Negative piezoelectric response of van der Waals layered bismuth tellurohalides

TL;DR

This study reveals a novel negative piezoelectric response in layered bismuth tellurohalides caused by internal charge redistribution, suggesting potential for robust ultra-thin film applications.

Contribution

It introduces a new mechanism for negative piezoelectric response based on charge redistribution in layered tellurohalides, confirmed through first-principles calculations.

Findings

Charge redistribution leads to negative piezoelectric response.

Internal structure remains weakly affected by external stress.

Negative response is robust in ultra-thin films.

Abstract

The polarization and piezoelectric response of the BiTe$X$ ($X$=Cl, Br, and I) layered tellurohalides are computed from first principles. The results confirm a mixed ionic-covalent character of the bonding, and demonstrate that the internal structure within each triple layer is only weakly affected by the external stress, while the changes in the charge distribution with stress produce a substantial negative piezoelectric response. This suggests a new mechanism for negative piezoelectric response that should remain robust even in ultra-thin film form in this class of materials.

Tables

Table 1. Table 1: Calculated structural parameters of BiTe X 𝑋 X ( X 𝑋 X =I, Br, Cl). V 𝑉 V is cell volume, a 𝑎 a and c 𝑐 c are in-plane and out-of-plane lattice constants, α 𝛼 \alpha and β 𝛽 \beta are Bi- X 𝑋 X and Bi-Te layer spacings, and P 𝑃 P is polarization. The prime on BiTeCl ′ denotes the results for the same P ​ 3 ​ m ​ 1 𝑃 3 𝑚 1 P3m1 structure as for X 𝑋 X  = I and Br; the unprimed version is for the ground-state P ​ 6 3 ​ m ​ c 𝑃 subscript 6 3 𝑚 𝑐 P6_{3}mc structure.

	$V$ (Å3)	$a$ (Å)	$c$ (Å)	$\alpha$ (Å)	$\beta$ (Å)	$P$ (C/m2)
BiTeI	111.5	4.343	6.823	2.104	1.721	0.069
BiTeBr	102.6	4.270	6.499	1.871	1.754	0.100
BiTeCl′	97.5	4.235	6.275	1.677	1.767	0.107
BiTeCl	195.5	4.239	12.563	1.667	1.765	0.099

Table 2. Table 2: Dipole moment of the triple atomic layer in bulk BiTe X 𝑋 X , in units of e 𝑒 e Å, for the three models in Fig.  2 . Last two columns give the values of δ X subscript 𝛿 𝑋 \delta_{X} and δ Te subscript 𝛿 Te \delta_{\rm Te} , describing the displacements of the p 𝑝 p -band Wannier centers from nearby ionic planes, in units of Å. BiTeCl ′ refers to the same P ​ 3 ​ m ​ 1 𝑃 3 𝑚 1 P3m1 structure as for X 𝑋 X  = I and Br, although it is not the observed ground state for X 𝑋 X  = Cl.

	$p_{\mathrm{DFT}}$	$p_A$	$p_B$	$p_c$	${\delta }_X$	${\delta }_{\mathrm{Te}}$
BiTeI	0.48	$-$1.34	2.96	0.43	0.24	0.54
BiTeBr	0.64	$-$1.64	2.75	0.56	0.19	0.55
BiTeCl′	0.61	$-$1.86	2.56	0.57	0.15	0.55
BiTeCl	0.61	$-$1.86	2.55	0.53	0.15	0.55

Table 3. Table 3: Calculated piezoelectric responses of BiTe X 𝑋 X materials: e 33 subscript 𝑒 33 e_{33} for relaxed-ion (RI), frozen-ion (FI), and frozen-triple-layer (FTL) cases; and proper, improper, and epitaxial d 33 subscript 𝑑 33 d_{33} piezoelectric responses. The in-plane-lattice constants are relaxed for proper and improper d 33 subscript 𝑑 33 d_{33} cases and fixed otherwise. BiTeCl ′ refers to the same P ​ 3 ​ m ​ 1 𝑃 3 𝑚 1 P3m1 structure as for X 𝑋 X  = I and Br, although it is not the observed ground state for X 𝑋 X  = Cl.

	$e_{33}$ (C/m2)			$d_{33}$ (pm/V)
	RI	FI	FTL	Prop.	Imp.	Epi.
BiTeI	$-$0.53	0.10	$-$0.57	$-$24.4	$-$23.0	$-$24.4
BiTeBr	$-$0.61	0.06	$-$0.60	$-$30.6	$-$28.6	$-$31.2
BiTeCl′	$-$0.57	0.04	$-$0.54	$-$34.6	$-$32.4	$-$35.9
BiTeCl	$-$0.47	0.09	$-$0.47	$-$27.6	$-$25.1	$-$29.7

Table 4. Table 4: Decomposition of the piezoelectric response into two contributions as given in Eq. ( 7 ). All quantities in units of C/m 2 .

	$-p_0/V_0$	$V_0^{-1}\left(\partial p/\partial \eta \right)$	$e_{33}$
BiTeI	$-$0.069	$-$0.460	$-$0.53
BiTeBr	$-$0.100	$-$0.507	$-$0.61
BiTeCl′	$-$0.107	$-$0.464	$-$0.57
BiTeCl	$-$0.099	$-$0.374	$-$0.47

Equations

$$e_{\alpha j}={-\frac{\partial {\sigma }_j}{\partial {\mathcal{E}}_{\alpha }}|}_{\eta }={\frac{\partial P_{\alpha }}{\partial {\eta }_j}|}_{\mathcal{E}},$$

$$d_{\alpha j}={-\frac{\partial {\eta }_j}{\partial {\mathcal{E}}_{\alpha }}|}_{\sigma }={\frac{\partial P_{\alpha }}{\partial {\sigma }_j}|}_{\mathcal{E}}.$$

$$p_{\mathrm{A}}=e(\alpha -2\beta ).$$

$$p_{\mathrm{B}}=e(\alpha +\beta /2)$$

$$p_{\mathrm{C}}/e$$

$$p_{\mathrm{C}}=p_{\mathrm{A}}+6e({\delta }_{\text{Te}}-{\delta }_X).$$

$$e_{33}$$

$$d_{33}^{\text{prop}}=\frac{1}{A}\frac{\partial \tilde{P}}{\partial \sigma }=\frac{P}{A}\frac{\partial A}{\partial \sigma }+d_{33}^{\text{imp}}.$$

Full text

I Introduction

Conventional soft-mode ferroelectric materials exhibit a structural transition from a paraelectric to a polar phase as the temperature falls through a critical $T_{\rm c}$, below which a polar phonon mode freezes in to generate a ferroelectric ground state.Haertling (1999); Resta (1994a); Dawber et al. (2005) In general, this polar phonon mode stiffens as the lattice constants are reduced, and consequently the magnitude of the polarization decreases with compressive strain. This corresponds to a positive piezoelectric response, $d_{33}>0$, where by convention we consider the polar variant with $P>0$. Recently, however, Liu and Cohen proposed a different mechanism by which materials can exhibit a negative piezoelectric response, and identified hexagonal $ABC$ ferroelectrics as a class of materials in which the lattice contribution to the piezoelectricity is positive, but the frozen-ion or electronic contribution is negative and larger.Liu and Cohen (2017) Negative piezoelectricity has also been experimentally demonstrated in ferroelectric polymersKatsouras et al. (2015) and in organic molecular ferroelectric materials.Urbanaviciute et al. (2019)

BiTe$X$ ($X$ = Cl, Br, and I) compounds have attracted considerable recent interest as strongly polar quasi-2D materials. The breaking of inversion symmetry results from the layer geometry, in which a central Bi layer is neighbored by a Te layer on one side and a halide layer on the other side, forming a triple layer (TL) as shown in Fig. 1. The TLs are bonded to each other by weak van der Waals interactions, implying easy exfoliation and a soft mechanical response under uniaxial stress. The TLs are stacked so that the Te layer is always on the same side of Bi, with one-TL periodicity resulting in a polar $P3m1$ space group for $X$ = Br and I, and two-TL periodicity resulting in a polar $P6_{3}mc$ space group for $X$ = Cl. Because of the strongly broken inversion symmetry combined with strong spin-orbit coupling, these materials are of interest for their large bulk Rashba effect, with potential spintronic applications. Ishizaka et al. (2011); Bahramy et al. (2011); Monserrat and Vanderbilt (2017) BiTeI has also been much discussed for its topological properties, since it has been predicted to undergo a topological phase transition to a strong topological-insulator phase under pressure, mediated by a narrow but topologically robust Weyl semimetal phaseMurakami (2007); Bahramy et al. (2012); Liu and Vanderbilt (2014); Murakami et al. (2017), and to exhibit an enhanced nonlinear Hall conductivity.Facio et al. (2018)

The broken centrosymmetry of the crystal also naturally suggests the possibility of ferroelectricity or piezoelectricity. The polarization is not associated with a polar distortion of a nearby high-symmetry reference structure, and is inherently not switchable, since the bonding within the TL is much too strong to allow a structural reversal under applied electric field. However, as these systems are mechanically soft, there is a marked change in structure under applied stress, which can be expected to result in a change in polarization and corresponding piezoelectric response.

In this study, we investigate the electric polarization and piezoelectric response of BiTe$X$ by using first-principles calculations. We compare the calculated dipole moments with two plausible models that anticipate opposite directions of the dipole moment, deciding in favor of the one that treats the BiTe unit as more covalently than ionically bonded. We will see that while structurally, the BiTe$X$ TLs behave as relatively rigid units, internal charge rearrangement under applied uniaxial strain or stress leads to a substantial negative piezoelectric response. This suggests a new mechanism of piezoelectricity that may be widely applicable to a broad class of insulating materials based on layered van der Waals stacking of polar constituents.

II Methods

The polarization and piezoelectric response of BiTe$X$ are determined from first-principles calculations carried out using the VASP package.Kresse and Furthmüller (1996a, b) The pseudopotentials are of the projector-augmented-wave type as implemented in VASP,Blöchl (1994); Kresse and Joubert (1999) with valence configurations 6$s^{2}$6$p^{3}$ for Bi, 5$s^{2}$5$p^{4}$ for Te, and 3$s^{2}$3$p^{5}$, 4$s^{2}$4$p^{5}$, and 5$s^{2}$5$p^{5}$ for Cl, Br, and I, respectively. The exchange-correlation functional is described by the modified Perdew-Burke-Ernzerhof generalized gradient approximation for solids (PBEsol). Perdew et al. (2008) The plane-wave cut-off energy is set to $400$ eV. The Brillouin zone sampling grid is 12 $\times$ 12 $\times$8 for the 1-TL periodic $P3m1$ structure and 12 $\times$ 12 $\times$4 for the 2-TL periodic $P6_{3}mc$ structure; relative energy differences between the two structures were obtained by computing both in the doubled-cell structure with the 12 $\times$ 12 $\times$4 grid. Spin-orbit coupling is included in all calculations. The structural coordinates are relaxed within a force threshold of 1.5 meV/Å. The electric polarization is computed using the Berry-phase method, King-Smith and Vanderbilt (1993); Vanderbilt and King-Smith (1993) and the Wannier charge centersMarzari et al. (2012) are obtained using the VASP-Wannier90 interface.Mostofi et al. (2014) The maximal localization of the Wannier functions is carried out separately for the $s$ and $p$ bands to avoid $sp_{3}$ hybridization.

For a crystal composed of weakly coupled molecules or layers, it is natural to compute the polarization from the dipole moment of the individual unit. For a periodic system, this value can be quantitatively obtained by computing the Berry phase polarization  King-Smith and Vanderbilt (1993); Vanderbilt and King-Smith (1993); Resta (1994b), where the branch choice arising from the quantum of polarization can resolved by choosing the value closest to that estimated by the dipole moment integral, or by using the Wannier center formulation and choosing the Wannier centers to be within the individual unit. In this work, we consider only $p=p_{3}$, the dipole moment per the unit cell measured along the stacking direction $\hat{e}_{3}$, and adopt the convention that the polarization $P_{3}$, electric field $\mathcal{E}_{3}$, strains $\eta_{3}=c/c_{0}-1$, and stresses $\sigma_{3}$ (Voigt notation for $\eta$ and $\sigma$), will also be written without the subscript for simplicity.

We calculate various piezoelectric responses following the standard definitions. Ballato (1995); Nye (2012); Wu et al. (2005) The piezoelectric stress tensor elements $e_{\alpha j}$ are defined in terms of the derivative of stress with respect to the electric field, or equivalently, polarization with respect to strain,

[TABLE]

while the piezoelectric strain tensor elements $d_{\alpha j}$ are related to the derivative of strain with electric field, or equivalently, the derivative of polarization with respect to stress,

[TABLE]

To calculate the piezoelectric stress response, the polarization $P$ is calculated on a grid of strains $\eta$ with the in-plane lattice constant fixed to the zero-stress value, and is fitted to a polynomial to obtain the derivative corresponding to the piezoelectric response $e_{33}$. For the same grid of strains $\eta$, we compute the optimized value of $a$ at each $\eta$, and then using the values of stress and polarization reported by VASP at each $\eta$, we fit the results in order to extract the value of the piezoelectric $d_{33}$ coefficient. In addition, we compute a mixed response $d_{33}^{\rm epi}$ by carrying out a similar fitting procedure but at fixed in-plane lattice constant.

III Results

III.1 Structure and polarization

BiTeI and BiTeBr crystallize in the hexagonal structure illustrated in Fig. 1(a), space group $P3m1$ (#156), with three atoms per cell. BiTeCl has the same internal layer structure, but alternate TLs are rotated 180∘ about $\hat{e}_{3}$ on an axis passing through the $X$ atom, as shown in Fig. 1(b), resulting in a doubled six-atom unit cell belonging to space group $P6_{3}mc$ (#186).

Our computed structural parameters for these three materials, together with the Berry-phase polarization $P$, are given in Table 1. In the case of $X$ = Cl, we carried out calculations in both the $P3m1$ and $P6_{3}mc$ structures; the former is designated with a prime (BiTeCl*′*) as a reminder that it is not the experimental ground-state structure. We see rather obvious trends in that the volume and the Bi-$X$ distance $\alpha$ shrink as $X$ becomes more electronegative, while the Bi-Te distance $\beta$ remains roughly constant. The trend in going to $X$ = Cl is most consistent when the same structure is assumed (first three rows of the table). The change to the doubled-cell $P6_{3}mc$ in the last row is generally small, showing that the stacking sequence does not have a strong effect on the structural parameters.

The calculated $c$ lattice constant of BiTeI is close to the experimental value of 6.854 Å.Shevelkov et al. (1995) We tested several different exchange-correlation potentials including some with van der Waals corrections, but we find that our use of PBEsol produces the closest agreement for the $c$ lattice constant, with an error of 0.5%, compared to the other ones we tested (5.9% for PBEPerdew et al. (1996), 3.7% for PBE+TSTkatchenko and Scheffler (2009), and 2.2% for SCANSun et al. (2015)). This is consistent with a previous theoretical reportMonserrat and Vanderbilt (2017) in which the PBEsol functional was found to give the most accurate prediction of the BiTe$X$ structure.

We computed the ground-state energies for each of the materials in both the $P3m1$ and $P6_{3}mc$ structures, finding that the doubled-cell $P6_{3}mc$ structure is higher in energy by 7.3, 4.8, and 5.1 meV for BiTeI, BiTeBr, and BiTeCl, respectively. This correctly predicts the 1-TL ground state structure for BiTeI and BiTeBr, but it does not account for the observation of the 2-TL structure of BiTeCl. However, the energy differences are small, and are near the limit of our first-principles resolution. We speculate that it may be necessary to take differences in vibrational entropy into account in order to explain the observed structure of BiTeCl. In any case, as noted above, a comparison of the BiTeCl and BiTeCl*′* results in Table 1 shows that the structural properties are not very sensitive to the choice of space-group structure, and we report results for BiTeCl in both structures.

Figure 1(c) shows two of the maximally localized Wannier functions constructed from the $p$ bands of BiTeI, rendered using the VESTA software package Momma and Izumi (2011). We see a somewhat asymmetric bond orbital composed of Bi and $X$ $p$ orbitals at bottom, and a somewhat more symmetric bond orbital made of Bi and Te $p$ orbitals at top. Both show significant covalent bonding, but the greater asymmetry of the Bi-$X$ bond orbital is consistent with a stronger ionic character, as expected from the stronger electronegativity of the halide $X$ atom. The trend in the strength of covalency is also evident from the values of $\alpha$ and $\beta$ reported in Table 1. Not surprisingly, the $\beta$ value (Bi-Te spacing) remains roughly constant, while the $\alpha$ value (Bi-$X$ spacing) shrinks significantly in going from $X$ = I to Br to Cl, with the increasing electronegativity of the $X$ ion.

The results for the electric polarization for each of the three materials, computed using the Berry-phase approach as described in the Methods section, are presented in the last column of Table 1. Not surprisingly, the polarization, being a dipole moment per unit volume, increases as the volume decreases, but this does not account for all of the variation. The interpretation of the sign and magnitude of the polarization is the topic of the next subsection.

III.2 Interpretation of the polarization

The computed polarization reported in Table 1 is positive, so the dipole moment of the layers points along the direction from the halide to the Te. Surprisingly, two simple models, shown in Fig. 2(a) and (b), predict opposite signs of the polarization. Model A, the fully ionic model shown in Fig. 2(a), assumes that the ions keep their nominal valence, in which case the dipole moment is estimated as

[TABLE]

(recall that $\alpha$ and $\beta$ are the Bi-$X$ and Bi-Te layer spacings respectively). From the figure, it is clear that the dipole moment would then point to the left (negative, in our convention) because of the excessive negative charge of Te*-2* compared to $X^{-}$.

Model B, shown in Fig. 2(b), assumes a strong covalent bond between neighboring Bi and Te layers, and treats this pair of layers as a single unit with an overall valence of $+1$. The dashed vertical line at right in Fig. 2(b) indicates an average position of this Bi-Te unit, taken to be midway between the Bi and Te planes. In this model the dipole is predicted to be

[TABLE]

which is clearly positive, in contrast with the prediction of the previous model.

Table 2 reports the values of the dipole moment as computed from first principles ($p_{\rm DFT}$), as well as the values computed from Models A and B. Neither of these models gives a value for the dipole that is close to the first-principles value. The first-principles result is not far from an average of the two, suggesting that Models A and B can be taken as describing two end points corresponding to extreme ionic and mixed ionic-covalent bonding states, respectively.

As described in Sec. II, the polarization is given exactly in terms of the coordinates of the ions and those of the Wannier centers of the occupied bands. The positions of the Wannier centers constructed from the occupied $p$ bands, shown earlier in Fig. 1(c), are indicated by the solid vertical lines in Fig. 2(c). This approach would be exact if the information on the Wannier centers constructed from the occupied $s$ bands were included as well, but we assume these to coincide with the atomic coordinates; this is a reasonable approximation since the $s$ bands are well separated and weakly hybridized with other bands. There are thus two additional parameters taken from first principles, namely the shifts $\delta_{X}$ and $\delta_{\rm Te}$, relative to the anion coordinates, of the Wannier centers constructed from the occupied $p$ bands. We then include the $s$ charge $-2e$ into our definition of the core charges, which become +4, +3, and +5 for Te, Bi, and $X$, respectively. The remaining twelve electrons form anion $p$-like Wannier functions whose Wannier-center positions are illustrated in Fig. 2(c). Six of these are associated with Wannier centers displaced by $\delta_{\textrm{Te}}$ from the Te centers, and the other six are centered a distance $\delta_{X}$ from the $X$ centers, measured along the $z$ direction. Accounting for all of the ionic and Wannier contributions shown in Fig. 2(c), the dipole moment is given by

[TABLE]

and comparing with Eq. (3), this is just

[TABLE]

Turning to the results given in Table 2, we see that this analysis agrees well with the full DFT results. We can now think of Model A as a limit in which $\delta_{X}\!=\!\delta_{\textrm{Te}}\!=\!0$, and Model B as corresponding to $\delta_{X}\!=\!0$ and $\delta_{\textrm{Te}}\!=\!5\beta/12$. Clearly, neither is a good approximation. From another point of view, we can say that an accurate picture of the dipole is given by modifying Model A according to Eq. (6). Note, however, that $\delta_{\textrm{Te}}$ is much larger than $\delta_{X}$ in Table 2, as expected given the stronger covalency of the Bi-Te bonding, as was discussed toward the end of Sec. III.1 when describing the Wannier functions shown in Fig. 1(c). Thus, Eq. (6) leads to a large positive correction to the prediction of Model A.

III.3 Piezoelectric response at fixed in-plane lattice constant

III.3.1 Piezoelectric stress coefficients

Figures 3(a-d) show the polarization of BiTe$X$ as a function of uniaxial strain at fixed in-plane lattice constant. These values are labeled “relaxed-ion” since they include full structural relaxation. The slopes at the equilibrium state correspond to the piezoelectric stress tensor elements $e_{33}$, which are reported as the relaxed-ion (RI) values in the first column of Table 3. The values are strongly negative for all three materials.

In the theory of piezoelectricity, it is a common practice to decompose the response into a “frozen-ion” component, defined by uniformly scaling the atomic positions in the unit cell as the strain state is changed, and a “lattice” contribution that comes from the change of internal atomic coordinates with strain. The polarizations predicted by a naive frozen-ion model (green circles) in Figs. 3(a-d) are far from correct, and the corresponding $e_{33}$ values shown in Table 3 are much too small, and have the wrong sign, compared to the full relaxed-ion results.

The above decomposition is in fact not appropriate for these quasi-2D systems. To see why, we first look at the changes of the ionic coordinates shown in Figs. 3(e-h). It is clear that the Bi-$X$ and Bi-Te interlayer spacings ($\alpha$ and $\beta$) are almost constant as a function of $\eta$, while the changes in the inter-TL spacing $\gamma$ tracks very closely with the $c$ lattice constant. This shows that a frozen-TL model, in which the internal coordinates of the TL are fixed while only the spacing between them changes, is an excellent approximation for these systems. The accuracy of this model can be understood as arising from the weak van der Waals bonding between TLs, in contrast to the much stronger covalent bonding within the TLs. The polarizations obtained from the frozen-TL model are shown as the triangles in Figs. 3(a-d), and the $e_{33}$ values are given in the ‘FTL’ column of Table 3. These are in excellent agreement with the full first-principles (‘RI’) results, both in sign and magnitude.

It is important to note that the frozen-TL model does not imply a fixed dipole for the TL, as electronic relaxation occurs self-consistently at each value of $c$. If the dipole of the frozen TL were independent of $c$, then the negative piezoelectric response would be described as coming from a simple volume effect, where the change in $P=p/V$ is mainly a result of the change in $V$, with the result that a compression or expansion simply concentrates or dilutes the polarization. More generally, the change in dipole can be taken into account by writing

[TABLE]

where $p_{0}$ and $V_{0}$ are the dipole moment and cell volume at $\eta=0$. If the polarization $p$ of the frozen TL were independent of $\eta$, then only the first term would be present. This purely mechanical model, based only on the change of volume, does correctly predict the negative sign of the piezoelectric response, but we find that it severely underestimates the magnitude of the effect.

To investigate this, we have calculated the dependence of the Wannier-center shifts $\delta_{\textrm{Te}}$ and $\delta_{X}$ on the uniaxial strain $\eta$, since these are the parameters that reflect the internal change of the TL dipole that is not captured by the structural coordinates alone. In the purely mechanical limit, $\delta_{\textrm{Te}}$ and $\delta_{X}$ would be independent of $\eta$, and the second term in Eq. (7) could be dropped. We find that $\delta_{\textrm{Te}}$ is indeed very nearly independent of $\eta$; it varies by only about 0.1% with a 1% change in $c$. By contrast, we find a much more significant change in the position of the centers of the Bi-$X$ Wannier functions, with $\delta_{X}$ changing by about 2.3, 2.6, and 2.1% for $X$=I, Br, and Cl respectively, for every 1% change in $c$. We can rationalize this change by noting that the halogen environment becomes more symmetric (less distinction between intra-TL and inter-TL neighbor distances), so that the Wannier center shifts toward the $X$ coordinate, as the TLs are pressed closer to each other. An inspection of the changes of the Wannier functions (not shown) does indicate a stronger $X$-Te hybridization across the van der Waals gap, consistent with a reduction of $\delta_{X}$, as the distance between TLs is decreased.

In short, we find that there is a substantial increase in the magnitude of the slab dipole with compression, which comes about because of the change in $\delta_{X}$ values as the TLs are pressed together. The consequences of this are summarized in Table 4, where the contributions of the first and second terms in Eq. (7) are given independently, clarifying their relative contribution to the piezoelectric response $e_{33}$. It now becomes clear that while the purely mechanical model, represented by $-p_{0}/V_{0}$, correctly gives the negative sign of the piezoelectric response, it underestimates the magnitude of the response by almost an order of magnitude. The modulation of the dipolar contribution of the Bi-$X$ bond with compression provides a large boost to the observed effect, and is responsible for the unexpectedly large piezoelectric effect that is revealed by our calculations.

III.3.2 Piezoelectric strain coefficients

Next, we consider the piezoelectric strain response $d_{33}$. Because this is defined under zero-stress boundary conditions, the unit cell area now varies with $\sigma$, and there is a distinction between proper and improper piezoelectric responses Vanderbilt (2000). The “improper” piezoelectric response $d_{33}^{\textrm{imp}}={\partial P}$/${\partial\sigma}$ simply describes the change of polarization under stress. However, the piezoelectric response is typically measured by tracking the stress-induced current flowing between top and bottom electrodes in a capacitor configuration, which corresponds to the change of surface charge per unit cell $\tilde{P}=AP$ with external stress, where $A$ is the cell area. This defines the “proper” piezoelectric response, where the two are related by

[TABLE]

Figure 4 shows the variation of $P$ with respect to the uniaxial stress $\sigma_{3}$; the slope at $\sigma_{3}=0$ yields $d_{33}^{\rm imp}$. A corresponding analysis of the dependence of $\tilde{P}$ on uniaxial stress gives $d_{33}^{\rm prop}$, and the results are summarized in Table 3. The improper response ranges from $-$23 pm/V to $-$32 pm/V, which systematically increases in magnitude from $X$=I to Br to Cl*′* in the same $P3m1$ structure. The change of structure (Cl*′* to Cl) gives a $\sim$20% reduction, resulting in the Br compound having the largest $d_{33}^{\rm imp}$ response. The correction term expressing the difference between $d_{33}^{\rm prop}$ and $d_{33}^{\rm imp}$ in Eq. (8), whose sign is negative (the cell area expands under uniaxial compression along $c$), enhances the negative piezoelectric responses slightly, by $\sim$5-9% compared to improper responses. As a result, the proper piezoelectric response of BiTeBr reaches $-$30 pm/V.

III.3.3 Epitaxial piezoelectric coefficients

Finally, we calculated the mixed response $d^{\textrm{epi}}$ (see the Methods section), defined under conditions of fixed in-plane strain and out-of-plane stress. Here there is once again no distinction between proper and improper responses. The results are given in the last column of Table 3. The changes are not very dramatic; $d_{33}^{\rm epi}$ is similar to $d_{33}^{\rm prop}$ for $X$ = I and slightly larger for the Br, Cl*′, and Cl cases. We have checked the effect of a fixed in-plane lattice constant that is set to a modified value, as in the case of coherent heteroepitaxy on a substrate. A compressive epitaxial strain is found to induce a negligible change, whereas a tensile epitaxial strain reduces the magnitude of the $d_{33}^{\rm epi}$ response somewhat, especially for $X$=Cl′* and Cl.

III.4 Discussion

To review, we have shown that the physics of the piezoelectric response in the BiTe$X$ system is very different from that of conventional ferroelectrics such as perovskite oxides. In those systems, the proximity to a polar instability, the associated soft polar modes, and the anomalously large dynamical effective charges generate very strong piezoelectric responses. Here, instead, we start with a system that is far from any structural phase transition, so it might be expected to show quite a small piezoelectric response.

Nevertheless, we find a substantial piezoelectric response in the BiTe$X$ system. Our theory shows that a model in which the internal structure of each TL is frozen, and only the spacing between them changes, gives an excellent account of the structural changes under applied uniaxial strain. If the dipole moment of the TL were also frozen, this would already account for the anomalous negative sign of the piezoelectric response. However, we find that the electronic charge redistribution within the TL plays a very important role, and is responsible for the surprisingly large $e_{33}$ values.

To be sure, our calculated $|e_{33}|$ values of $\sim$0.50 C/m2 are smaller, by an order of magnitude or more, than those of well-known perovskites such as PbTiO3 and PZT, which have values in the range of 4-12 C/m2. However, it is still comparable to that of wurtzite semiconductors (0.02-1.5 C/m2) and $ABC$ ferroelectric materials (0.4-1.5 C/m2). Bernardini et al. (1997); Liu and Cohen (2017) AlN and LiMgAs are reported to have the largest $e_{33}$ responses of $\sim$1.5 C/m2 among each class of materials,Bernardini et al. (1997); Liu and Cohen (2017) which is slightly larger than the classical wurtzite piezoelectric material ZnO having $e_{33}\simeq$ 1.2 C/m2,Wu et al. (2005); Catti et al. (2003) and only a factor of three larger than that of BiTe$X$. The most negative $e_{33}$ among the $ABC$ ferroelectrics is found in NaZnSb as $e_{33}$ = $-$1.04 C/m2, only twice larger than BiTe$X$. A theoretical investigation of the negative piezoelectric responses in the $ABC$ ferroelectrics has revealed that the frozen-ion $\overline{e}_{33}$ gives a negative contribution that dominates the total response. Liu and Cohen (2017) This is in a sharp contrast with BiTe$X$, where the frozen-ion contribution is rather small and positive.

Moreover, the piezoelectric response of BiTe$X$ is further magnified when converting to the piezoelectric strain coefficient $d_{33}$ because of the softness of the interlayer van der Waals interaction, which implies a large strain per applied uniaxial stress. This makes the comparison of the $d_{33}$ values of the BiTe$X$ materials even closer to being competitive with other piezoelectrics. Our calculated $|d_{33}|$ values, in the range of 24-36 pm/V, compare well to those of LiMgP (25 pm/V) and LiMgAs (29 pm/V), which have the most positive $d_{33}$ among the $ABC$ ferroelectric materials, even though their $e_{33}$ values are a factor of three larger as discussed above. Liu and Cohen (2017) The relative enhancement of $d_{33}$ relative to $e_{33}$ in BiTe$X$ is also evident by comparison with BaTiO3, Wu et al. (2005) where $|e_{33}|=4.44$ C/m2 is an order of magnitude larger than for BiTe$X$, while $|d_{33}|=14.7$ pm/V is a factor of two smaller than for BiTe$X$.

Recent studies have shown that ultrahigh piezoelectric responses have been achieved with $d_{33}$ up to 2,800 pm/V in PMN-PT Park and Shrout (1997); Zhou et al. (2014) and 640 pm/V in BaTiO3-based ceramic systems, Praveen et al. (2015); Wei et al. (2018) and a negative piezoelectric response of $d_{33}\!=\!-$690 pm/V has been reported in the classe of ferroelectric polymers based on polyvinylidene difluoride (PVDF). Ghosh et al. ; Wei et al. (2018) We note, however, that these ultrahigh piezoelectric responses are a result of careful compositional tuning of the system. For example, the simple $\beta$ phase of PVDF exhibits a $d_{33}$ of $-$50 pm/V, which is only a factor of two larger than for BiTe$X$. Soin et al. (2015); Wei et al. (2018)

We also note that the $d_{33}$ responses of BiTe$X$ are comparable to the values reported for some thin-film PZT samples (21.3 pm/V) Guo et al. (2013), although still an order of magnitude smaller than for thick-film PZT (457 pm/V). Because the mechanism of piezoelectricity in the BiTe$X$ system is completely independent of any soft-mode transition, there is no reason to expect our computed responses to suffer from the kind of finite-size effects that suppress the piezoelectricity in thin-film geometries for conventional materials. Taken together with our encouraging estimates of the size of the responses reported above, these results suggest that the BiTe$X$ materials could provide a promising alternative material system to use as a basis for microactuator and other specialized applications.

IV Summary

We have used first-principles methods to calculate the polarizations and piezoelectric responses of BiTe$X$ for $X$=Cl, Br, I. The piezoelectric response is found to be negative. The change in structure under uniaxial stress is described to an excellent approximation by a “frozen triple-layer” model in which the BiTe$X$ unit is internally rigid, while the spacing between these units is modulated by the applied uniaxial strain. However, the dipole moment of the TL is not frozen, and changes with stress due to electronic relaxation are found to dominate the piezoelectric response. The piezoelectric responses are an order of magnitude smaller than those of commercial bulk piezoelectric materials, but the mechanism can be expected to survive in the thin-film limit where standard piezoelectrics tend to degrade. Thus, BiTe$X$ could be a promising alternative material for thin-film piezoelectric devices. The recent explosion of interest in stacked van-der-Waals–bonded heterostructures provides opportunities for BiTe$X$ as a piezoelectric component in such systems.

Acknowledgements.

This work is supported by the ONR Grants N00014-16-1-2951 and N00014-17-1-2770.