Electric field driven insulator-to-metal phase transitions
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Electric field driven insulator-to-metal phase transitions
M. Nardone
Department of Environment and Sustainability, Bowling Green State University, Bowling Green, OH 43403, USA
V. G. Karpov
Department of Physics and Astronomy, University of Toledo, Toledo, OH 43606, USA (Dated: October 18, 2018)We show that strong enough electric fields can trigger nucleation of needle-shaped metallic embryosin insulators, even when the metal phase is energetically unfavorable without the field. This generalphenomenon is due to the gigantic induced dipole moments acquired by the embryos which causesufficient electrostatic energy gain. Nucleation kinetics are exponentially accelerated by the field-induced suppression of nucleation barriers. Our theory opens the venue of field driven materialsynthesis. In particular, we briefly discuss synthesis of metallic hydrogen at standard pressure.
PACS numbers: 05.70.Fh, 64.60.qe, 64.60.Bd, 82.60.Nh
The topic of metal-insulator transitions has long beenestablished [1, 2] and remains active. Application as-pects aside, it concentrates mostly on the underlyingmicroscopic mechanisms, such as those induced by dis-order (the Anderson transition), intra-atomic interac-tion (Mott-Hubbard), band crossing, and some others.The common feature of these transitions is that they aredriven by changes in some material parameter: degree ofdisorder, doping concentration, etc.Here, we introduce the concept of electric field driveninsulator-to-metal phase transitions. They start withneedle-shaped metal embryos forming in an insulatorwhen the system is immersed in a strong enough field.We argue that, even when the bulk metallic phase assuch is energetically unfavorable, increasing the field willeventually cause the transition to occur. Furthermore,assuming a fixed electric field, the final state of the sys-tem will be a uniformly metallic needle-shaped body.This concept holds regardless of the microscopic mech-anism of the transition (densification, crystallization,electron solvation, or others [1, 2]); it applies equallyto solids and liquids. For example, it predicts conduc-tive needle-shaped crystallites forming in an insulatingglass under strong enough fields. As another example,strong electric fields will trigger nucleation of liquid Si(metallic) needle-shaped inclusions in a semiconductingSi host, even at temperatures well below melting. A moreprovocative example, briefly discussed in this Letter, isthe field-induced synthesis of metallic hydrogen understandard pressure.The fact that symmetry-breaking electric fields candramatically affect nucleation processes was recently re-alized [3] while studying nucleation of highly conduc-tive filaments in chalcogenide glasses of phase changememory [4]. Other related phenomena can include bias-induced metal-insulator transitions in resistive randomaccess memory [5], and dielectric breakdown in thin-filmdevices [6]. Another category is non-photochemical laserinduced nucleation [7]. The theory herein takes a significant additional stepby predicting that sufficiently strong fields can triggertransitions to states that would not be stable withoutthe field; they will remain metastable upon field removal.As a possible candidate we mention bias-induced switch-ing from insulating to highly conductive states, such asin vanadium dioxide (VO ) [8] and chalcogenide glasses[4], where the conductive phase disappears upon field re-moval. Other known phenomena could be pertinent, suchas e. g. dielectric breakdown in thin oxides. However,mainstream understanding of the latter refers to a build-up of defects produced by stress, eventually forming theonset of a percolation path across the oxide [6]. Thesetypes of mechanisms are beyond the present framework.We start with a brief introduction to the electric fieldeffect in classical nucleation theory (CNT). The free en-ergy of a new particle in the presence of the field is, F = Aσ − Ω µ + F E . (1)Here A and Ω are the particle surface area and volume, σ is the coefficient of surface tension, and µ is the chemicalpotential difference between the two phases, taken to bepositive when the bulk new phase is energetically favor-able. Eq. (1) does not specify the type of transition. Theelectrostatic term has the form [9], F E = − εE Ω8 πn , (2)where ε is the electric permittivity of the host insulat-ing phase and the effect of particle geometry is embodiedin the depolarizing factor, n . For a sphere, n = 1 / A = 4 πR , and Ω = 4 πR /
3. In zero field, the maximumof F ( R ) from Eq. (1) provides the nucleation barrier W = 16 πσ / (3 µ ) and radius R = 2 σ/ | µ | with typi-cal magnitudes near 1 eV and 1 nm. Maintaining theassumption of spherical geometry, the field reduces thenucleation barrier and radius according to [10], W sph = W (1 + E /E ) and R sph = R E /E , (3)where E = 2[ W / ( εR )] / is typically in the range ofseveral MV/cm.In CNT, µ must be positive in Eq. (1), implying ametastable host phase. Another important assumptionis that of spherical symmetry. We show next how phasetransitions are possible even for the case of energeticallyunfavorable bulk new phase (negative µ ) when a strongelectric field is applied and the constraint of sphericalshape is relaxed.In our concept, the free energy of Eq. (1) has twodegrees of freedom: spherical symmetry is broken and,when the field is sufficiently strong, needle-like conduc-tive particles aligned with the field become energeticallyfavorable. That can be understood by comparing theelectrostatic energy contribution of a sphere to that of aprolate spheroid of height H and radius R , for which thedepolarizing factor is [9], n = ( R/H ) [ln (2 H/R ) − ≡ ( R/H ) L. (4)Considering particles of equal volume, the electrostaticcontribution is greater for a prolate spheroid by a hugefactor of approximately ( H/R ) ≫ H / R F E / W E (a) (b) FIG. 1: (a) Absolute, normalized value of the electrostaticenergy contribution, F E /W , to the free energy as a functionof nucleus aspect ratio; H/R = 1 corresponds to a sphere.A value of
E/E = 0 .
25 and nucleus volume of 4 πR / The exact shape of the elongated nucleus is not known,but modeling with either spheroidal or cylindrical parti-cles leads to differences only in numerical coefficients [3].We opt for the mathematically concise cylinder shapewith A = 2 πRH , Ω = πR H , and free energy, F cyl = W (cid:18) RHR ± R HR − E E H R (cid:19) , (5) with the approximation n ≈ ( R/H ) for H ≫ R in Eq.(4). As illustrated in Fig. 2, the free energy landscapeexhibits a range of low nucleation barriers at small R . (a) (b)0.0 0.2 0.4 0.6 0.8 1.00.01.02.03.0 R/R H / R R/R H / R α H i gh ba rr i e r pa t h
0_ + 0_ + α FIG. 2: Free energy landscape of FIMS as a function of nu-cleus radius,
R/R , and height, H/R [from Eq. (5) with E/E = 0 . F/W ∼ .
1. Regionsof positive and negative free energy are separated by the zerocontour (red). In (a), the new phase is stable in the bulk[negative sign in Eq. 5], while in (b) the new phase would beunstable [positive sign in Eq. 5] if the electric field were notpresent (note the difference in scale). The contours show thatwhen the field is strong enough, nucleation pathways withmuch lower barriers become available for elongated embryos,regardless of bulk phase stability without the field.
In the second term of Eq. (5), we allow µ (included in R ) to be negative for a new phase that is energeticallyunfavorable in the bulk. Eq. (5) predicts needle-shapedsecond phase particles to be energetically favorable pro-vided that, HR > s (cid:18) R R (cid:19) E E . (6)Figs. 2 shows, indeed, that in a strong field the systemlowers its energy more easily by forming elongated par-ticles, regardless of the sign of µ . That is the generalmechanism by which the field can drive the transition.While post-nucleation growth is beyond the presentscope, the final state can be readily described in the sameframework. If the final state is a stand-alone metallicbody in the same field E , then the obvious modificationsto Eqs. (5) and (6) will be as follows: µ will have themeaning of the chemical potential of that metal, σ willstand for its surface tension, and ε should be set to unityassuming that the body is in vacuum. Therefore, thetransformation will result in a uniformly metallic needle-shaped body.Eq. (5) suggests that nuclei with R → R must be greater than someminimum value determined by extraneous requirements,such as sufficient conductivity to support a large dipoleenergy or mechanical integrity. Based on relevant data,it was estimated [3] that R min = αR , where α ∼ . R min in therange of molecular size. The free energy in the region R < R min is substantially larger than described by Eq.(5) because the energy reducing effect of the electric fieldcannot be manifested by such thin particles; this can beapproximated by a potential wall. With the latter inmind, the maximum of the free energy in Eq. (5) (with R = αR ) yields the nucleation barrier [3], W cyl = W α / E E . (7)The associated critical aspect ratio is, H c /R min = E / ( Eα / ) ≫ E > E c = α / E . Correspondingly, nucleation of needle-shapedparticles is vastly accelerated by electric fields underwhich spherical particle nucleation would be practicallyunaffected [cf. Eq. (3)], as illustrated in Fig. 3. More-over, for the case of µ <
0, there exists a field range(
E < E ) where spherical nucleation is not possible andnucleation only occurs via needle-shaped embryos (left ofthe vertical line in Fig. 3). E/E c W / W FIG. 3: Normalized nucleation barrier,
W/W , as a functionof the applied field, E , relative to the critical field E c . Thebarrier for nucleation of elongated particles (red, solid line)[Eq. (7)] is compared to that of spheres [Eq. (3)] for thecases of: a stable new phase [ µ > µ < E c < E < Ec/α ), where α = 0 .
1. Nucleation in the region
E > E c /α is uncertaindue to the requirement of ultra-small nuclei. In the region tothe left of the vertical line, the field can drive the transitionto a phase that would be unstable without the field; that isthe region where otherwise unobtainable materials may besynthesized. It should be noted that while significant as the phasechange driver, the electric polarization here remainssmall with respect to the total charge distribution inthe needle-shaped embryo. Indeed, the charge movedto the embryo ends is estimated as q E ∼ EH c , with H c given below Eq. (7). That should be compared tothe total charge of the embryo q ∼ eH c R min /a , where e is the electron charge and a is the characteristic in-teratomic distance: q E /q ∼ E / ( E at √ α ) ≪
1, where E at ∼ e/a ∼ V/cm is the characteristic atomic field.A more accurate analysis in Ref. [9] (p. 17) shows thatthe ratio q E /q is further reduced by a factor of 1 /L , where L ≫ T c ), pressure, or concentration,such as e. g. the bulk phase transition between the insu-lating and conductive phases of VO at T c = 340 K. Us-ing the standard approximation µ = µ (1 − T /T c ), where µ is the chemical potential difference between the twophases at zero temperature, results in the correspondingrenormalization [cf. Eq. (3)], W ∝ (1 − T /T c ) − , R ∝ (1 − T /T c ) − , E ∝ (1 − T /T c ) / . Because R min is determined by the microscopic struc-ture and remains practically independent of T , we ob-serve that α ∝ (1 − T /T c ). As a result, the barrier W cyl is temperature independent. That conclusion isin striking contrast to the prediction of CNT that thenucleation barrier is strongly temperature dependent, W ∝ µ − ∝ (1 − T /T c ) − . Thus, we observe that field-induced nucleation becomes exponentially more effectivethan the classical nucleation of spherical particles in theproximity of bulk phase transition. It can dominate evenunder relatively weak fields E > E α / (1 − T /T c ) ,where E and α are the zero-temperature values of E and α . This effect can be rather substantial. Forexample, 1 − T /T c ≈ . at roomtemperature yields E >
100 V/cm. We note that needle-shaped nuclei in polycrystalline VO have been observed[11], which can be attributed to nucleation in the internalfields induced by the grain boundaries.Field driven phase transitions would also be enhancedin laser or dc fields that are sufficiently strong to ionizethe material. Indeed, that process would generate freecharge carriers, thereby increasing the system polariz-ability and its related trend toward the transformation.As a provocative example, consider next the synthe-sis of metallic hydrogen (MH). Predicted by Wigner andHuntington [12] in 1935, solid MH has not yet been ob-served under static pressures of up to 342 GPa [13, 14].Dynamic compression beyond 200 GPa has also been em-ployed [15, 16]. The only direct evidence thus far was thebrief observation [17] of a highly conductive liquid phaseunder a shockwave pressure of 140 GPa and temperaturearound 3000 K. We will now attempt a rough estimateof the electric field range under which MH could be syn-thesized under standard pressure.We use µ ∼ . σ ∼ R ∼
10 ˚A and W ∼ µR ∼ eV. Assuming α = 0 .
1, and ε ∼ E c = α / E ∼ V/cm, or equivalent laser intensity I c ∼ W/cm . Therefore, the practical window forfield-induced synthesis of MH is 10 < E ≪ V/cm.That field range could be made even lower in the proxim-ity of the bulk phase transition (e.g. close to the criticalpressure). Investigation could also be conducted withhydrogen rich alloys, such as CH (or other paraffins) orSiH (H ) [19]. We note that while MH particles can befield-induced according to our estimates, they will remainmetastable and will exist for only a finite time upon fieldremoval due to the inequality in bulk chemical potentials( µ <
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