Prediction of first-order martensitic transitions in strained epitaxial films

Epitaxy is an important concept for the fabrication of films with good crystalline quality such as overlayers, multilayers, compound materials, and ordered alloys. Such films are technologically important materials with adjustable electronic, magnetic, and optical properties [1]. Epitaxial growth also allows us to study fundamental aspects of low-dimensional structures and interfacial effects, and it allows us to produce structures that are unstable in the bulk [2]. For example, body-centered cubic (BCC) Co does not exist as a bulk phase but was stabilized by epitaxial growth on a GaAs substrate [3].

In coherent or pseudomorphic film growth, the in-plane film lattice parameters are determined by the substrate-in-plane lattice parameters. This allows us to expose a material to static, non-isotropic but homogeneous strain, which is a valuable means to influence its intrinsic properties [4]. Experiments on ferromagnetic bulk-like films showed that the magnetic moments, the Curie temperature, and the magnetic anisotropy can be tuned in a quasi-continuous manner by varying the substrate [5, 6]. Using piezoelectric substrates, even a continuous variation of the lattice parameters and of the related film properties was achieved [7, 8].

Here, we predict by density-functional calculations scanning 30 non-magnetic metallic elements, that bulk-like epitaxial films of at least six metals show first-order martensitic transitions upon variation of the epitaxial strain. For example, we find jumps in the lattice parameter of La by 0.83 Å and in the superconducting critical temperature of V by 20%.

The examples considered in the present work include only the simplest case of tetragonal non-magnetic mono-atomic films. We, however, emphasize that our main result, the possibility of isostructural phase transitions [9, 10] under epitaxial strain and a related discontinuous dependence of intrinsic film properties on the interface lattice parameters, is not restricted to certain structures, compositions, or ground states.

We model the structures of the considered films by the epitaxial Bain path (EBP) [11]. The EBP model applies to the bulk part (i.e., the interior) of a coherently grown thick film [12]. It assumes that the film can be described by a bulk structure with periodicity in all three dimensions. Due to epitaxial coherency, the in-plane film lattice parameters are rigidly tied to the substrate lattice parameters in the substrate-film interface via the underlying epitaxial relationship. On the other hand, the film is free to relax its geometry perpendicular to the interface due to the absence of forces. Summarizing, the model neglects all surface and interface effects except the determination of the in-plane film lattice parameters by the substrate.

In the past, the EBP model was used to predict metastable states in transition metals [11, 13–16], to investigate the magnetic order in strained overlayers [17–20], and to identify the mother phases of strained bulk-like films [11, 21–23]. In the same context, we recently suggested Ru, Os, and U films to be ferromagnetic in certain ranges of epitaxial strain [24].

Focusing on the important case of substrates with four-fold surface symmetry and body-centered tetragonal (BCT) films with (001) orientation, we denote the in-plane film lattice parameter by a and the out-of-plane lattice parameter by c; see figure 1. The mentioned absence of forces perpendicular to the substrate-film interface allows for a relaxation of c minimizing the total film energy $E(a,c)$ at any fixed value of a [11, 12]. Hence, the EBP of a certain BCT material is the set of points $\{(a,{{c}_{{\rm min} }})\}\subset \{(a,c)\}$, for which

$$\begin{eqnarray}&&{\rm EBP}\mathop{=}\limits^{{\rm def}}\left\{ (a,{{c}_{{\rm min} }})\mid E\left( a,{{c}_{{\rm min} }} \right)=\mathop{{\rm min} }\limits_{c}\;E(a,c) \right\}.\end{eqnarray} \tag{ 1 }$$

The side condition determines the relaxation of c for a given a and defines the relation ${{c}_{{\rm min} }}(a)$. Quantities traced along the EBP only depend on a, e.g., the total energy ${{E}_{{\rm EBP}}}(a)\mathop{=}\limits^{{\rm def}}E(a,{{c}_{{\rm min} }}(a))$. The difference between ${{E}_{{\rm EBP}}}(a)$ and the ground state energy of the film material, E₀, corresponds to the general definition of an epitaxial strain energy [12].

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A strict implementation of the notion 'relaxation of c minimizing $E(a,c)$' leads to relevant, until now disregarded consequences for all materials to which strained coherent epitaxy applies. As we show, the function $E(a,c){{\mid }_{a}}$ can have two equal-valued global minima at $c_{{\rm min} }^{1}\ne c_{{\rm min} }^{2}$ for certain values of $a={{a}_{{\rm crit}}}$. Thus, ${{c}_{{\rm min} }}(a)$ can be discontinuous at ${{a}_{{\rm crit}}}$ in a first-order transition. This includes, by epitaxial coherency, also the dependence of ${{c}_{{\rm min} }}$ on the substrate lattice parameters in the interface plane. In this way, the possibility to tune overlayer-material-specific properties can be restricted to a discontinuous manner. However, one can switch these properties on purpose, if one succeeds to control the lattice parameter a in a narrow interval around ${{a}_{{\rm crit}}}$, e.g., by a piezoelectric substrate.

In the following, we verify the existence of degenerate global minima in $E(a,c){{\mid }_{a}}$ by high-precision density-functional theory (DFT) calculations, carried out with the full-potential local-orbital scheme FPLO-7.00–28 [25]. Using the local-density approximation according to [26] (PW92) and a scalar-relativistic mode for elements with atomic number $\lt 49$, a full-relativistic mode otherwise, we scanned the EBP of 30 elements with the atomic numbers 20–23, 29, 30, 38–48, 56, 57, 71–80, and 92 in a wide range of parameters a [16]. The convergence of numerical parameters and the basis set were carefully checked [16]. In order to converge the total energy per atom at a level smaller than 0.3 meV, linear-tetrahedron integrations with Blöchl corrections were performed on a $24\times 24\times 24$ mesh in the full brillouin zone, apart from the elements Cu, Ru, Ir, and Pt, for which a denser $48\times 48\times 48$ mesh was required. all detected degenerate minima in $E(a,c){{\mid }_{a}}$ were cross-checked by additional calculations employing the Perdew–Burke–Ernzerhof (PBE96) parameterization of the exchange-correlation functional [27] (results listed in table 1).

Table 1.  List of elements with discontinuous EBPs in PW92, which were cross-checked in PBE96. The exchange-correlation functional (XC), the position of the discontinuity, ${{a}_{{\rm crit}}}$, the related out-of-plane lattice parameters $c_{{\rm min} }^{1/2}$, the change of ${{c}_{{\rm min} }}$ across the discontinuity, $\Delta {{c}_{{\rm min} }}$, the relative volume change $\Delta V/V=2\Delta {{c}_{{\rm min} }}/(c_{{\rm min} }^{1}+c_{{\rm min} }^{2})$, and the strain energy at ${{a}_{{\rm crit}}}$ with respect to the energy of the ground state are tabulated.

		${{a}_{{\rm crit}}}$	$c_{{\rm min} }^{1}$	$c_{{\rm min} }^{2}$	$\Delta {{c}_{{\rm min} }}$	$\Delta V/V$	${{E}_{{\rm EBP}}}({{a}_{{\rm crit}}})-{{E}_{0}}$
Element	XC	[$\overset{\circ}{A}$]	[$\overset{\circ}{A}$]	[$\overset{\circ}{A}$]	[$\overset{\circ}{A}$]	%	[meV/atom]
V	PW92	2.637(3)	3.97	3.39	0.58	16	313
	PBE96	2.698(2)	3.96	3.51	0.44	12	280
V	PW92	3.345(5)	2.54	2.37	0.17	7	323
	PBE96	3.447(2)	2.59	2.46	0.13	5	330
Nb	PW92	2.933(2)	4.38	3.76	0.62	15	390
	PBE96	2.988(2)	4.41	3.87	0.54	13	354
Ru	PW92	3.010(1)	3.26	3.03	0.23	7	685
	PBE96	3.062(2)	3.31	3.05	0.26	8	627
La	PW92	4.043(1)	4.50	3.67	0.83	20	178
	PBE96	4.232(2)	4.65	3.91	0.74	17	103
Os	PW92	3.063(1)	3.28	3.02	0.26	8	907
	PBE96	3.104(1)	3.39	3.03	0.36	11	900
Ir	PW92	3.077(2)	3.23	2.89	0.24	11	630
	PBE96	3.118(1)	3.31	2.93	0.38	12	818

Vanadium is one of the elements for which EBP was studied in several investigations [11, 15, 28, 29]. This provides us with the following, and as we will argue incomplete, picture. The EBP of vanadium includes two energy minima: one corresponds to the stable BCC phase; the other one at a = 2.400 Å, ${{c}_{{\rm min} }}/a=1.84$, is a local minimum; see figure 2. This local minimum corresponds to an unstable BCT bulk state [15, 16], since its energy is lowered by a shear deformation. Tian et al [22, 29], however, succeeded in stabilizing coherently grown vanadium films on Cu{001} and Ni{001}, the structures of which were determined to be strained derivatives of this BCT state. The face-centered cubic (FCC) structure was found at a saddle point of $E(a,c)$ and assigned to a maximum of ${{E}_{{\rm EBP}}}$ [11, 15, 28, 29].

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By symmetry, the BCC structure ($c/a=1$) and the FCC structure ($c/a=\sqrt{2}$) are stationary points of $E(a,c)$ [15, 28], but the side condition in definition 1 determines whether they are located at the EBP or not. In contrast to previous reports [11, 15, 28, 29], we find that the FCC structure does not belong to the EBP of vanadium, because it is unstable against a tetragonal distortion at fixed $a={{a}_{{\rm FCC}}}$. We note that in terms of linear elasticity theory this instability of FCC vanadium with respect to a relaxation of c corresponds to a negative elastic constant ${{c}_{zzzz}}\lt 0$, since ${{c}_{zzzz}}\propto {{{\rm d}}^{2}}\;E/{\rm d}{{c}^{2}}$. This is obvious from the plot of $E({{a}_{{\rm FCC}}},c)$ in figure 3. There, $E({{a}_{{\rm FCC}}},c)$ has a local maximum at ${{c}_{{\rm FCC}}}=\sqrt{2}{{a}_{{\rm FCC}}}$ and the global minimum is situated at ${{c}_{{\rm min} }}=3.35\;\overset{\circ}{A}$, ${{c}_{{\rm min} }}/a=1.27$.

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The characteristic feature of this instability is the occurrence of a double well in $E({{a}_{{\rm FCC}}},c)$ versus c with the global minimum at ${{c}_{{\rm min} }}$ and a local minimum. According to figure 3, double wells also occur for other values of a in the vicinity of ${{a}_{{\rm FCC}}}$. The energies of both minima vary with a such that the two minima are degenerate at a critical value $a={{a}_{{\rm crit}}}$. All in all, the behavior shown in figure 3 is typical for a first-order phase transition. One can readily read off the energy barrier of somewhat less than 10 meV per atom, an upper limit for the hysteresis of $\Delta {{a}_{{\rm hyst}}}\approx 0.015$ Å, and the discontinuity of ${{c}_{{\rm min} }}$, $\Delta {{c}_{{\rm min} }}=\mid c_{{\rm min} }^{1}({{a}_{{\rm crit}}})-c_{{\rm min} }^{2}({{a}_{{\rm crit}}})\mid \approx 0.58$ Å.

In general, a saddle point in $E(a,c)$ (here, at $({{a}_{{\rm FCC}}},\sqrt{2}{{a}_{{\rm FCC}}})$) and a discontinuity in the EBP do not have the same a-coordinates. In the case of vanadium, the already-described (first) discontinuity occurs at $2.635\;\overset{\circ}{A} \lt {{a}_{{\rm crit},1}}\lt 2.640\;\overset{\circ}{A} \lt {{a}_{{\rm FCC}}}$. The associated change of ${{c}_{{\rm min} }}/a$ amounts to $\Delta {{c}_{{\rm min} ,1}}/{{a}_{{\rm crit},1}}\approx 0.2$. Surprisingly, the EBP of vanadium shows yet another discontinuity at $3.34\;\overset{\circ}{A} \lt {{a}_{{\rm crit},2}}\lt 3.35\;\overset{\circ}{A}$ with $\Delta {{c}_{{\rm min} ,2}}/{{a}_{{\rm crit},2}}\approx 0.05$.

Figure 4(a) depicts a contour plot of $E(a,c/a)$ of vanadium together with the EBP. The two minima of $E(a,c/a)$ correspond to the two minima of ${{E}_{{\rm EBP}}}$ from figure 2 and the FCC structure is the saddle point as demanded by symmetry. The contour diagram (see inset) reveals the following: there are pairs of lines of constant energy, $E\lt {{E}_{{\rm FCC}}}$, where each line of the pair encloses one of the two minima and the two lines overlap in their a-coordinates in the vicinity of the saddle point. The double wells in figure 3 arise from this overlap. A similar peculiarity of the energy landscape at $a\sim 3.35$ Å causes the second discontinuity and the related kink of ${{E}_{{\rm EBP}}}$ visible in figure 2.

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While the specific energy landscape in the vicinity of the saddle point leads to a discontinuous EBP, the mere presence of a saddle point—be it symmetry dictated or not—does not automatically lead to a discontinuous EBP. For example, if the abscissa and ordinate axes in figure 4(a) were interchanged, determining the EBP would require searching for minima along horizontals, which would result in a continuous EBP, including the FCC structure.

In order to analyze the origin of the discontinuities, we consider the electronic DOS at the Fermi level, $N({{E}_{{\rm F}}})$; see figures 2 and 4(b). The transition at ${{a}_{{\rm crit},1}}$ is accompanied by a pronounced change in $N({{E}_{{\rm F}}})$ due to a significant change of the unit cell volume. This change depends monotonously on c; see figure 4(b). Further analysis of the total energy shows that the discontinuity at ${{a}_{{\rm crit},1}}$ is mainly driven by the kinetic energy contribution. Since there is no obvious peculiarity in $N({{E}_{{\rm F}}})$, the double well has to be assigned to an integral effect of the whole DOS. At the position of the second transition ${{a}_{{\rm crit},2}}$, however, there is a distinct peak of $1.96\;{\rm states}/({\rm eV}\cdot {\rm atom})$ in $N({{E}_{{\rm F}}})$ along the EBP (figure 2). The same characteristic peak appears as a tooth-like feature in the large-a region at approximately $c/a=0.73$ in figure 4(b). The EBP of vanadium, which crosses this area, becomes discontinuous in order to avoid the large $N({{E}_{{\rm F}}})$ (Jahn–Teller behavior). At the local maximum of the related double well of $E({{a}_{{\rm crit},2}},c)$ (not shown), $N({{E}_{{\rm F}}})=2.4\;{\rm states}/({\rm eV}\cdot {\rm atom})$.

We searched the EBPs of the 30 elements listed above for the occurrence of first-order transitions in the same way as described for the case of vanadium [16]. The search space was restricted to $0.7\lt c/a\lt 2.1$ and had a resolution $\Delta a=0.05$Å ($\Delta a$ was subsequently reduced to pinpoint ${{a}_{{\rm crit}}}$). Seven transitions were found in this way; see table 1. Apart from vanadium, the group-5 element niobium (but not tantalum) and the elements ruthenium, lanthanum, osmium, and iridium exhibit discontinuous EBPs. For La and Ir, we identified a negative c_zzzz of the BCC structure as the driving force of the instability, while in Nb the FCC structure has ${{c}_{zzzz}}\lt 0$. BCC Ru and Os possess small positive c_zzzz but the discontinuities of the EBP occur in the vicinity of the BCC structure at a lattice parameter being slightly smaller and larger than ${{a}_{{\rm BCC}}}$, respectively. No second instability was found for any element except vanadium. All seven first-order transitions were also found in additional calculations relying on PBE96; see table 1. Compared to the PW92 results, ${{a}_{{\rm crit}}}$ is shifted to larger values in PBE96.

The present work focuses on non-magnetic elements, where we found first-order martensitic transitions under epitaxial strain due to either a large peak in $N({{E}_{{\rm F}}})$ or a more subtle interplay of all contributions to the total energy in the vicinity of a saddle point. In order to find out whether the epitaxial relation of a particular substance is continuous or not, one usually has to calculate this relation in detail. An FCC or BCC structure with negative c_zzzz always results in a discontinuous EBP, but a (small) positive c_zzzz does not exclude a discontinuity.

In the case of magnetic systems, high-spin to low-spin transitions [9] or a change of the ground state magnetic order [17] are usually accompanied by discontinuities of the lattice geometry. An example might be the a-dependence of the magnetic ground state of BCT Fe found by Qiu et al [17]. These authors did not, however, discuss the consequence of a transition into a different magnetic state for the lattice parameter c of coherent films.

We now return to the case of vanadium to demonstrate how strongly the structural discontinuity may alter another particular intrinsic property, namely, the superconducting critical temperature ${{T}_{{\rm c}}}$. Vanadium is a well-known superconductor and possesses one of the highest ${{T}_{{\rm c}}}$-values among all elements, ${{T}_{{\rm c}}}=5.4$ K, at normal pressure [30]. A significant change of ${{T}_{{\rm c}}}$ may be expected at ${{a}_{{\rm crit},1}}$, since $N({{E}_{{\rm F}}})$ jumps by $0.2\;\;{\rm states}/({\rm eV}\cdot {\rm atom})$; see figure 2.

The calculations for ${{T}_{{\rm c}}}$ of BCT vanadium were carried out with the ABINIT-DFT package in the version 6.12 [31, 32] employing a pseudopotential (PP) generated with the FHI98PP code [33] for PW92. With this PP, we found slightly shifted values of ${{a}_{{\rm BCC}}}=2.963\;\overset{\circ}{A}$ and of ${{a}_{{\rm crit},1}}=2.623\;\overset{\circ}{A}$ compared with the FPLO results. The phonon properties were obtained through density-functional perturbation theory [34] as implemented in ABINIT [35, 36]. To compute the superconducting properties of V, we used a plane-wave-kinetic-energy cutoff of 650 eV and a $52\times 52\times 52$ mesh in the full Brillouin zone for linear tetrahedron integrations. The phonon quantities are sampled by eight wave vectors in the full Brillouin zone.

We verified the accuracy of our approach for BCC vanadium for which we obtained an electron–phonon coupling constant $\lambda =0.95$ and a logarithmic-averaged phonon frequency ${{\omega }_{{\rm log} }}=288\;{\rm K}$. For comparison, tunneling experiments found $\lambda =0.82$ [37]. We derived a reasonable value of ${{T}_{{\rm c}}}=3.2$ K for BCC vanadium according to the Allen–Dynes modified McMillan formula [38]. This equation contains one parameter, the effective Coulomb repulsion ${{\mu }^{*}}$, which we set to a value ${{\mu }^{*}}=0.13$ commonly used for transition metals [39].

${{T}_{{\rm c}}}$ across the discontinuity ${{a}_{{\rm crit},1}}$ was computed for the points ($(a,{{c}_{{\rm min} }})$ coordinates) ($2.595\;\overset{\circ}{A} ,4.128\;\overset{\circ}{A}$) and ($2.646\;\overset{\circ}{A} ,3.545\;\overset{\circ}{A}$). λ changes by $\Delta \lambda =0.20$ and ${{T}_{{\rm c}}}$ by $\Delta {{T}_{{\rm c}}}=0.6\;{\rm K}$ or 20% across ${{a}_{{\rm crit},1}}$ (from 3.5 K at $a=2.595\;\overset{\circ}{A}$ to 4.1 K at $a=2.646\;\overset{\circ}{A}$). The actual value of ${{T}_{{\rm c}}}$ may depend on the film thickness, as this was found in experiments for BCC vanadium films [40, 41], but we expect that the relative change will not be substantially affected by this dependence.

We have shown that coherently grown epitaxial films can exhibit a discontinuous relation between in-plane and out-of-plane lattice parameters in a strain-driven, first-order phase transition. In particular, we predict that BCT bulk-like films of the elements V, Nb, Ru, La, Os, and Ir undergo martensitic lattice transformations induced by epitaxial strain with related volume changes up to 20%. As a consequence, the intrinsic film properties can be switched, if one succeeds in controlling the substrate lattice parameter appropriately. We demonstrated this possibility with the example of vanadium, where the superconducting critical temperature changes by about 20% at a critical strain value.

Since the mechanisms of the discussed instabilities may be present in arbitrary metallic systems, our results are not limited to mono-atomic overlayers, BCT films or specific ground states. Verification of the predicted transitions should be possible by advanced diffraction methods applied to coherently grown overlayers on a series of substrates with quasi-continuous lattice spacings. An indirect verification could be achieved by measuring any other structure-dependent physical property of the film material.

We thank K Dörr, S Gemming, S Haindl, S Fähler, and A Möbius for discussions.