Anomalous Hall effect in the spin-valley-polarized phase

Figure 5.1: Magnetic bands, according to Eq. 5.25, for (a) the normal phase,D^>U <1, and (b) the magnetic phase,D^>U >1. In (b) we have assumedξ >0.

we can determine the solutionsη^A_↑,Uandη^B_↓,U in the broken symmetry phase, yielding

η_↑,U^A =−η^B_↓,U =|µ0| −∆^>

U . (5.35)

These solutions confirm our ansatz: they are proportional (by a factor ofD^>U) to the variation of charge density in each cone and, indeed, such a variation is smaller in magnitude that the total charge density of a cone.

Finally, note that the same results obtained with the formalism used can be obtained us- ing a variational mean-field approach based on the Gibbs-Bogoliubov-Feynman inequality, as discussed in Sec. D.1 of Appendix D.

needed. This should be clear from Fig. 5.1: due to depletion of charge carriers in a given valley, the other valley will have its charge carriers energetically closer to the conduction band, at the distance of the band gap. Thus, shining properly tuned light is enough to excite valley-polarized charge carriers.

Moreover, the topological character of the Dirac cones has direct and observable conse- quence. As discussed in Sec. 3.2.1, the expression for the valley Hall conductivity was derived and revealed that this class of materials is topologically trivial, manifested in the fact that contri- butions of both valleys cancel out identically. However, in a valley-polarized phase, that will no longer be the case, and an anomalous Hall effect can be expected. To see this, we retrieve the expression of the Berry curvature, Eq. (3.31), written for the spin-split valence band,

Ω^στ_µν(k) =εµν

τ 2

a²t² (∆^στ)²

"

1 + a²t²k² (∆^στ)²

#−3/2

, (5.36)

and the general expression for the anomalous Hall conductivity, Eq. (B.37), written for the mag- netic bands,

σ_µν^v = e²

~ X

σ,τ

ˆ d²k (2π)²n_F

ε^τ_k,σ−µ₀−σU ξU

2

Ω^σ,τ_µν .

The contributions from the low-lying cones cancel identically, keeping only contributions from the imbalanced high-lying cones. Taking theT →0limit for the Fermi distribution,

σ_µν^v =e² h

m^>

~²

ˆ −∆^>

µ0−U ξU/2

dEεµν

2 a²t² (∆^>)²

"

1−2m^>a²

~² t

∆^>

2

E+ ∆^>

#−3/2

.

The integral can be solved directly, yielding

σ^v_µν =εµν

e² 2h

1 q

1−^2m>a2

~² t

∆^>

²

E+ ∆^>

E=−∆^>

E=∆^>−2|µ0|

=εµν

e² 2h



1− 1 q

1 +^{8πD√ >}

3 t

∆^>

2

(|µ0| −∆^>)



, (5.37)

where Eqs. (5.30) and (5.34) were used to substitute for the band mass in terms of the DOS.

Furthermore, noting that the second term in the square root is essentially the magnetization density, Eq. 5.34, and assuming that the system is doped only slightly below the maximum of the high-lying cones, we can expand the square root to 1st order, yielding

2h σ^v_AH e² = 1−

"

1 + 4π

√3 t

∆^>

² m

#−¹₂

= 2π

√3 t

∆^>

²

m+O(m²) , (5.38)

withσ_µν^v =εµνσ_AH^v . This phase displays a valence band anomalous Hall response linear in the magnetization density, for small doping. Not only is this conductivity transversal, it implies also a spin-polarized response, which can be inferred from Figure 5.1: while the contributions from the spin-up cones cancel, the contributions from the spin-down cones do not, and ultimately the re- sulting conductivity stems from this imbalance. Note also that there is an additional longitudinal conductivity, due to the charge pocket in the high-lying spin-down cone. As such, this response will be spin polarized, as well.

Experimental realization

The existence of an anomalous Hall conductivity in the broken symmetry phase provides an experimentally accessible test to its realization. There is, however, a limitation regarding the doping capacity, in the sense that not all doping regimes may be experimentally accessible due to fundamentally physical limitations.

Due to the geometry of 2D materials, such as single-layered TMDCs, it is possible to induce charge carriers by electric field effect. That is, setting up the single-layer to be one of the plates plate of a parallel plane capacitor, thus applying an external electric fieldE. This is the electric field inside the parallel plane capacitor, which is known to be

E=eρ ε ,

wheree >0is the absolute electron charge,ρis the hole surface density andεthe permittivity of the medium. This implies a gate voltage such thatV_g=E l, wherelis the inter-plane distance.

The set-up consists, essentially, of the capacitor composed of the single-layered TMDC and of a wafer ofSi⁺. The inter-planar medium is aSiO2wafer of dielectric constantεr= 3.9and with a typical thicknessl∼300 nm[120]. A schematic is shown in Figure 5.2.

Figure 5.2: Schematic of the electric-field effect apparatus.

Although this method has the advantage of inducing charge carriers without introducing disorder in the system, it has a limitation regarding the scale of the gate voltage and, thus, of the in-plane electric field, since beyond a certain threshold the material sample, as well as the medium, will be ionized and, ultimately, annihilated. The upper bound for this threshold is set in terms of the ionizing electric field magnitude for hydrogen. The magnitude of this electric field

can be estimated by

EH = e 4πε0a²₀,

where a₀ is the Bohr radius. In turn, this implies a hole surface density and a gate voltage respectively given by

ρH = εr

4πa²₀ = 1.11×10¹⁶cm⁻², Vg,H= el

4πε0a²₀ = 1.54×10⁵V,

whereεr is the relative permittivity of the medium. We have previously determined the thresh- oldsµAandµBfor the onset of the intermediate and high doping regimes, respectively,

|µA|= ∆, |µB|= ∆^< = ∆ +λ .

Within the validity of the parabolic approximation, the relation between hole surface densityρ and the Fermi levelµ0is given by

ρ= 2D^>(|µ₀| −∆^>) Auc

, where Auc = a²√

3/2 is the area of the unit cell. Note that the terms in numerator are the number of holes per unit cell (which have, so far, been referred to simply as densities). We can thus estimate the thresholdµ_Ain terms of hole surface density by

ρ_A= 2D^>λ Auc

. Numerical values for this threshold are shown in Table 5.2.

Table 5.2: Hole surface densityρAfor the intermediate threshold for the different TMDCs. (Units:

cm⁻²)

M X₂ MoS2 WS2 MoSe2 WSe2

ρA(×10¹⁵) 1.23 2.41 1.66 2.69

Clearly, these values are only one order of magnitude below the order ofρH, so that we can expect the intermediate and high doping regimes to be experimentally untenable. However, it is interesting, if only academically, to get a sense of the physics of the two further doping regimes, to which we now proceed.