Herzfeld versus Mott transition in metal-ammonia solutions. G.N. Chuev(1,2), P. Qu´emerais(3)1 1(1) Institute of Theoretical and Experimental Biophysics, 7 0 Russian Academy of Science, Pushchino, Moscow Region, 0 2 142290, Russia and (2) School of Physics, n a The University of Edimburgh, Mayfield Road, J 7 Edinburgh EH9 3JZ, United Kingdom 1 (3)Institut N´eel, CNRS, BP 166, 38042 Grenoble Cedex 9, France ] l e - Although most metal-insulator transitions in doped insulators are generally vie- r t s wed as Mott transitions, some systems seem to deviate from this scenario. Alkali . t a m metal-ammonia solutions are a brilliant example of that. They reveal a phase sepa- - d rationintherangeofmetalconcentrations whereametal-insulatortransitionoccurs. n o Using a mean spherical approximation for quantum polarizable fluids,we argue that c [ the origin of the metal-insulator transition in such a system is likely similar to that 1 v proposed by Herzfeld a long time ago, namely, due to fluctuations of solvated elec- 2 0 trons. We also show how the phase separation may appear : the Herzfeld instability 4 1 of the insulator occurs at a concentration for which the metallic phase is also uns- 0 7 table. As a consequence, the Mott transition cannot occur at low temperatures. The 0 / t proposed scenario may provide a new insight into the metal-insulator transition in a m condensed-matter physics. - d n keywords : Solvated electrons; Mott transition; Phase separation o c : v i In the first theory of metallization, Herzfeld [1] has considered dipolar fluctuations of X r neutral atoms as the origin of a metal-insulator transition (MIT). He emphasized that the a restoring force ofanelectron boundto anatomcollapses andthesubstance becomes metallic at increased densities due to local field effects. Could the Herzfeld idea beapplied to describe the MIT in real systems? Despite some successful examples of such applications [2, 3, 4, 5], serious doubts arose because most of the experiments gave evidence in favor of the Mott scenario [6]. The latter focused on the screening of the long-range Coulomb potential by a stable electron gas, which prevents the formation of bound electrons. As far as this metal 2 Fig. 1: Phase diagram of Na-NH . The experimental data on the locus of the phase separation 3 are indicated by the square [9] and diamond symbols [10], respectively. The triangles show the change in sign of the derivative of the conductivity coefficient dσ/dT, which is used to estimate the locus of the MIT [11]. The solid curve corresponds to our calculations of the locus of the polarization catastrophe. All dotted curves are guides for the eyes, except the dashed horizontal line at T = −80o C indicating the solidification temperature of ammonia. [1(MPM) ≈ 2 · 1020 cm−3]. hallmark process runs, the system remains metallic, but when it ceases at lower densities, a MIT occurs. It is very important to realize that the two theories are based on different effects, since Mott considered the MIT from the metallic side of the transition, whereas Herzfeld investigated the same phenomenon from the insulating one. No comprehensive connection between them has been provided up to now. Recent studies on a Wigner crystal formed by large polarons have however opened new perspectives for the Herzfeld idea. It was shown [7, 8] that this Wigner crystal of polarons loses its stability owing to the dipolar interactions between polarons, yielding a polarization catastrophe, which provokes the onset of metallization. This phenomenon is a quantum version of the classical Herzfeld scenario. Motivated by these results, we argue that a modified Herzfeld approach may provide a key to understand the MIT in certain real systems. In particular, we focus in this paper on the metal-ammonia solutions (MAS). Although it is a century-old problem [9], the phase diagram of MAS (see Fig. 1) has remained mysterious up to now. Many studies have been performed and a large volume of experimental data have been accumulated about this fascinating system (for review, see [11, 12]). Once an alkali metal is dissolved in liquid ammonia, it immediately dissociates to give two separated entities with unlike charges : the solvated ions and the excess electrons. At low metal concentration, the solution remains non metallic (electrolytic) and has an intense blue colour independently on the type of alkali metal. Jortner [14] argued that due to short-range interactions with ammonia molecules, an excess electron forms a cavity free of solvent in which it localizes with the help of the polarization carried by the surrounding ammonia molecules. This process results in a trap formation similar to that for the polarons in solids. The radius of the cavity has been estimated to be r ≈ 3.2 ˚A [14]. Modern theories c 3 basedonpathintegralsimulations[15],oronthedensity functionalapproach[16,17]provide an evaluation of the microscopic structure around solvated electrons but they all yield the same physical picture as that described above. At large enough metal concentration, the MAS becomes a liquid metal with a typical bronze coloration. However, at concentrations varying from 1 to 10 mole percent of metal (MPM), a separation between the low density blue phase and the higher density bronze one takes place, resulting in a miscibility gap below a critical temperature (Fig. 1). Importantly, the phase separation occurs for Li, Na, or K, but was not observed in the case of Cs. However, for all type of alkali metal, many experimental data reveal the presence of a MIT in the same range of densities [11]. This is reported in Fig. 1 for the case of Na. Earlier models considering the Mott mechanism [13] or involving an association of loca- lized electrons in clusters [18] were not able to explain the whole phase diagram observed in MAS satisfactorily. What is the reason? Let’s give an outlook on the complexity of the problem. From an electrostatic point of view, solvated electrons behave more or less like some solvated anions, the counterpart of the solvated metal cations. The Debye screening length is found to be about ˚A at 4 MPM (taking into account the static dielectric constant ǫ ∼ 20 of ammonia), which makes the MAS a strong electrolyte in this concentration range. s The static Coulomb interactions are thus essentially already screened when the MIT occurs, and therefore cannot be its origin. The short-range interactions between electrons are also unlikely to be responsible for the MIT and the phase separation, because the mean dis- tance between electrons is still about 12 ˚A at the relevant concentration 4 MPM, which is enough to neglect any overlapping between the wave-functions of the electrons localized in their ground state. Finally, the occurrence of a phase separation at low temperatures in the concentration range where the MIT occurs, gives serious doubts about the possibility of a Mott transition. We will come back to this point. The origin of the MIT must be found elsewhere, and a reasonable hypothesis is that it results from quantum momentum fluctuations of the solvated electrons. These ones are self-trapped quantum particles, whose dipolar momentum effectively fluctuates due to their quantum nature, with a characteristic frequency ω (T). This frequency corresponds to elec- 0 tronic transitions of the electrons between two states bound in their own trap potentials. They are experimentally detected by ordinary optical absorption measurements. The lat- ter reveal a broad absorption line peaked at ω (T) ∼ 0.9 eV at low concentration and 0 4 Fig. 2: The density of state (DOS) of the collective polarization modes of interacting dipoles. The DOS is drawn in arbitrary units, for various metal concentrations at T = −35◦C. The lower edge ω−(T,n) drives the stability of the system (see text). Fig. 3: Concentration dependencies of the dielectric constant (a) and the locus of the maximum of optical absorption (b) in MAS. In a) the square symbols correspond to the experimental data [27] on dielectric constant at T = +20oC, the circle symbols to that at T = −35oC [28]. The solid and dashed curves show our results at T = +20oC and −35oC, respectively. In b) the triangle symbols indicate the experimental data on the absorption maximum in Na-NH at T = −65oC 3 obtained from [29] and square symbols from [30], while the solid curve shows our results at the same temperature. T = −70◦C, with a tail extending in the visible region, and providing the blue colour of diluted MAS. Fora given temperature, themaximum of opticalabsorptionshows a pronoun- ced red shift at metal concentration above 0.1 MPM, which cannot be caused by short-range or static Coulomb interactions as it has been discussed above. The frequency ω (T) charac- 0 terizing the solvated electron state, is a significant phenomelogical parameter of our theory. It is associated to the static polarizability α (0) = e2/mω2 ∼ 10−22cm3 of a solvated electron 0 0 (m is the electron mass). That is a huge polarizability with respect to the one of a single ammonia molecule α ∼ 2.8·10−24cm3 [19] or that of a sodium ion α ∼ 2·10−25cm3 NH3 Na+ [20]. Therefore, these quantum fluctuations of isolated solvated electrons and their induced dipole-dipole interactions may have a dominant role in the MIT. Moreover, since ω (T) is 0 much higher than the inverse relaxation time τ−1 of the solvent (typically in the THz range), the dipolar interactions between solvated electrons are only screened by the high frequency dielectric constant ǫ∞ of ammonia. The key idea of Herzfeld was to evaluate the effect of the local field, with the help of the Clausius-Mossoti relation, as a function of the density of a substance and the polarizability of its constituents. Nevertheless, his calculation did not take into account the fact that the particles interact. However, these interactions induce collective modes of polarization, which substantially modify the generalized susceptibility χ(ω) with respect to the non-interacting case. This susceptibility gives the response to an external field E , i.e. P = χ(ω)E , where 0 0 P is the polarization. Taking into account both the local field and the interactions effects, 5 we generalize the Clausius-Mossoti relation as ǫ(T,ω)/ǫ (T,ω)−1 4π NH3 = χ(ω), (1) ǫ(T,ω)/ǫ (T,ω)+2 3 NH3 where ǫ (T,ω) is the temperature- and frequency-dependent dielectric function of pure NH3 ammonia, ǫ(T,ω) is the similar quantity of the solution. The susceptibility may be expressed in terms of an effective dynamical polarizability α(ω) of a single solvated electron, by the relation χ(ω) = nα(ω)/ǫ∞. Hence, the problem focuses on the calculations of α(ω). The spectraldensity-functional theory[21]seems toprovide arigorousbasisforsuchcalculations. However, despite recent progresses in this direction [22], the complexity of the microscopic structure and the presence of disorder in MAS prevent accurate numerical calculations of the electronic properties. To overcome this barrier we choose a simple semi-analytical model suitable to calcu- late α(ω) with a reasonable accuracy. We consider MAS as a fluid of quantum particles localized in cavities with diameter σ = 2r , which interact through induced dipole-dipole c interactions. They can be treated as a set of quantum Drude oscillators with isolated pola- rizability α (ω) = e2/m(ω2 − ω2). In the simplest approximation the interactions between 0 0 particles can be cut off at small distances by the cavity size, whereas short-range details are ignored. Similar models of quantum polarizable fluids have been extensively studied [23, 24, 25, 26], and we use the results which were previously obtained. In particular (see Methods), the problem is reduced to solve a quadratic equation, whose roots are complex, ′ ′′ i.e. α(ω) = α(ω)+iα (ω). The imaginary part is non zero only in a finite range of frequency ω−(T,n) < ω < ω+(T,n) that corresponds to the dispersion of the collective polarization modes, regarded as a distribution of eigenvalues. Their density of state (DOS) is given by D(ω) ∝ ωα′′(ω) [25]. Fig. 2 illustrates our calculation at T = −35◦C. At low concentrations, the DOS is peaked at ω (T), whereas the spectrum broadens progressively as the density n 0 increases, indicating the drastic effect of the interactions. Since the squared eigenfrequencies of the collective modes are to be positive, the low edge ω2(n,T)drives thestability ofthesystem. Generalizing theHerzfeldcriterionofpolarization − catastrophe, we define the critical density n of the MIT as c1 ω−(T,nc1) = 0. (2) We have reported in Fig. 1 the calculated critical densities obtained with the use of Eq.(2). It indicates that the MIT occurs between 2 and 5 MPM depending on the temperature, 6 Fig. 4: Reduced isothermal compressibility of the electron gas as a function of concentration at T=−70o C .The solid and the dashed curves correspond to the case of Na and Cs counterions, respectively. The arrow shows the locus of the dielectric catastrophe for the solvated electrons, indicating a miscibility gap between n and n (Na) for Na, which does not occur in the case of c1 c2 Cs ions (see text). which is quite comparable to the experimental data. The evaluation of the original Herzfeld critical density, i.e. without taking into account the interactions, provides the MIT located at about 14 MPM at T = −70◦C. That shows how important the effect of the interactions is to correctly evaluate the MIT. Another consequence of the polarization catastrophe is that the low-frequency dielectric constant ǫ(T,ω) diverges at n as it is experimentally observed c1 in MAS [27, 28]. Comparing the calculated data with the experimental ones, we find a good agreement between them (Fig. 3a). We also have calculated the real and the imaginary parts of the dielectric constant and evaluate the optical absorption coefficient A(ω). Again the calculated concentration dependence of A(ω) at the locus ω of its maximum, agrees max well with the experimental data [29, 30] at concentrations below n (Fig. 3b). c1 Coming at the issue from a different angle, let’s consider the question : could the dipo- lar fluctuations lead to the phase separation experimentally observed in MAS? Above the critical concentration n , the localized electrons are not stable. Hence, the behaviour of the c1 system above n depends on thermodynamics of the metallic state. But the homogeneous c1 electron gas is known to be unstable at sufficiently low densities due to occurrence of a negative compressibility. Therefore, if the polarization catastrophe occurs at a lower den- sity than this instability, it should provoke a phase separation. The MAS seems to be just this case. To reveal it, we have calculated the electronic part of the compressibility for the metallic state with the use of a modified model of stabilized jellium [31] (see Methods) for several alkali metals. Fig. 4 shows the two curves obtained for Na and Cs respectively. In the case of Na, the compressibility κ diverges at a critical density n ≈ 6 MPM, whereas F c2 the dielectric catastrophe of the solvated electron state occurs at n ≈ 5 MPM. This is the c1 origin of the miscibility gap and the associated phase separation : it exists a range of density n ∈ [n ,n ] for which both states are unstable. Another consequence of this phenomenon c1 c2 is that a Mott mechanism for the MIT, which requires a stable electron gas at the critical density, appears impossible since the experimental MIT occurs at lower concentration than 7 n . The second curve in Fig. 4 is for Cs. It is seen in that n < n , contrary to the case of c2 c2 c1 Na. No miscibility gap is thus expected in the case of Cs. This result is also coherent with the experimental facts. Although our estimations give upper and lower bounds for n and c1 n respectively, underestimating the instability range, they reveal a correct trend in the c2 dependence on the size of ions, namely, a decrease of the instability range for heavier ions, due to scattering of delocalized electrons on ion cores. The later may decrease n enough c2 to destroy the miscibility gap. In conclusion, following the Herzfeld idea and using the hard-sphere models for quantum polarizable fluids, we have evaluated peculiarities of MIT in MAS, namely, the anomalies of dielectric response and concentration changes in the absorption maximum. Our estimations of the behaviour of the insulating and the metallic phases have revealed an instability range at low temperatures. Although the predicted miscibility gap is sufficiently smaller than the experimental one and is sensitive to variations of the model parameters (for example, 10% decrease in ǫ∞ enhances twice this range), our calculations tell us that MAS system may deviate from the usual Mott scenario. The MIT seems dominated by the old Herzfeld mechanism. Our model is simple and does not take into account phenomena which may influence the transition such asdisorder effects onthe cavity formationof localized electrons, clusterization of electrons, influence ofionized states of metalatoms, andso on.These effects may be important in the vicinity of the transition, nevertheless simple estimations yield their energy scales sufficiently lower than that of the dipolar interactions between solvated electrons. Long-range nature of these interactions and low density of excess electrons in the solution support our game with the semi-analytical calculations. We also believe that the proposed scenario may be applied to some other systems. The reason of our optimism is the following. The behaviour of the quantum polarizable particles with respect to the homogeneous electron gas is controlled by two dimensionless parameters α σ−3 and nσ3. The concentrations of electrons are to be low to provide the instability of 0 the electron gas, and, hence, nσ3 is to be small. But the parameter α σ−3 is also small 0 in the case of ordinary polarizable fluids. For such systems, the present scenario remains unlikely. However in our case of self-trapped quantum particles, α σ−3 may be large enough 0 to provoke a phase separation (it is about of 0.5 for MAS). Hence, our scenario is general and may take place in other systems, depending on the origin of the polarizable particles. For instance, in alkali metal-alkali halide solutions, solvated electrons, phase separation and 8 dielectric anomalies were experimentally observed [35].Another example is the case of doped polar solids such as oxides where formation of large polarons occurs. For such materials α σ−3 ∼ ǫ−1 − ǫ−1 and the origin of the unusual behaviour is essentially due to different 0 ∞ s scales of screening at various frequencies ǫs >> ǫ∞. We have previously revealed it in the case of a Wigner crystal of polarons [7] which, from this point of view, behaves similarly like MAS. Finally, we would like to emphasize that our scenario opens a window in the general understanding of the MIT, facing a situation when the metallic and the insulating states are both unstable in a finite range of density. This stimulates the further challenging question. Can the dipolar interactions of localized quantum particles provoke a transition to a superconducting state? Such a possibility was already discussed in [36], and, perhaps, a careful analysis of former experiments on superconductivity in frozen MAS [37, 38] could help to find the answer. Methods Evaluations of the effective polarizability In the case of interacting quantum Drude oscillators the problem is to evaluate the effec- tivepolarizabilityα(ω).Duetodipolarinteractionsbetweentheoscillators,thepolarizability α(ω) is modified with respect to the non-interacting case and is given by the self-consistent equation : ǫ∞/α(ω) = ǫ∞/α0(ω)−2E(α(ω)/ǫ∞), (3) where the last term accounts the correlations between induced dipoles. Eq.(3) has been derived in [23], we only modify it by taking into account the high-frequency screening by the solvent (the use of ǫ∞ in Eq.(3)). Formally the quantity 3αE(α)/β equates with the dipolar part of the internal energy per particle of a classical liquid of nonpolarizable particles with permanent dipole momentum (3α/β)1/2 [23, 26], where β is the inverse temperature. Once the function E(α) is known, all the physical properties of the system can be evaluated by solution of Eq.(3). The simplest method to obtain E(α) is the Pad´e approximation [23, 24], which is an interpolation between the case of low and large polarizability α(ω). Adapting this method to our case, we get I (nσ3)nα(ω) 0 E(α(ω)/ǫ∞) = , (4) ǫ∞σ3 +I1(nσ3)α(ω) 9 where I (x) and I (x) are analytical functions depending on dimensionless density x = 0 1 nσ3 [26]. Replacing Eq.(4) in Eq.(3) leads to a quadratic equation, which allows complete calculations of both the real and the imaginary part of α(ω). Our input phenomenological parameters are 1) σ = 2r = 3.2 ˚A [14] which remains fixed for all our calculations, 2) c ǫ∞=1.76 [33], and 3) ω0(T) extracted from experimental data. This last parameter induces an implicit temperature dependence of α(ω). From [32], we take ω (T = −70oC) = 0.9 0 eV and ∂ω (T)/∂T = −2.2 · 10−3 eV/K for higher temperature (−70oC < T < +70oC). 0 The static dielectric constant in Fig. 3a is calculated by taking ǫ (T = −70oC)=25, and s ∂ǫ /∂T = −0.1 K−1 [33]. s Stability of delocalized states In a metallic state, the solution represents a plasma consisting of a degenerate electron gas strongly coupled with the ions dissolved in ammonia. Although a microscopic study of the system is still beyond possibilities of current methods, low temperatures and low metal concentrations of MAS simplify our analysis. We treat the influence of the solvent simply as a screening effect of the interactions, but we use different dielectric constants for the interacting electron gas and the ionic potential, because the latter are additionally screened by the orientational polarization of solvent molecules. We characterize the plasma by dimensionless parameters rs = (4πn/3)−1/3a−B1, Γe = βe2/ǫ∞aBrs, and Γi = βe2/ǫsaBrs, wherea = ~2/me2.Wecanexpressthechangeofthefreeenergycausedbythedissolutionof B metal atoms in terms of the dimensionless parameters, and write the change ∆f per electron (or per metal atom) as the sum of the electron, the ion, and the electron-ion contributions ∆f(n) = ∆f (Γ )+∆f (Γ )+∆f (Γ ,Γ ,r ,a ), (5) e e i i ie e i s i where a is a parameter related with the short-range electron-ion pseudopotential, which i takesintoaccountdeviationsfromCoulombinteractionsbetweenelectronsandions.Because Γ ≫ Γ , it may be checked that the electron gas gives the main contribution to ∆f(n), e i whereas the ionic contribution is only a correction. We can thus ignore thermal effects for simplified evaluations. We take the expression obtained for the stabilized jellium model as in [31] : 3k2 3k C ∆f(n) = F − F +e (n)+ M +a n, (6) c i 10 4πǫ∞ ǫsrs 10 where k = (3π2n)1/3a is the Fermi wave vector, and C ≈ −0.89774 is the Madelung F B m constant in atomic unit (a.u.). The first, the second, and the third terms in Eq.(6) are the ki- netic,theexchange, andthecorrelationcontributions tothetotalenergy ofthehomogeneous electron gas with a positive jellium background. The last two terms are corrections which take into account the atomic nature of the cations. The only difference with the stabilized jellium model [31], resides in the use of the dielectric constant in the relevant contributions. The local density approximation [34] was applied to calculate e (n). The main difficulty is to c evaluate a . In simple metals, the parameter a may be estimated as a˜ ≈ 2πR2/3, where R i i i is the ion-core radius related with the atomic number of the ion. Advanced models treating smooth continuous pseudopotentials [31] yield numerical corrections to this trend. We apply a˜i derived from [31] and take the screening effect of solvent into account, i.e. ai = a˜i/ǫ∞. Importantly, we here use ǫ∞ because ai represents the short-range part of the interaction between electrons and ions. As a result, we find a =10.8 and 26.1 a.u. for Na and Cs res- i pectively. Numerical deviations of about 10 percent from these values do not change our results significantly. With the use of Eq.(6) we calculated the electronic part of the reduced isothermal compressibility κ ∼ [n∂2(nβ∆f)/∂n2]−1 and found the concentration n below F c2 which the electron gas is unstable (Fig. 4). Acknowledgments G.N.Ch. thanks the Leverhulme Trust for partial support of this work. P.Q. thanks B.K. Chakraverty and D. Mayou for many discussions concerning MAS. E-mail : pas- [email protected] and [email protected] [1] Herzfeld, K. F. On atomic properties which make an element a metal. Phys. Rev. 29, 701-705 (1927). [2] Ross, M. On the Herzfeld theory of metallization. Application to rare gases, alkali halides, and diatomic molecules. J. Chem. Phys. 56, 4651-4653 (1972). [3] Berggren,K.-F.OntheHerzfeldtheoryofmetallization. Applicationtodopedsemiconductors, electron-hole liquids in semiconductors, and expanded states of simple metals. J. Chem. Phys. 60, 3399-3402 (1974).