The density functional theory (DFT) was used for the interface electronic struc- ture calculations of the heterostructure in figure 1b. Of particular interest are interfaces between the magnetic half-metallic layer (CrI3) and the semiconduc- tor layer (CrGeTe3). The CrI3-CrGeTe3 junction highly important in spintronic applications. Studying the surfaces contaction manner of two active layer with thickness of dielectric inserted between these layers is likely similar to the growth in the z-axis. The tunneling probability monotonically increases as the barrier thickness shrinks. The atoms Te, Ge, and I with s and p valence electrons were used for slab formation, in addition to the highly spin-polarized magnetic atom Cr of 6 valence electrons in s and d. The two ideal slabs in figure 1b were grad- ually brought into close contact until the inter-atomic range (z= 27.15, 22.7167,18.2834, 13.8501, 9.4168, 4.9835 and 2.765 ˚A).
Figure 2a indicates the relation between the slab separation (z) and the spin magnetic moment. The moment gradually decreases as the two slabs CrI3 and CrGeTe3 approach each other, attaining the minimum value for the moment at distance z=18.3 ˚A; the moment then increases as z decrease further. Hence, the magnetic properties of the system drastically affected by the layer’s separations, with magnetic inversion at critical slab separation z=18.3 ˚A, where the system attained the ability to acquire and maintain an internal dipole moment before and after that critical separation. These is nearly the same as the dipole moment associated with the displacement of atoms away from the center of the point group in the crystal structure.
Figure 2b displays the system energy as a function of z. The energy of the system is nearly constant in the studied separation range. However, the inter- action energy between the slabs begins to decrease at approximately 23 ˚A.
3.1 Electronic Structure
Electronic structure calculations were conducted for each slab, CrI3 and CrGeTe3, separately under the same conditions considered for the CrI3/CrGeTe3 junction. A semiconductor gap was observed for both channel directions of the CrGeTe3 layer, as shown in figure 6(c, d). Additionally, CrI3 indicates a half metallicity with semiconductor behavior for the majority of the channel, as shown in figure 6(a, b). Cr-3d bands indicated by the symbol d in figure 6a, which extended from -0.3 to -0.6 eV down the Fermi surface at the Γ point and nearly at the surface of Brillouin zone, are indicated by the high symmetry points M, K and A.
The electronic structure calculations of the CrI3/CrGeTe3 system show a
conductive behavior for different separation values between the slabs. The major shortcomings for the conductive behavior of the junction Cr3/CrGeTe3 are as follows:
- The heterostructure led to the creation of a conductive Kohn-Sham state with energies within the insulator and semiconductor gap for Cr3 and CrGeTe3, respectively. Where the surfaces charges interaction rearrangement the charges in the semiconductor layer (CrGeTe3) producing new energy states in the band structure, as shown in figure 3. This corresponds with the theory of quantum tunneling, which is a partial propagation of the particle wave function into a forbidden region (energy gap), i.e., the wave function in the barrier region dose not equal zero.
- The heterostructure layers yield bonds inside the slab that partially sat- isfy the bonding requirements described for the solids with s and p valence electrons. The elements Ge, Te and I with a 4, 6 and 7 valence electrons, respectively, may form a hybridization or covalent-like quasi bonding between the s and p orbitals or pure directed p bonding that participates in the band structure.
- The difference in Fermi energy between the CrI3 layer and the CrGeTe3 layer generates overlapping between the slab states such that the band structure for the successive z values may appear as conductive.
The challenge of controlling conductivity by electric fields was achieved by the band topology changes as a function of z in figure 3. These z-changes strongly modify the electronic structure near EF as follows: The three dashed states red, black and orange mostly derived from Cr-3d, as z=0.16238 to z=0.36238 have no crossing to the Fermi level at the Γ-point. The location at which the d-band crosses the Fermi level indicates that the bands constitute a low-energy dispersion relation of fermions with d-components. In addition, a free electron
state appeared for z=0.36238, 0.46238 and 0.56238, as indicated by the dashed orange d-state in the valance band near EF . This means that z=0.36238, which
equivalent slabs separation by 13.85˚A, is preferable for free to move carriers.
In the upward motion of z values until z=0.21238, a group of Cr-3d bands are separated by a clean slab field and extend to -0.8 eV at the Γ point. This group shows a large dispersion for low and high z values.
The significant role of Cr-3d in the magnetic properties and performance of the studied junction led to the analysis of the Cr-3d changes with z inside the junction. Figure 4 indicates the projected DOS changes for Cr-3d with different values of z. The Cr-3d states have a sharp peak at approximately 0.5 eV below the Fermi level. The figure indicates a ferrimagnetical spin arrangement between the valence states and conduction states. This arrangement decreases the bonding energy because of the difference in the strength of the hybridization between the majority and minority channels. As a result, the Cr-3d states are polarized like f -states. These totally spin polarized valence and conduction bands led Cr-3d to produce an allowable spin current, which coincides with the
experimental observations . This spin wave current is maximum at z=18 ˚A, as shown in figure 4. Additionally, figure 4 indicates a DOS spike of fixed potential for Cr-3d under the Fermi level; its width is about 0.3 eV. The majority narrow d-bands of the spike are indicated by the symbol d on the energy axis of figure 6a at the Γ point.
The remaining set of states split from the d-band around point M in valance band, is for s-band which is indicated by the ellipses in figure 6a. These ellipses are closer to the Fermi level for CrGeTe3 in figure 6c than CrI3 in figure 6a. When the junction is formed, the surface field acts like a breakdown voltage, destroying these electronic pockets and producing new electronic states. These states have energy in the semiconductor gap energy range. Where, all the s- and
d-levels that have the same K values are close together, and that would explain the pesky ability of CrI3 in tunneling applications.
3.2 DOS of CoMnI3, CrI3, MoI3 and WI3
The distinct magnetic properties of the CrI3 layer led to an examination of other similar layers to explain the magnetic superiority of CrI3.
The following results are presented in figure 5, which shows the main DOS differences between the CoMnI3, CrI3, MoI3 and WI3. When the orbital d, which is responsible for the magnetic properties, changes from Cr to W, the sharp peak density decreases, and its energy changes. This trend is due to the band filling of d-electron sub-bands, which increase the d-states, splitting with respect to the Fermi level. The figure also presents the increase in DOS near the
Fermi level, which is the reason for the fascinating magnetic properties of CrI3 in contrast to the other layers. That remarkable moment in DOS of CrI3 layer support an increase in the surface field, which in turn explains the effectiveness of the CrI3 layer in tunneling applications. Additionally, the increase in DOS is the result of the bonding and antibonding character of d-electron pairs around the Cr-atom. The DOS shape of the MoI3 layer demonstrates fairly interesting magnetic properties that are nearly similar to those of the CrI3 layer. The peak intensity and the interval between the valance and conduction peaks of the MoI3 layer are close to those of the CrI3-layer, as shown in figure 5. The highest peak of Mo shifted somewhat towered the low energy in the conduction and valance bands.
3.2.1 Surface Moment and Fermi Level
When the junction was formed, the partial DOS (not shown here) explained the imbalance of the directed p-orbital electronic charge of Ge, Te and I in the immediate neighborhood of the interface, creating a dipole moment associated with the presence of the CrI3/CrGeTe3 junction. That surface p-dipole moment becomes stronger with the formation of the CrI3/CrGeTe3 junction due to the addition of the moment produced by the Cr-3d, as shown earlier in figure 4. These dipole moments equate to a Fermi level on either side of the junction
because the Fermi level is adjusted by the electrical neutrality of the two layers.
In the case of a large gap, the dipole moment is not sufficient to produce elec- tronic equilibrium. Figure 6 shows a small gap for CrGeTe3 and half metallicity for CrI3, which could be suitable for the Fermi surface to be balanced on both sides of the studied junction. That would neatly point toward, the properties of the studied heterostructure depend on the changes in energy gap along the reciprocal directions, as indicated in figure 7b. Thus, considering figure 6 for the semiconductor layer CrGeTe3, the equilibrium condition may occur at high symmetry points with a small gap, such as at M or K, but not at Γ or A. This in turn explains the tunnelling behavior of the studied junction.
3.2.2 Free Electron Bands
Consider the ellipses of the valence band electronic pockets at the high symmetry point M in figure 6. These pockets consist of a very small pieces of surfaces that surround the occupied levels and represent a narrow cross section of the Fermi surface. Interestingly, a weak potential would be enough to destroy those pockets because the energy bands at the high symmetry point M of those pockets take the form of the free electron parabola, as shown in figure 6.
By arguing against conventional band structure, figure 8, which displays a free electronic state, is separated from the band structure in figure 6a for CrI3. Figure 8 presents a particular way to show the original free electron parabola of the s-level disconnected from the two parabolic sectors that split and form fairly large direct energy gaps in the interior of the Brillouin zone at about k=0.5ΓM and k=0.7ΓK. In the same way, most energy states in figure 6 can
be given a parabola representing the levels of the free electron. Hence, the free electron parabola states prevail in the band structure. Therefore, the application of external weak potential will beat on, not necessary all, the parabolas of free electrons. These results match the experimental observations . On the other hand, according to the Schottky model, the barrier voltage is proportional to
the metal work function. The CrI3 barrier may be small due to the appearance of nearly free electronic states in the band structure, which is key for the most promising area of the practical use of the CrI3-layer. Additionally, the band structure of the magnetic metal layer CrI3 in figure 6(a,b) indicates a difference for the majority electrons work function than minority.
The CrI3-layer shows a characteristic band structure for the majority channel at the Cr-3d orbital, whose width of about 0.3 eV has less conductive activity and nearly has the same behavior at all K values. Where, Cr-3d localized states work as a fixed potential under the Fermi level. However, the set of low lying s-electrons bands under the d-orbital has higher activity from -1.5 eV to -0.4 eV and behaves as a free electron for nearly all zone faces K values, as indicated in
figure 6. The distinction between the d- and s-bands on one side and the dipole moment field, produced by occupied and empty states, on the other could be the cause of the special properties of the CrI3-layer [34, 15, 35, 36, 37, 38, 39,40, 41, 42, 43].
When a junction forms, the excitation of electrons from the valence band to the conduction band of the semiconductor layer CrGeTe3 occurs through new, quasi-bound energy states, as in figure 3. These states are mostly referred to the half-metallic layer CrI3 for most values of k. Further, since the conductivity mainly depends on the states around the Fermi level, conductivity depends on the electronic pockets that produce semi-free electrons moving under the influence of the field of the dipole moment, which is associated with the occupied and empty states of Cr-3d (figure 4). The band structure diagram shows that those nearly free electrons are allowed in the ΓM and ΓK directions in reciprocal space or the  and  directions in real space, as shown in figure 7b.
In summary, Fermi level equilibrium and the small gap and small work func- tion in the KM direction both diminish the conductivity, except in the ΓM and ΓK directions, which in turn produce the tunneling effect, as indicated on the Brillouin zone of figure 7b. Additionally, the picture outlined above may present a disturbance in the oversimplified Schottky model view of barrier width.
3.2.3 Effective Mass (m∗)
In addition, the effective mass (m∗) is an important parameter for the tunnel- ing current and plays an important role in the dynamics of both electrons and holes, [44, 45, 46]. Additionally, m∗ has a complicated dependence on the crys- tallographic directions depending on parabola and non-parabola states on band structure, as in figure 6. Moreover, m∗ relies on moment (figure 4) and state’s energy as indicated for the Cr-d states in figure 6a. It can also have a different magnitude and even different signs, as indicated by figure 7a. Figure 7a depicts the effective mass of electron (me∗) considered by its value at the maximum or at minimum for small electronic or small holes pockets. For the conduction electronic state interval from 0.5k until the high symmetry point M and the valance path M −→ K in figure 8, each has the same behavior as in figure 7a.
Consider figure 6 and figure 7a; the band structure of the two layers shows behavior that is the opposite of the effective mass as follows: the (me∗) of the CrGeTe3-layer, for most conduction states near the Fermi level (Ef ), is negative (m∗ < 0), and the opposite is true for valance states where m∗ > 0. However, the case is different for CrI3, where (me∗) is positive in the valance band. This contrast in the (me∗) signal facilitates electrical conductions and is one of the unique characteristics of a CrI3/CrGeTe3 junction.
Near the bottom of the nearly free electron band, the effective mass is ap- proximately constant and can be calculated by the Schr¨odinger equation:
After that, m∗ increases somewhat at the inflection point by the Cr-3d magnetic field, which is indicated in figure 4, and even becomes negative at the high symmetry points M and K.