Thermal and magnetic properties of landau quantized group VI dichalcogenide carriers in the approach to the degenerate limit

This work is concerned with determination of the thermal and magnetic behavior of the Group VI Dichalcogenides in a quantizing magnetic field. Our discussion includes analysis of the Grand Potential, Helmholtz Free Energy, partition functions (grand and ordinary) and the entropy and specific heat: their dependencies on temperature and magnetic field strength are carefully examined, particularly in the degenerate regime and the approach to zero temperature. The joint dependence on magnetic field and temperature is also determined for the Dichalcogenide magnetic moment in the degenerate statistical regime, replete with de Haas–van Alphen (dHvA) oscillatory phenomenology and also above the zero–temperature limit.

1. In the list of key words, 'magnetic field' should be replaced by 'de Haas-van Alphen oscillations'.
The publisher sincerely apologises for any inconvenience caused by this error and can confirm that the final results of the paper remain unaffected.

Introduction
This paper addresses some fundamental physical properties of 'Dirac-like' materials, in particular the Group VI Dichalcogenides. Starting with Graphene [1][2][3][4][5][6], such materials have been at the focus of research attention since the discovery of the extraordinary electrical conduction and detection properties of Graphene about fifteen years ago. Additional materials of this type include Silicene [7], Topological Insulators [8], as well as the Dichalcogenides [9] (and some others). All are under intense investigation worldwide in all science and engineering disciplines for their potential to succeed Silicon as the material of choice for the next generation of electronic devices and computers. Recognition of the importance of these materials has been underscored by the award of the 2010 Nobel Prize in Physics to Geim and Novoselov for their pioneering work on Graphene. The fact that the low energy carrier spectrum of 'Dirac-like' materials mimics that of relativistic electrons/positrons (with energy linearly proportional to momentum) also heightens intellectual interest in them as accessible solid state laboratories of relativistic physics, albeit with different parameters. Much has already been done in regard to experimental and theoretical studies on Graphene, and considerable work on the other 'Dirac-like' materials is now filling the scientific and engineering literature. Here, we address the magnetic response and the thermodynamic properties of Group VI Dichalcogenides [9] to study their behavior subject to Landau quantization in a high magnetic field, particularly in the degenerate regime and also above the zero-temperature limit. It should be remarked that all the results obtained here are directly applicable to other pseudospin-1/2 Dirac materials, Graphene in particular, by setting the energy shift  E 0 s z and  g 0 (section 2), and appropriately adjusting the numerical value of the characteristic speed γ of the underlying linear low energy band structure approximation.
In regard to the thermal properties of matter and the calculation of their temperature dependence, the entropy is of central importance: determination of the entropy is essential for evaluation of the specific heat, which we present as an example for the Landau-quantized Dichalcogenides. As the associated Dichalcogenide spectrum has an unusual unbounded negative energy component, we verify that the usual positivity of entropy and its vanishing at zero temperature still apply, and stand as guiding requirements on the validity of approximate procedures for calculations of temperature dependence of the statistical thermodynamic Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
functions. Here, we examine these functions in the degenerate statistical regime, including determination of their temperature dependence in the approach to zero temperature. In view of the great importance of the magnetic field as an agent for probing the properties of matter and also modifying them [10,11], particularly with Landau quantization of orbits, we address its role in the statistical thermodynamics of the Group VI Dichalcogenides jointly with that of temperature, as reflected in the entropy and specific heat of these systems. Furthermore, the results are applied to the analysis of the magnetic moments of the Group VI Dichalcogenides and their de Haas-van Alphen (dHvA) oscillatory phenomenology due to Landau quantization.
The basic formulation of our study of statistical thermodynamics employs Green's functions, as set forth below in this section. The pertinent Dichalcogenide Green's functions are reviewed in detail in section 2. Section 3 presents our calculations of the temperature and magnetic field dependencies of the Grand Potential, Helmholtz Free Energy and magnetic moment in the degenerate regime, including the approach to the zero temperature limit. Entropy and the specific heat of the Landau quantized Dichalcogenides are analyzed in the degenerate regime in section 4, again including temperature dependence in the approach to zero temperature. Some qualitative features of our results are discussed in section 5, including de Haas-van Alphen oscillatory phenomenology of the Group VI Dichalcogenides.
Our use of a retarded Green's function G ret in the determination of statistical thermodynamic functions is based on the fact that its trace in position-time representation produces the ordinary ('classical') partition function ( ) T is Kelvin temperature, and Tr denotes the trace. This is readily verified using the definition of the Green's function in terms of a time translation operator [12]. Actually, it is the logarithm of the grand partition function Z(β) for Fermions that is required to determine the Grand Potential, Ω, and the Helmholtz Free Energy, F, and Wilson's book [13] reports a clever way to obtain it from the ordinary partition function ( )  b Z , as follows (μ is chemical potential, N is particle number and E γ represents the single-fermion energy spectrum): and writing the E γ -summand (B(E)) as an inverse Laplace transform ( ò ds c represents integration over the inverse Laplace transform contour in the complex s-plane) This expresses Ω and F in terms of the ordinary partition function (  Z ), or alternatively, the Green's function. Noting that rewriting equation one may employ a useful special case of the convolution theorem for Laplace transforms [14] to obtain we have the convenience of dealing directly with the temperature dependent Fermi distribution f 0 (E) (rather than B(E)):

Retarded Green's function of group VI dichalcogenides in a magnetic field
As indicated above, our approach to the determination of the role of a quantizing magnetic field in the magnetic moment and statistical thermodynamic functions of charge carriers of the Group VI Dichalcogenides in the low energy regime (in which their Hamiltonian is 'Dirac-like' with energy proportional to momentum) is undertaken using the retarded Green's function; In earlier work [15], the associated Landau-quantized Green's function matrix was derived with full account of its pseudospin-1/2 and spin-1/2 features in the presence of a high magnetic field; and the pertinent diagonal elements of its retarded Green's function pseudospin matrix G ret 11 22 are given in 2D-position ( z with Δ as the energy gap without spin splitting. Also, L n represents the Laguerre polynomials and   n 2 is given by ( and γ is an effective speed determined by the tight binding hopping parameter and lattice spacing. It is useful to rewrite the trace of equation (2.1) in the following form (for use below): This trace encompasses sums over the spin index =  s 1 z , valley index n = 1, pseudospin index±, the signature ¢ of exponentials constituting sine and cosine functions and the Laguerre sum index =  ¥ n 0 . For the problem at hand, where the (area) factor arises from the 2D  dx -integration; this area factor will henceforth be taken as unity, and Ω, F,  Z , and entropy S, magnetic susceptibility M and specific heat C V are to be understood on a per-unit-area basis, with  = N n density. It is also of interest to describe the thermodynamic Green's function matrix, ) instead of retardation. It may be written in terms of its 'greater' ( > G ) and 'lesser' ( < G ) constituents as [12,15] , ; , ; .

2.5
The matrix Green's function constituents > G and < G define a corresponding spectral weight matrix A as all satisfy the homogeneous counterpart of the Green's function equation; and all involve the same Peierls phase factor The constituent parts of the thermodynamic Green's function matrix may be determined from the spectral weight function matrix in frequency representation as , ; may be determined from the structure of the retarded Green's function G ret in frequency representation using the relation [12] ( leading to the result for its diagonal elements , ; 11 22 as From this, the diagonal elements of the thermodynamic Green's function may be determined for the Dichalcogenides in a magnetic field using equation (2.8). In particular, we obtain 3. Temperature dependence of the grand potential and magnetic moment: degenerate regime and also above the zero-temperature limit The analysis of temperature dependence devolves upon a careful evaluation of the Grand Potential Ω and Helmholtz Free Energy F, and we examine this in the approach to the degenerate regime, mb  ¥, by rewriting equation (1.9) with a change of variable,   Noting that the contour of z-integration along c is a straight line from = -¥ + + z i 0 to + ¥ + + i 0 , we consider closing the contour [17]  This indicates the behavior at very low temperature to be given approximately by The results exhibited above in equations (3.8, 9) for the Grand Potential and Helmholtz Free Energy in the degenerate regime clearly exhibit the effects of the quantizing magnetic field jointly with those of finite temperature. To elaborate further on the role of the magnetic field we evaluate the magnetic moment, M, which may be obtained from the free energy as (per unit area) where The last term of ( ) m - ¶ ¶ M n B may be evaluated by differentiating equation (2.12), with the result In regard to the final magnetic moment contribution from m - ¶ ¶ n Bin the degenerate limit of zero temperature, we have 4. Entropy and specific heat of the group VI dichalcogenides: temperature dependence in the degenerate regime and also above the zero-temperature limit The entropy, S, is determined by a variation of the Helmholtz Free Energy, F, in the thermodynamic relation 2D), μ is chemical potential, ¢ T is Kelvin temperature, N is number and V is volume (area in 2D)). Holding N and V constant, the entropy (per unit area) may be identified as , , , it should be noted that only the explicit dependence of F(b m , ) on β contributes, to the exclusion of implicit dependence on β through μ (as determined by the expression for density n(β, μ)), since such a contribution would be of the form by definition of the density n. In this context the μ-dependence of F cannot contribute to ¶ ¶ ¢ F T , so the term m - ¶ ¶ ¢ n T in equation (4.1) must be understood to vanish. As we are dealing with the statistical thermodynamics of an unusual system, which has an unbounded negative component of its energy spectrum (in addition to the more usual unbounded positive component), it is of interest to verify that the usual basic features of the entropy still apply to the system under consideration. The first feature to verify is that the entropy vanishes in the zero temperature limit, b  ¥: this is readily verified from equation (3.3), whose low temperature limit is given by An alternative, fully general, proof based on equation (1.2) follows: for either sign of ( ) m g E . Moreover, at finite temperatures, the summand of equation (4.3) given by is positive for all x in the range [ ] -¥  ¥ , assuring the positivity of the entropy even with the negative component of the energy spectrum. (A plot of the function readily verifies this.) [17] Furthermore, we evaluate the entropy S Deg for the Dichalcogenides in the approach to zero temperature in the degenerate regime: employing equations It is readily verified (again) that the summand of equation (4.6) vanishes in the limit of zero temperature for energies above and below the Fermi level, including negative energies. It should be noted that the entropy is important in the calculation of thermal properties. In particular, the specific heat at constant volume is given by [19]  The specific heat is of particular interest as a measure of the ability of the material to assist in the management of dissipated heat, an issue of importance in electronic device operation and transport. Specific heat also plays an important role in a standard characterization technique employed to understand the underlying physics of the Dichalcogenides, as has been emphasized by Stewart [20] and Geballe's group [21][22][23][24][25].

Discussion
The key thermodynamic function considered here in the presence of a quantizing magnetic field, the Grand Potential, , has been examined in the degenerate statistical regime for the Group VI Dichalcogenides (which have both positive and negative unbounded energy branches) with an evaluation of its temperature dependence in the approach to the zero temperature limit. The results for the degenerate regime in equations (3.8, 9) include its zero-temperature contributions from energy levels E γ below the Fermi energy μ>E γ (summand terms given by ( E ) and finite temperature corrections from energy levels both above and below the Fermi energy ( , its contribution can be substantially larger than the zero-temperature counterpart in the ratio The zero-temperature terms describe de Haas-van Alphen oscillatory phenomenology, in which abrupt changes in statistical thermodynamic (and other) functions are introduced when a varying magnetic field forces successive displaced/split Landau levels across the Fermi energy, inducing abrupt population/depopulation of states: This is represented mathematically by the activation/deactivation of a succession of associated Heaviside unit step functions, ( ) h mg + E , as the magnetic field changes. As indicated above, the ratio [ ] m b -E 1 j provides a relative measure of the role of finite temperature in this process. Of course, the introduction of scattering/ disorder will moderate this. In regard to the dHvA oscillatory phenomenology, it should be noted that the Dirac-like Landau energy levels are proportional to the square-roots of integers multiplying the magnetic field ( ( ) )   g = + = ¢ + , so the simple periodicity of the dHvA oscillations of the nonrelatavistic case no longer applies to the present 'relativistic' case involving more complex oscillatory behavior. Furthermore, it is important to bear in mind that the contributions of higher Landau-level-index-n terms in equations such as equations (3.8, 9) correspond to Landau eigenstates of correspondingly higher energies, and when those energies approach and exceed the limits of validity of the approximate low-energy 'Dirac-like' Hamiltonian under consideration (due to the curvature of the underlying band structure), such contributions must be discarded, constituting an effective 'cut off' terminating the 'n'-series summation.
The remarks above also generally apply to Again, de Haas-van Alphen oscillatory structure is induced by variation of the magnetic field in E γ causing abrupt activation/deactivation of the Heaviside step functions ( ) h mg + E as successive energy levels E γ cross the Fermi energy μ; and these oscillations are not simply periodic because the Dirac Landau levels are proportional to the square-roots of integer multiples of the magnetic field B. Our dHvA results may be useful in the interpretation of magnetic field experiments on the Dichalcogenides to help identify physical parameters such as E s z , g, γ, etc.
On the other hand, finite temperature contributions involve all Landau levels, not just those crossing the Fermi energy (equation (3.12)), where the summand has the form