Thermodynamics of intercalated layer crystals at low temperatures

All the layer crystals (LC) have common peculiarities, namely, anisotropy of chemical binding. It is very closely connected with processes of intercalation-deintercalation [1–3]. Due to these processes LC can be widely used in the systems of hydrogen accumulation, sources of high energy [2], superlattice structures [3]. To know the free energy changes is very important in understanding the thermodynamics of the intercalation processes [4–6]. Thus we study the dependence of free energy of an electron subsystem on the microscopic parameters describing the following processes:


Introduction
All the layer crystals (LC) have common peculiarities, namely, anisotropy of chemical binding.It is very closely connected with processes of intercalation-deintercalation [1][2][3].Due to these processes LC can be widely used in the systems of hydrogen accumulation, sources of high energy [2], superlattice structures [3].To know the free energy changes is very important in understanding the thermodynamics of the intercalation processes [4][5][6].Thus we study the dependence of free energy of an electron subsystem on the microscopic parameters describing the following processes: (i) interlayer mixing in LC, (ii) energy state of intercalant and its mixing with LC, (iii) changes in electron mixing between the atoms of adjacent lattice layers close to the intercalate.
We assume the host material as the recipient of the guest species, the intercalate as the guest species resident in the host material, and the intercalant as the guest species impurities prior to intercalation [6].
The character of free energy changes (its sign) ∆F can be used as the test of thermodynamic advantage (or disadvantage) of the system at certain parameters, and thus, one can confirm the possibility or the impossibility of intercalation processes.Even in the case of thermodynamic disadvantage, the character of ∆F changes depending on i-iii parameters is important.Interlayer mixing stands out especially, because it can be easily changed under hydrostatic pressure or pressure along the anisotropic axis.Another opportunity is the choice of an atom-intercalant (ii).Theoretically, the system transformation from thermodynamic disadvantageous state to the advantageous one is possible in the case when there exists the tendency of ∆F decrease.
The value of ∆F is important, if the system "intercalated LC" is stable, because it determines the degree of intercalant-intercalate binding.The application of intercalation phenomenon is tightly connected with the deintercalation process, and thus ∆F magnitude determines energetic losses during the intercalation-deintercalation process.

Theory
We consider the changes in the density of electron states for the model layer crystal with layers in the XOY plane [7,8].The atoms will be characterized by their radius-vectors with coordinates ( n, α) where n = (n x , n y ); integer α fixes the layer number.We consider one-impurity one intercalate atom case.Let the impurity be located between land l + 1 layers which is characterized by radius-vector n 0 .
In one-electron approximation in site representation, the Hamiltonian is as follows: where the first sum describes the crystal lattice (c n α is Fermi operator of annihilation on n α site; the term in straight brackets is a sum of the energy of the electron on the intercalate and the energy of the electron overlapping of the intercalate with the nearest crystal atoms in the l and l + 1 layers; the term in figure brackets considers the change in overlapping from t(nl/nl + 1) to γ(nl/nl + 1), which is the result of the interlayer distance change at intercalation.As it was mentioned above, t − γ can be both positive (in the case of intercalation dichalcogenide transition metals: organic molecules or Li in TlGaS 2 ) or negative (in the case of intercalation alkali metals).
Hamiltonian in momentum representation is of the following form [7,8] Such a dispersion law reflects the difference between chemical bonds in different crystallographic directions of layer crystals.Normally to layers one can use tightbinding approximation because here the electrons are bound by weak overlapping of their atomic orbitals but in the layer the effective mass approximation is acceptable.Simultaneous choice of an effective mass approximation (for the electron motion in the layer plane) and the tight binding one (along c-axis) is justified by the difference in chemical binding along different crystallographic directions in a layer crystal.We use two-time retarded Green's function a 0 |a + 0 for intercalate.The equation of motion for operator a 0 is of the form Then From the equation of motion for the Green function c χk |a + 0 , we obtain where Expression ( 5) is an integral equation with degenerated kernel.Thus, we have where Using ( 6) and ( 4) we obtain From the equation of motion for c χk |c + s , where s is three-dimensional vector, with Equation of motion for a 0 |c + s gives and thus with ; In the case t = γ

Calculations
Density of electron states of intercalated LC is found from imaginary part of Green's function To avoid bulky expressions, the density of electron states of intercalated LC and intercalate is given here only for the case where and Density of states may be presented by ρ LC = ρ 0 +∆ρ, where ρ 0 = ρ p +ρ (i) (ρ p and ρ (i) are densities of states of pure LC and intercalant, respectively), ∆ρ describes the distortion in density caused by interaction between LC and intercalant.We neglect the contribution of the renormalized intercalate density of states which in the one-impurity case is of O(1/N) order, N is the number of crystal sites.
The change of free energy ∆F , caused by intercalation at T = 0 equals to where ω is counted off from the conduction band bottom.We calculated ∆F numerically depending on: (a) ε F , (b) degree of the electron intercalate-LC mixing, V 0 (which was changed in the range 0.05 − 0.15 eV; the magnitude of energy parameter V 0 was chosen according to the results of the calculations of resonance levels and occupancies of an atom or molecule on the crystal surface carried out in terms of a model Hamiltonian of Anderson's type [9]), (c) changes of the electron hopping constant along c-axis due to intercalation, t − γ.As it was shown in [10], intercalation can be accompanied by either the increase or the decrease of the latter parameter.Therefore t − γ is negative in the first case and positive in the second one (|t − γ| 0.04 eV), while the different impurities were considered as intercalant.Hence, their energetic position was changed in the range −0.2 ε 0 0.2 (eV) (value t = 0.1 eV was chosen), including special points in density of electron states of ideal LC; point A was the fold point and point B was the point of non-analyticity (figure 1).

Discussion
As it follows from figure 2, at fixed V 0 and t − γ, ∆F is a nonmonotonic function and the higher is ε F the higher ∆F will be.At ε 0 = 10 −5 eV, ∆F is negative only at ε F 0.002 eV.Dependence of ∆F = f (ε F ) < 0 for ε 0 > 10 −5 eV has the similar type independently on ε F , the peak is shifted in the direction of higher ε F at t − γ increase.If the intercalate does not change electron mixing between atoms from the nearest layers (t − γ = 0) ∆F is almost monotonous weakly increasing ε F function with a small peak at ε F = 0.03 eV (figure 2).Comparison of ∆F = f (ε F ) dependences for different V 0 and ε F ∈ [0 ÷ 0.23] eV (figures 3a, 3b) shows the existence of two intervals of V 0 , at which instead of the monotonous increase, a peak is observed.In the case of anisotropy increase (t − γ = 0.04 eV) the monotonic dependence takes place when V 0 ∈ [0 ÷ 0.1] eV (figure 3a), whereas for t − γ = −0.04eV such a dependence is observed for V 0 ∈ [0 ÷ 0.03] eV (figure 3b).
At t − γ = −0.04eV and V 0 = 0.05 eV, ∆F = f (ε F ) has two maxima.At t − γ increase, the second maximum vanishes, and the first one increases the shifting to the large values of ε F decrease (figure 4).For positive t − γ, only one peak near ε F = 0.05 eV exists, it vanishes at t − γ = 0.04 eV and ∆F = f (ε F ) becomes a monotonic function.The existence of maxima ∆F at ε F ≈ 0.1 eV, 0.23 eV (figure 4) permits to determine the range of parameter values for which the intercalation-deintercalation process is the most stable.

Conclusions
Calculations of ∆F as a function of ε F at certain V 0 and t − γ show two intervals of V 0 values for which ∆F increases with ε F (a) in the monotonic way; (b) with peaks.In the case of anisotropy increase (t − γ > 0), a monotonic region occurs at V 0 ∈ [0 ÷ 0.1] eV, for the anisotropy decrease -at V 0 ∈ [0 ÷ 0.03] eV correspondingly.
It is shown that thermodynamic stability (that is ∆F < 0) of the intercalated LC occurs in case: • When intercalant electron level is in the conduction band, |∆F | is growing, being however still negative if ε F increases; • The behaviour of ∆F with the change of ε F is different, it depends on anisotropy increase or decrease.

Figure 1 .
Figure 1.Schematic density of states of the intercalant-crystal system: 1 is density of states of ideal LC ρ 0 , 2t is bandwidth along anisotropy axis; 2 is the position of intercalate energy level ε 0 .