value of α but also the difference in α between the P þ r and the P À r states (Δα), i.e., Δα ¼ αðP þ r Þ À αðP À r Þ , decreases, indicating both a reduced contribution of the surface states to the conductance and a suppression of the effects of polarization switching on surface states with increasing temperature. Because of enhanced inelastic scattering with increasing temperature, the electron phase coherence length ' ϕ for both polarization states decreases with increasing temperature from 2 to 50 K. The relationship between ' ϕ and temperature T can be rather well described by a power law, ' ϕ ∝ T −p/2 , with p = 1 for both polarization states, constituting yet another signature of the surface states in the 6-nm film. 20,44 One should note that polarization switching from P þ r to P À r has resulted in reducing ΔG at temperatures T < 20 K and in its enhancement for T > 20 K (Fig. 4b). Near the crossover temperature (T = 20 K), the polarization switching has almost no effect on ΔG. According to Eq. (3), this opposite effects of polarization switching on ΔG above and below T = 20 K may be related to the opposite trends in ' ϕ induced by polarization switching above and below T = 20 K (Fig. 5). For T < 20 K, ' ϕ for the P þ r state is larger than that for the P À r state while for T > 20 K ' ϕ for the P þ r state is smaller than that for the P À r state. Such opposite changes in ' ϕ are probably related to the polarization switching-induced changes in the inelastic scattering of bulk states, as presented in Tables S1, S2, S3, S4 (Supplementary Materials SII), where the inelastic scattering coefficients γ in are seen to decrease as polarization is switched from P þ r to P À r . In summary, our work reports the first fabrication of CBS/PMN-PT topological ferroelectric field-effect devices and the effects of ferroelectric polarization switching on the electronic properties of CBS films with different thicknesses (3-80 nm). The results demonstrate that the carrier density, Fermi level, conductance, magnetoconductance, conductance channel, phase coherence length, screening factor of electron-electron interaction of the 6nm CBS film can be tuned in a reversible and nonvolatile manner by polarization switching at room temperature. Our findings show that the polarization switching has opposite effects on the magnetoconductance and the phase coherence length ' ϕ above and below T = 20 K for the 6-nm CBS film, suggesting a competition between the surfaces states and the bulk states. The combination of a topological insulator with ferroelectrically and piezoelectrically active PMN-PT would not only provide an alternative effective and convenient approach for tuning the Fermi level and electronic properties of a variety of topological insulators in a reversible and nonvolatile manner, but also may pave a way toward exploring topological insulator/ferroelectric (piezoelectric) hybrid multifunctional devices that can not only take advantage of the ferroelectric field-effect of the structure but also the unique properties of respective materials.

PMN-PT single-crystal growth and ferroelectric and piezoelectric measurements
A rhombohedral PMN-PT (PT = 29%) single crystal with a size of ϕ 50 × 80 mm 3 was grown by the modified Bridgman technique at the Shanghai Institute of Ceramics. The single crystal shows excellent ferroelectric properties (2P r = 76 μC/cm 2 ) (Fig. 1e) and an ultra-high piezoelectric coefficient d 33 ∼ 2500 pC/N. The single crystal was cut into (111)-oriented rectangular plates with a size of 5 × 5 × 0.5 mm 3 and carefully polished on one side to root-mean-square roughness R q < 1 nm. Such polished PMN-PT crystals were used as substrates to grow the Cr x Bi 2−x Se 3 (x = 0.11) films. Ferroelectric hysteresis loops of PMN-PT substrates were measured using a Precision Multiferroic test system (Radiant Technologies, Inc., USA) at a frequency of 1 Hz at T = 300 K. The piezoelectric coefficient d 33 was measured using a d 33 -meter (ZJ-3A, China). Piezoresponse force microscopy (PFM) images of PMN-PT were measured using a Veeco NanoScope-IV Multimode scanning probe microscope.
Thin-film growth CBS films were grown on PMN-PT single-crystal substrates by the pulsed laser deposition from a high-purity CBS target prepared by the spark plasma sintering (SPS) under 50 MPa at 450°C. The target was ablated by a KrF excimer laser (248 nm) with a pulse energy density of 1.2 J/cm 2 and a repetition rate of 1 Hz. The film growth was carried out at a substrate temperature of 350°C in 35 Pa Ar atmosphere, followed by in situ annealing for 30 min and then cooled to the room temperature. Film thicknesses were controlled by the number of laser pulses.

Composition and structural characterization
The phase purity and growth direction of CBS films were characterized by a high-resolution Bruker D8 Discover X-ray diffractometer using CuK α1 radiation (λ = 1.5406 Å). Transmission electron microscopy (TEM) and energy-dispersive X-ray spectroscopy (EDS) measurements were performed using the Tecnai G2 F20 S-Twin transmission electron microscope. EDS results show that the Cr doping level is~11 at% and distributes homogeneously in the films (Supplementary Figure S1).

Electronic transport and magnetic property measurements
The electronic transport properties of CBS films were measured using a Physical Property Measurement System (PPMS-9, Quantum Design), with the direction of the magnetic field perpendicular to the film plane. We particularly note here that, except for the data shown in Fig. 1f, all the other electronic transport data were obtained by measuring the resistance of the CBS films after the removal of the electric fields so that the effects of electric-field-induced lattice strain due to the electrostriction could be ruled out. Magnetic properties were measured using a SQUID magnetometer (Quantum Design). Using the conducting CBS film and the Ag film (∼100 nm) as top and bottom electrodes, respectively, the poling of PMN-PT substrates was achieved by applying a positive or negative dc poling voltage of 330 V (∼6.6 kV/cm) to the PMN-PT layer along the thickness direction (Supplementary Figure S2).  5 Temperature dependence of the conductance channel α and the phase coherence length ' φ for the 6-nm CBS film when the PMN-PT substrate is in the P þ r and P À r states, respectively. The thick cyan line is the fitting to ' φ using with p = 1