Entropy per particle spikes in the transition metal dichalcogenides
V. O. Shubnyi V. P. Gusynin S. G. Sharapov A. A. Varlamov
cond-mat.mes-hall





























poliabstract[1]
   
   








Department of Physics,
Taras Shevchenko National University of Kiev,
6 Academician Glushkov ave.,
Kiev 03680, Ukraine


Bogolyubov Institute for Theoretical Physics, National Academy of Science of Ukraine, 14-b
        Metrolohichna Street, Kiev 03680, Ukraine


Bogolyubov Institute for Theoretical Physics, National Academy of Science of Ukraine, 14-b
        Metrolohichna Street, Kiev 03680, Ukraine


CNR-SPIN, University “Tor Vergata”, Viale del Politecnico 1, I-00133 Rome, Italy














We derive a general expression for the entropy per particle as a function of chemical potential, temperature and
gap magnitude for the single layer transition metal dichalcogenides. The electronic excitations in these materials can
be approximately regarded as two species of the massive or gapped Dirac fermions.
Inside the smaller gap there is a region with zero density of states where the dependence of the entropy per particle on the
chemical potential exhibits a huge dip-and-peak structure. The edge of the larger gap is accompanied by the discontinuity
of the density of states that results in the peak in the dependence of the entropy per particle on the chemical potential.
The specificity of the transition metal dichalcogenides makes possible the observation of these features at rather
high temperatures order of 100K. The influence of the uniaxial strain on the entropy per particle is discussed.



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Entropy per particle spikes in the transition metal dichalcogenides
    A.A. Varlamov
    December 30, 2023
===================================================================






§ INTRODUCTION
 

We devote our work to the  memory of Alexei Alexeyevich Abrikosov. One of the topic of his research
at the end of the last millennium <cit.> was the unusual magnetoresistance, linear in magnetic field
and positive, observed  in nonstoichiometric silver chalcogenides. His approach was
based on the assumption that these substances are gapless semiconductors with a linear energy spectrum
discovered by himself with coauthors in sixties <cit.>.

This work of Abrikosov had drawn attention of two authors of the present work (VG and SS) to the various
realizations of the Dirac fermions in condensed matter systems. It was impossible to foresee that
the discovery of graphene in 2004 would make the Dirac fermions in condensed matter one of the hottest
topics of research for decades.

Another lesson that one may learn studying the scientific heritage of
Alexei Abrikosov is to focus on the theoretical results that are closely related to experiment.
He always taught that the article must be finished by the formula, which can be checked by experimentalist.



Following his advice here we present a study of the entropy per particle s =
∂ S/ ∂ n (S is the entropy per unit volume and n is the
electron density) for which a witty approach for experimental measurement was
discovered by Kuntsevich et al. <cit.>.
In spite of the fundamental character of entropy  that  characterizes thermodynamics, heat transfer, thermoelectric properties
of  many-body systems, it is always hard to measure it directly.



The recent experiment <cit.> is
not an exception as the quantity measured directly in a 2D electrong gas is the
temperature derivative of the chemical potential, ∂μ / ∂ T.
The key idea of the authors of experiment <cit.>  is that
modulation of the sample temperature changes the chemical
potential and, hence, causes recharging of the gated structure,
where the 2D electrons and the gate act as two plates of a capacitor.
Therefore, ∂μ /∂ T is directly determined in the experiment from
the measured recharging current.
The Maxwell relation is then allows to equate both derivatives

    s = ( ∂ S/∂  n)_T = -( ∂μ/∂ T)_n.


It was theoretically predicted <cit.> that in
a quasi-two-dimensional electron gas (2DEG) with parabolic dispersion, the entropy per electron
exhibits quantized peaks when the chemical potential crosses the size quantized levels.
The amplitude of such peaks in the absence of scattering depends only on the
subband quantization number and is independent of material parameters,
shape of the confining potential, electron effective mass, and temperature.

Very recently we studied <cit.> the behavior of s as a function
of chemical potential, temperature and  gap magnitude for the gapped Dirac materials.
A special attention was paid to low-buckled Dirac materials <cit.>, e.g.
silicene <cit.> and germanene <cit.>.
The dispersion law in these materials writes

    ϵ_η s (k) = ±√(ħ ^2 v_F^2 k^2 + Δ_ησ ^2),

where η = ± 1 and σ = ± 1 are the valley and spin indices, respectively.
Here  v_F is the Fermi velocity,
k is the wavevector, and the valley- and spin-dependent gap,
Δ _ησ=Δ _z-ησΔ _SO, where Δ _SO is the
material dependent spin-orbit gap caused by a strong intrinsic spin-orbit interaction.
It can have a relatively large value, e.g. Δ _SO≈4.2meV in silicene and
Δ _SO≈11.8meV in germanene.
The adjustable part of the gap Δ_z = E_z d, where 2 d is the separation between the two sublattices situated in different  planes,
can be tuned by applying an electric field E_z.
Accordingly, the density of states (DOS) reads

    D( ϵ) =f(ϵ)∑_i=1^N θ(ϵ^2-Δ_i^2),

where the function f(ϵ) is assumed to be a continuous even function of energy ϵ
and
in the case of the discussed materials N=2 and f(ϵ) =  |ϵ|/(πħ ^2 v_F^2).
The DOS (<ref>) has  4 discontinuities at the points ϵ = ±Δ_i, where
i=1 corresponds to η = σ = ± 1 with Δ_1 = | Δ _SO - Δ _z | and
the second one with i=2 corresponds to η = -σ = ± 1 with Δ_2 = |Δ _z+ Δ _SO |.

One of the main results predicted in <cit.> is that for μ = ±Δ_2
(Δ _SO ,Δ _z >0 was assumed) there is a peak of the height s = ± 2 ln 2/3
in entropy per particle when T → 0.
The calculation of <cit.>  shows that a peak at μ = ±Δ_2
can still be seen for the temperature, T ∼ 10^-2Δ_1 for Δ_2 = 2 Δ_1.
Taking Δ_2 ∼Δ _SO, one estimates that the necessary temperature is the order of a few Kelvins.


Layered transition-metal dichalcogenides (TMDCs)  represent another class
of materials that can be shaped into monolayers, where similar effects might be observed.
Single layer TMDCs  with the composition MX_2
(where M = Mo, W is a transition metal, and X = S, Se, Te is a chalcogen atom)
are truly two-dimensional (2D) semiconductors with a large band gap of the order of 12eV (see, e.g. Refs. <cit.>).
Consequently, one may expect that the peaks in entropy per particle can be seen at much higher temperatures.


The paper is organized as follows.  We begin by presenting in
section <ref> the model describing single layer TMDCs.
Since the full description of strained TMDCs is very complicated,
the effect of a uniform uniaxial strain is taken into account only via
scalar potential  spin-independent parts of the Hamiltonian.
In section <ref>  we discuss the DOS
and present an analytical expression for the entropy per particle
in TMDCs.  The results for the obtained behaviour of  the entropy per particle
are discussed in section <ref> and
conclusions are given in section <ref>.




§ MODEL



The low-energy excitations in monolayer TMDCs can be described by the following model Hamiltonian density
<cit.>

    H     = ∑_τ = ± 1 H_τ, 
    
     H_τ    = H_D^τ + H_2

where τ=± 1 is the valley index,
H_D^τ is the linear in momentum in Dirac-like part <cit.>
and H_2 is the quadratic part. The Dirac Hamiltonian contains free massive Dirac fermion, H_0^τ,
and  spin-orbit term H_SO^τ, H_D^τ = H_0^τ + H_SO^τ.
The first term is

    H_0^τ = ħ v_F(τ k_xσ_x+k_yσ_y)+Δ/2σ_z ,

σ are the Pauli matrices acting in the 2 × 2 “band” space,
σ_0 is the unit matrix, the Fermi velocity
v_F = a t/ħ∼0.5d6m/s with t being the effective hopping integral
and a is the lattice constant, the major band gap Δ∼12eV.
The inversion symmetry breaking results in the
the spin-orbit part of the Hamiltonian

    H_SO^τ=λ_vτσ_0-σ_z/2s_z + λ_cτσ_0+σ_z/2s_z,

where s_z is the Pauli matrix for spin,
2λ_v ∼150500meV is the spin splitting at the valence band top
caused by the spin orbit coupling, 2 λ_c is the spin splitting  at
at the conduction band bottom. The DFT calculations <cit.> show that
absolute value 2λ_v ≫  |2λ_c |∼350meV and the sign
of λ_c depends on the compound, λ_c >0  for MoX_2 and λ_c < 0 for WX_2 compounds.

The quadratic part of the Hamiltonian, H_2, contains the following diagonal terms

    H_2 =ħ^2 k^2/4 m_e (ασ_0 + βσ_z),

where m_e is the free electron mass, and α≠β are constants of the order of 1.
Finally, as discussed in <cit.>
more accurate approximations also include the trigonal warping terms.


The spin-up and spin-down components are completely decoupled,
thus the spin index σ=±1 is a good quantum number.
Neglecting the quadratic term (<ref>) we obtain
the dispersion laws for conduction and valence bands

    ϵ_c,v(k)     =  λ_v+λ_c/2τσ
       ±√(ħ^2 v_F^2 k^2+ (Δ-(λ_v-λ_c)τσ)^2/4).

This spectrum closely resembles  that of described by Eq. (<ref>) of massive  fermions
in low-buckled Dirac materials except to the first
valley- and spin-dependent term in Eq. (<ref>).
In the first approximation one can neglect the conduction band splitting and take λ_c =0 to arrive at the
simplest model <cit.>, where  the conduction bands remain spin degenerate at 𝐊 and 𝐊^'
points and have small spin splitting quadratic in 𝐤, whereas the valence bands are completely split,

    ϵ_c,v(k)=λ_v/2τσ±√(ħ^2 v_F^2 k^2+(Δ-λ_vτσ)^2/4),



Single layer TMDCs  can sustain deformations higher than
10%  <cit.>. The experimental possibility
to tune the band gap with strain has been proven for MoS_2 in
<cit.>
and in WS_2 <cit.>.
The full description of strained TMDCs is much more involved than that of graphene
and includes five different fictitious gauge fields as well as scalar potentials
entering spin-independent and spin-dependent parts
 of the Hamiltonian <cit.>. Below we restrict ourselves by a qualitative estimate of the strain
effect on the properties of TMDCs and consider only the scalar potential term in the
spin-independent Hamiltonian (<ref>), viz.

    H_str = D_+(ε) + D_- (ε)/2σ_0 +
    D_+ (ε) - D_- (ε)/2σ_3,

where ε is the strain tensor.
The explicit expressions for the diagonal terms D_± are provided in <cit.> and here
we only keep the linear in strain contributions neglecting the higher order terms

    D_± = α_2^± (ε_xx + ε_yy) ,

with α_2^+ = -3.07eV and  α_2^+ = -1.36eV.
The corresponding parameters 
for the spin-dependent part are smaller by the three orders of magnitude, so that
the corresponding term can be safely neglected.
Assuming that the strain is a uniform uniaxial one, we can express D_± via  ε≡ε_xx
(ε >0 for tensile strain) and the
Poisson's ratio, ν, <cit.>  as follows D_±= α_2^±ε (1- ν).
Thus in the present toy model the effect of strain is reduced to renormalization of
the chemical potential,

    μ→μ - ε (1-ν)(α_2^+ + α_2^-)/2

and the gap

    Δ→Δ + ε (1-ν)(α_2^+ - α_2^-).

Setting ν =0 one may estimate that 1% tensile strain shifts μ by 22meV and
Δ by -17meV, respectively.





§ ENTROPY PER PARTICLE



As it was mentioned above, the entropy per particle is directly related to the
temperature derivative of the chemical potential at the fixed density n
(see Eq. (<ref>)). The latter can be obtained using the thermodynamic
identity

    ( ∂μ/∂ T)_n = - ( ∂ n/∂ T)_μ( ∂ n/∂μ)_T^-1.

At thermal equilibrium, the total density of electrons is

    n_tot (T,μ)  =∫ _-∞^∞d ϵ D (ϵ ) f_FD(ϵ -μ/T),

where f_FD ( x ) = 1/[exp(x)+1 ] is the Fermi-Dirac distribution function and we set k_B=1.
Note that in the presence of the electron-hole symmetry
it is convenient to operate with the difference n between the densities of electrons and holes instead of the total
density of electrons, as usually done for graphene <cit.>.

One can show that in a close analogy with graphene and low-buckled Dirac materials
the DOS for TMDCs described by the approximate spectrum
(<ref>) is

    D(ϵ)=1/π(ħ v_F)^2∑_i = ± 1| ϵ- ϵ_i |
    θ[(ϵ- ϵ_i)^2- Δ_i^2].

Here we denoted
ϵ_i = i(λ_v + λ_c) /2 and
Δ_i = [Δ-i(λ_v - λ_c)]/2 with
i=+1 corresponding to τ = σ = ± 1 and
i=-1 corresponding to τ = -σ = ± 1.

Obviously  for λ_c =0  the resulting DOS corresponds to the spectrum
(<ref>). The DOS (<ref>) differs from the one described by the
equation (<ref>) by the presence of the energy shift, ϵ_i, in the modulus
and in the argument of the θ-function.
As a consequence the quantization of the entropy per particle, s = ± 2 ln2/3,
obtained in <cit.>  for the low-buckled Dirac materials
does not occur in TMDCs.


The behavior of the DOS given by Eq. (<ref>) is illustrated in  Fig. <ref>.
To be specific, we took the values Δ = 1.79eV, 2λ_v= 0.43eV corresponding to
the compound WS_2.
The constant 2λ_c  for WS_2  is -0.03eV <cit.>.
In order to demonstrate the role of this parameter we choose the larger value of λ_c.
Furthermore, we consider three possible cases: λ_c =0
is shown by  the dash-dotted (red) line, long dashed (green) line is for λ_c = 0.05eV,
dotted (blue) line is for  λ_c = -0.05eV.
Note that in general ab initio density functional theory calculations <cit.> predict that
λ_c >0 and  λ_c <0 correspond to MoX_2 and WX_2  compounds.

Going from the negative to positive energies we observe the first discontinuity of the DOS
at ϵ_-1^- = - Δ/2 - λ_v = -1.11eV. It linearly goes down until  the second discontinuity
that occurs at  ϵ_1^- = - Δ/2 + λ_v = -0.68eV. Their positions are independent of the value
of λ_c.
The DOS is zero inside the gap between ϵ_1^- and ϵ_-1^+ =  Δ/2 - λ_c.
Then it increases linearly until the discontinuity at the energy,
ϵ_1^+ =  Δ/2 + λ_c.
Obviously for λ_c =0 the last two discontinuities  become degenerate ϵ_-1^+ = ϵ_1^+ = 0.895eV.
For a finite λ_c  their ordering depends on the sign of λ_c.

The peculiarities of DOS  in  TMDCs beyond the Dirac approximations are discussed
in <cit.>. The quadratic part of the Hamiltonian  (<ref>) results in the
curving of the linear in energy pieces seen in Fig. <ref>.
Such curving is not essential and the  does not change the discontinuous character of the DOS function
that is responsible for the peaks in s(μ).

An advantage of the linearized approximation is that it resembles the case of gapped
graphene and allows to obtain rather simple analytical results. For example, one can derive the analytical expression
for the particle density (carrier imbalance)  <cit.> and find the derivative ∂μ/∂ T
using Eq. (<ref>). Its generalization  for the low-buckled Dirac materials was made in
<cit.> (see also <cit.>). The expression for the particle density in
TMDCs beyond the Dirac approximation is discussed in <cit.>, but it is not very practical for
obtaining the derivative ∂μ/∂ T.

Differentiating Eq. (<ref>) with respect to T and μ and shifting the variable of integration
ϵ→ϵ + ϵ_i for each term in the DOS (<ref>)
 one obtains

    ( ∂ n_tot/∂ T)_μ
    = ∫ _-∞^∞d ϵ  (ϵ - μ)  D (ϵ )  /4 T^2 cosh^2 ϵ - μ/2T
    = ∑_i = ± 1 n_T (μ_i,Δ_i,T)

and

    ( ∂ n_tot/∂μ) _T
    = ∫ _-∞^∞d ϵ D (ϵ )/4 T cosh^2 ϵ - μ/2T
    = ∑_i = ± 1
    n_μ (μ_i,Δ_i,T),

where μ_i=μ-ϵ_i is the shifted chemical potential. Since the corresponding integrands in
Eqs. (<ref>) and (<ref>) become formally the same as in the case of the low-buckled
Dirac materials <cit.> we arrive at the final expressions

    n_T     (μ,Δ,T)  =  1/πħ^2 v_F^2[
    Δ/Tμsinh (Δ/T) + Δsinh (μ/T)/cosh (Δ/T) + cosh (μ/T). 
        + 2 T _2 (-e^- μ+ Δ/T) - 2 T _2 (-e^μ- Δ/T)
    + 2 Δμ/T
       . - (μ - 2 Δ)  ln( 2 coshμ - Δ/2T) . 
       .  -
    (μ + 2 Δ) ln( 2 coshμ + Δ/2T)
    ].


    n_T (μ,Δ,T) =  1/πħ^2 v_F^2[ 2Δln1+exp( μ -Δ/T
    ) /1+exp( -μ +Δ/T) . 
        +2T_2( -e^-μ +Δ/T)
    -2T_2( -e^μ -Δ/T) 
        -μln( 2coshμ -Δ/2T) -μln( 2cosh
    μ +Δ/2T)   
       + . Δ/Tμsinh (Δ /T)+Δsinhμ /T/
    coshΔ /T+coshμ /T]

and

    n_μ (μ,Δ,T) =  1/πħ
    ^2v_F^2[ Δ/2( tanhμ -Δ/2T
    -tanhμ +Δ/2T) .   
       +. T( ln( 2coshμ -Δ/2T) +ln( 2coshμ +Δ/2T) ) ].

_2 (x) in Eq. (<ref>) is the dilogarithm function.
As one can see, Eq. (<ref>) is symmetric with respect to the transformation μ→ -μ
or Δ→-Δ. On the other hand  Eq. (<ref>) is antisymmetric under change μ→ -μ and
symmetric under Δ→-Δ. The last property is checked using the identity for the dilogarithm function

    Li_2(-1/z)=- Li_2(-z)-1/2ln^2(z)-π^2/6.





§ RESULTS



Basing on obtained Eqs. (<ref>), (<ref>) and (<ref>) one can investigate
the dependence s(μ) for the different cases.


Fig. <ref> is computed for the material parameters Δ, λ_v and λ_c
chosen for WS_2 compound. The dependence s(μ) is shown for three values of the temperature:
the solid (red) line is for T = 20K, the dashed (green) line is for T = 40K and
the dotted (blue) line is for T = 80K. The vertical lines are
at the values of chemical potential
μ = ϵ_-1^-, ϵ_1^-, ϵ_1^+, ϵ_-1^+
that correspond to the discontinuities of the DOS.

Comparing Fig. <ref> with the results presented in <cit.>, one can
see that overall shape of s(μ) is similar for TMDCs and low-buckled Dirac materials,
although the details are different. For example, inside the gap for μ∈ [ϵ_1^-, ϵ_1^+]
the dependence of s on the chemical potential exhibits a huge dip-and-peak structure in the
temperature vicinity of the point μ = (λ_v +λ_c)/2. (The value i=1 corresponds to the smaller gap in
Eq. (<ref>)). This feature is
even more pronounced and sharp in TMDCs than in the other materials due to the larger ratio Δ/T.
However in the low-buckled Dirac materials this structure was present in the temperature vicinity of the Dirac point,
μ =0, because the whole dependence s(μ) was an antisymmetric function of μ. This is obviously not the case of  TMDCs.
As discussed in <cit.> the peak inside the gap is mainly due to the specific dependence
of the chemical potential on the electron density.

The presence of the second larger gap, Δ_2 > Δ_1, in silicene and similar materials
results in the emergence of the peak in s(μ) near the points μ = ±Δ_2. Similarly
the discontinuities of the DOS given by Eq. (<ref>) at μ = ϵ_-1^-,  ϵ_-1^+  associated with
a larger gap i=2 also result in the peaks in s(μ). They are shown in the inserts in Fig. <ref>,
because they are much smaller in height. As explained above the value of s at the peaks in the low temperature limit is not
equal to the quantized value ± 2 ln 2/3 expected for the low-buckled Dirac materials <cit.>.
It is essential that both peaks can still be seen at rather high temperatures. The peak on the right starts to smear
at T = 80K, while the peak on the left can still be seen.


It is shown in Fig. <ref> that for λ_c =0 the two discontinuities of the DOS merge at
μ = ϵ_-1^+ = ϵ_1^+. Then the positive peak in s(μ) disappears as can be seen on the
dash-dotted (red) in in Fig. <ref>.
As in Fig. <ref> the vertical lines correspond to the singularities of the DOS.
There is only one singularity for the dash-dotted (red) line at μ = ϵ_-1^+ = ϵ_1^+ = 0.895eV.
For nonzero λ_c there are two singularities shifted from this point to the left and right by |λ_c| = 0.05eV.
In this case the peak at the larger energy μ = Δ/2 + |λ_c| is restored as can be seen on the
dotted (blue) line for λ_c < 0 and long dashed (green) line for  λ_c >0.


Finally we consider how a uniform uniaxial strain would affect the results shown in Fig. <ref>.
We use Eqs. (<ref>) and (<ref>) to model the dependence of chemical potential and gap Δ
on the strain, respectively.

The dependence s(μ) is shown for three values of the strain:
the dotted (green) line is for ε=0,
the dashed (red) line is for ε =2 % and
the solid (blue) line is for ε =4 %. As expected, the presence of strain results
in the movement of the peaks in s(μ).





§ CONCLUSION
 

In the present work we had derived a general expression for the entropy per particle  as a function of the chemical potential,
temperature, and gap magnitude for the single layer transition metal dichalcogenides subjected to the  uniform uniaxial
strain. The spectrum of quasiparticle excitations of these materials is similar to that of the
low-buckled Dirac materials, viz. there is the valley- and spin-dependent gap
Δ_τσ = [Δ- τσ(λ_v - λ_c)]/2 in the spectrum.
The difference from the latter is that the whole spectrum is also shifted by a
valley- and spin- dependent constant ϵ_τσ = τσ (λ_v + λ_c) /2. This
introduces the hole-electron asymmetry in the band structure of TMDCs and makes the  resulting DOS (<ref>)
asymmetric function of the energy.
When a small spin splitting at the conduction band bottom, λ_c, is taken into consideration
the DOS (<ref>) has  4 discontinuities: 2 for the negative  and 2 for the positive
energies. The positions of these discontinuities are not just at the energies ± |Δ_τσ| with
τ = σ = ± 1 and τ = -σ = ± 1 due to the energy shift ϵ_τσ.
It is demonstrated that inside the smaller gap there is a region with zero density of states where the dependence of
the entropy per particle on the chemical potential exhibits a huge dip-and-peak structure. The edge of the larger gap is
accompanied by the discontinuity of the density of states that results in the peak in the dependence
of s on the chemical potential. The specifics of the transition metal dichalcogenides makes the found  features
to be of the  “high temperature” nature, since they can be observed at rather high temperatures up to 100K.

Since the Seebeck coefficient is related to the temperature derivative of the chemical potential, the strong peaks in the
entropy per particle also indicate the same kind of singularities in the Seebeck coefficient in these materials. The latter
can be expected at the edge of the gaps and has the origin similar  to the electronic topological transitions
<cit.>.






We thank A.O. Slobodeniuk for illuminating discussion.
We acknowledge the support of EC for the RISE Project CoExAN GA644076.
V.P.G. and S.G.Sh. acknowledge a partial support from the Program of Fundamental Research of the Physics
and Astronomy Division of the NAS of Ukraine No. 0117U00240.






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