On the Importance of Fluctuations in Weakly First-Order Metal–Insulator Transitions

Jonathan Paras Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA (Dated: May 21, 2021) Metal-insulator transitions (MITs) —remain theoretically mysterious. In addition to the lack of an obvious order parameter, these phase transformations demonstrate singular behavior in their response functions in addition to large discontinuities in some extensive variables like the entropy. This has made their classification as either first or second-order transitions difficult. It is hypothe- sized that the emergence of pseudoscaling behavior at an otherwise first order transition may be due to the addition of long-range interactions for systems that are near tricritical points. Therefore, the importance of fluctuations around the saddle point for otherwise first-order phase transitions will be investigated. Renormalization group techniques will be used to examine the effects of the addi- tion of long-range interactions near the phase transition. We will demonstrate that such long-range interactions may alter the order of the phase transition upon renormalization.

I. INTRODUCTION in some high-dimensional Monte Carlo models [9]. Pre- vious experimental work has sought to avoid this trouble Metal-insulator transitions occur when electrons delo- all together by either doping or applying pressure to oth- calize from insulating states into otherwise free charge erwise first-order MITs until they become second order carriers. This transition often manifests as a change in so that they may be evaluated using classical field theory the magnitude of the electrical conductivity from oxide tools[10–12]. (<10-2 Ω-1m-1) to metallic (>102 Ω-1m-1) [1]. Such tran- Because of the inherent difficulties in providing a quan- tum mechanical picture of the transition, a Landau the- sitions notably occur in the insulating oxides VO2 and ory could prove useful in providing a thermodynamic pic- NbO2 solely as as function of temperature and are con- sidered first order because they have large discontinuities ture for the near singular behavior of the free energy of in the entropy at their phase transition[1–3]. It is worth the metal and insulating phases as they approach the first mentioning that other extensive quantities, like the vol- order phase transformation. ume, change as little as 0.2%. To establish the possibility that there may exist a pseu- The theoretical picture has been muddied by signifi- doscaling universality class for such transitions (and in cant ongoing debate over whether MITs are driven by the absence of a clear order parameter) we propose to changes in the electronic degrees of freedom, the phonon examine mean heat capacity and entropy data for two ox- density of states from a concomitant structural transfor- ides insulting oxides that undergo MITs. We will apply mation, or both [1, 4–8]. Recent work suggests there may mean field Landau theory, consider fluctuations about exist a description of such transitions in the previously the saddle point in the limit that we are near a tricriti- unevaluable electronic entropy from electronic transport cal point, and examine the effects of the addition of long property measurements [8]. Our study will thus confine range interactions under renormalization. itself to systems for whom this thermodynamic descrip- tion is valid. II. ELECTRONIC ENTROPY Because the theoretical descriptions of metals and insulators can vary dramatically, an appropriate (and quantitatively useful) Hamiltonian to describe the sys- The electronic entropy can be accessed from measur- tem is lacking. Taken together, this has made the identi- able electronic transport properties. It has been demon- fication of a relevant order parameter for theses systems strated that the partial molar entropy of a conduction difficult. electron can be related to the Seebeck coefficient from There are several other important features of these [13]. transitions that have hindered theoretical work. The range of transition temperatures is significant, magnetite  dS  undergoes the famous Verwey MIT in Fe3O4 at 125 K, = −αF (1) dne T,P,n NbO2 has a Tc of roughly 1100 K. This essentially rules j out electronic structure calculation methods as a viable where α is the Seebeck coefficient and F is Faraday con- way to probe the thermodynamics of these transitions. stant. These materials also tend to exhibit seemingly singular The integral form was implemented for metal-insulator behavior in their response functions at the phase transi- transitions by [8] and resulted in the electronic state en- tion temperature. This type of behavior would indicate tropy: if not a diverging correlation length ,ξ, at-least a large, finite ξ near the phase transition, as has been observed Se = −neeαe (2) 2

TABLE I. Electronic entropies of transition for VO2 and NbO2. T elec Material Tc ∆S ∆S (K) (J/molK) (J/molK) VO2 340 14.7[14] 9.2[8] NbO2 1100 10 [3] 7.9 (this work)

TABLE II. Critical exponents of the heat capacity near the first-order transition temperature from above (+) and below (-). Material α+ α- VO2 -0.32 -2.5 NbO2 -0.22 -0.24 where e is the fundamental charge constant, n is the num- ber of free charge carriers (here electrons). The electonic entropy has been evaluated for VO2 and NbO2. In both cases, the electronic entropy calculated using Eq. (2) ac- counts for 60-70% of the total observed entropy and is listed in Table 1. This suggests that the fundamental nature of these transitions may be similar.

III. CRITICAL BEHAVIOR FIG. 1. Variation of the heat capacity of NbO2 as a function of the temperature relative to the critical point. Reproduced The starting point for an evaluation of pseudoscaling from ref. [3] behavior of the electronic free energy is the critical expo- nent of the heat capacity given in the equation a 2-4-6 polynomial which should satisfy these require- ments as: α Csingular ∝ (T − Tc) (3) Z K t The critical exponent of the heat capacity ,α, is inde- Ψ(m) = ∇2(m) + m2 + um4 + vm6 (4) 2 2 pendent of the chosen order parameter, therefore serving as a useful starting point for examining the universal na- where the coefficients to the order parameter m are ture of the MIT. The critical behavior of the oxides of phenomenological parameters. interest were examined from data on both sides of the A first order phase transition occurs for the phase transition in Table 2, with the data coming from conditions[16]: below the VO2 largely relying on extrapolation [3, 15]. Table 2 otherwise suggests that the listed oxides may ex- u2 hibit some similarities in their pseudoscaling behavior. u < 0, t¯= (5) 2v A plot of the heat capacity as NbO2 approaches the critical point is presented in Fig.1.The shape of the heat However it can be demonstrated that in the absence capacity as a function of temperature is reminiscent of of fluctuations, such a formulation yields a divergence a λ transition, although it does not seem to belong to in the heat capacity only from below with an effective the superfluid He universality class, as the scaling law α = 0.5 and no such divergence coming from above, at coefficient in that case is αsuperfluid ≈ 0.01. odds with experimental observation. Additionally, the case in which u = 0 describes a tricritical point. While it may be simple to say that a weakly first-order transition involves both u < 0 and limu→0, in addition to evaluat- IV. MEAN FIELD THEORY ing the effects of fluctuations, we will examine the pos- sibility that long-range interactions may be driving an Previous work has examined MITs from the perspec- otherwise strongly first order transition towards second tive of second-order phase transitions, and employed 2-4 order. We believe this is relevant considering the mag- polynomials for the Landau free energy[11]. We propose nitude changes in the entropy are quite large for these 3 transitions to be considered ”weak”. We will implement both Gaussian theory and momentum-space renormaliza- tion group methods to examine the relevancy of u with the introduction of such long-range interactions. n X α m(x) = (m ¯ + φl(x))e ˆl + φ(x)t eˆα (6) α=2 V. FLUCTUATIONS

Transverse and longitudinal fluctuations are intro- Keeping fluctuation contributions to second order, we duced as find expressions for the free energy to be:

" n # Z K φ2 X K (φα)2  f(φ , φα) = f(0, 0) + ddx (∇φ )2 + l (t + 12m ¯ 2u + 30m ¯ 4v) + (∇φα)2 + t (t + 6m ¯ 4v) (7) l t 2 l 2 2 t 2 α=2 ( d n R d q ln(kq2 + t) if t > 0 = f(0, 0) + 2 2πd (8) n R ddq 2 2 2πd ln(kq − 4t) if t < 0

Where we have converted to Fourier representation of VI. LONG RANGE INTERACTIONS the free energy in equation (8). The heat capacity is d2f given by Cv = −T dT 2 and so therefore the contributions We postulated earlier that the addition of long range from fluctuations and the saddle point are given by interactions near the phase transition may be modulat- ing the weakly first-order behavior and perhaps the rel- −d/2 d −2 evancy of u. We shall explore this possibility utilizing Cfl. = C − Cs.p. ∝ K |t| 2 (9) momentum-space renormalization. An example of such 1 Cs.p. ∝ (−vt) 2 (10) an interaction is given by:

Where s.p. refers to the saddle point result and fl. to the fluctuations. We have induced critical behavior on Z Z K m(x) · m(y) both sides of the transition and note a reduction in the ddx ddy (11) |x − y|d+σ upper critical dimension, du, from 4 in the case of the second-order transition to 3 in the first order. The value of the fluctuations are still given by α = 0.5 in d = 3. Where K is a constant interaction parameter between While the qualitative behavior has been repaired, this spin states.The K associated with the long-range inter- still does not offer a satisfying explanation for the emer- action will be reffered to as Kσ going forward to avoid gence of singular like behavior near the phase transition confusion with equation (4). This inclusion alters the in otherwise first-order systems. Fourier space representation of the hamiltonian as

Z ddq t + Kq2 + K qσ + ... βH = σ m(q) · m(−q)+ (2π)d 2 Z ddq ddq ddq u 1 2 3 m(q )m(q )m(q )m(−q − q − q )+ (12) (2π)3d 1 2 3 1 2 3 Z ddq ddq ddq ddq ddq v 1 2 3 4 5 m(q )m(q )m(q )m(q )m(q )m(−q − q − q − q − q ) (2π)5d 1 2 3 4 5 1 2 3 4 5

Under renoramlization, we can define the rescaling pa- tions from equation (12) as : 0 0 rameters as q = bq and the order parameter m = m/z. This allows us to read off the first order recursion rela-  0 −d 2 t = b z t  0 −d−2 2 K = b z K  0 −d−σ 2 Kσ = b z Kσ (13)  0 −3d 4 u = b z u  0 v = v−5dz6v 4

The interesting behavior relevant to us occurs when u 6= 0 and in which we are at a fixed point where Kσ 2 d+σ is the scale invariant interaction, leading to z = b t ∝ a0(T − Tc), ao > 0 (15) 0 2σ−d d and u = b u. In d = 3, σ > 2 suggests the Gaus- sian results are invalid and that u remains relevant upon For materials with electron dominated conductivity, renormalization, whereas the converse demonstrates that this results in an imaginary order parameter. However, u is relevant (and necessary) for the transition to remain considering our expansion in even powers of m, this can first-order. This suggests that long-range interactions be overlooked. What is more concerning is that the can push us from a first to second-order transition, and metallic phase can be considered to have the nonzero that it is possible that weakly first-order transitions may value of eq (2)., but that the order-parameter should fall correspond to the crossover case of σ = 2 upon which the to zero as would eq (2) for an insulator. We consider magnitude of such an interaction may become important this to be a sign that order parameters reminiscent of and compete directly with other short range interactions the electrical conductivity (at least in the Drude approx- like K. imation) while seemingly the natural choice for such a problem, do not provide a useful description of the order One can now try to imagine how such interactions may parameter for metal-insulator transitions. emerge during a first order transition. Perhaps these Topological treatments akin to the XY model have also interactions occur at the meso-scale and are associated been used to replicate MITs, but they focus on purely with nanometer sized phases in close contact with each second-order phenomena[16].It may be worth examin- other. We can only conjecture about the mechanism be- ing what kinds of perturbations are necessary to cause hind this interaction, but if we examine the case of σ a weakly first-order transition in such models, in which =2 and in which u is marginal, the real space representa- case the order parameter again may becomes more clear. 1 tion of the potential would be proportional to r5 , slightly longer ranged than a typical Lennard-Jones. This type of interaction has been speculated about but not widely VIII. CONCLUSION treated, perhaps because it only becomes relevant near a first-order transition[17]. We have examined the weakly first-order nature of A more sophisticated treatment should explore the metal-insulator transitions and demonstrated the impor- thermodynamic effects of the finite size of such phases. tance of fluctuations near a tricritical point in recover- This may be possible numerically with q ≥ 3 Potts mod- ing some pseudoscaling behavior of the heat capacity. els, which have demonstrated large (but finite) correla- Through the use of first-order renormalization, we de- tion lengths in otherwise first-order phase transforma- rived recursion relations for the Landau phenomenolog- tions for d = 3 [18]. ical parameters and demonstrate that long-range inter- actions may weaken otherwise strong first-order phase transformations. Such interactions arising near the crit- ical point may account for the nearly singular beheavior in the heat capacity in otherwise first order transitions.

VII. ORDER PARAMETER IX. ACKNOWLEDGEMENTS It was originally hypothesized that eq (2). could prove useful in providing an order paramter because it can be We would like to thank Professor Mehran Kardar and related mathematically as Mr. Alexander Siegenfeld for an intellectually stimulat- ing semester. Any insight generated in this article is the consequence of good teaching. Any mistakes are borne 1 entirely by the author. ∆S = −n eα = − m2a (14) e e 2 o

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