**3. Spin gapless semiconductors**

Particular cases of half-metallic ferromagnetic materials are the spin gapless semiconductors, where a semiconducting band gap is formed in one spin channel and a pseudo-band gap in the other one. Such a pseudo band gap is often called zero or closed band gap because the maximum energy of the valence band is very close to the minimum energy of the conduction band. The Zr2MnAl compound presents a typical behavior of spin gapless semiconductors and may allow a tunable spin transport (see **Figure 4**). The Fermi level, located at 0.04 eV below the conduction band minimum, in case of Zr2MnAl, falls into a typical spin gapless semiconducting band gap of 0.41 eV in spin-up channel, according to Ref [25]. In the spin-down channel, a zero band gap is reported around the Fermi level. In both spin channels, the significant contribution to density of states between −4.5 and −1.5 eV comes from the 3d electrons of Mn, while the 4d electrons from Zr atoms have contribution only above the Fermi level.

**Figure 5** presents the contribution of double and triple degenerated states (deg and dt2g, respectively) of Zr and Mn atoms, calculated around the Fermi level, at optimized lattice parameters. In Zr2MnAl compound, the highest bonding states from valence band, below the EF, belong to triple degenerated states of manganese

**95**

**Figure 5.**

*dotted, red dashed, and blue solid line, respectively.*

**Figure 4.**

*lattice parameter.*

*Zr-Based Heusler Compounds for Biomedical Spintronic Applications*

dt2g, while the lowest anti-bonding states from conduction band come from the triple degenerated states dt2g of Zr1, Zr2, and Mn. As a result, the energy gap from spin-up channel results due to Zr-Mn hybridization. The Zr2MnAl alloy presents

*The densities of states of double and triple degenerated states of Zr and Mn atoms, around the Fermi level, calculated at optimized lattice parameters for Zr2MnAl. The Fermi level, deg and dt2g, are illustrated with black* 

*Partial and total density of states (PDOS and TDOS) of spin gapless semiconductor Zr2MnAl at equilibrium* 

*DOI: http://dx.doi.org/10.5772/intechopen.93372*

*Zr-Based Heusler Compounds for Biomedical Spintronic Applications DOI: http://dx.doi.org/10.5772/intechopen.93372*

**Figure 4.**

*Magnetic Materials and Magnetic Levitation*

**Figure 3.**

*a Ref. [22]. b Ref. [21].*

**Table 1.**

*lattice parameter.*

**3. Spin gapless semiconductors**

tion only above the Fermi level.

Particular cases of half-metallic ferromagnetic materials are the spin gapless semiconductors, where a semiconducting band gap is formed in one spin channel and a pseudo-band gap in the other one. Such a pseudo band gap is often called zero or closed band gap because the maximum energy of the valence band is very close to the minimum energy of the conduction band. The Zr2MnAl compound presents a typical behavior of spin gapless semiconductors and may allow a tunable spin transport (see **Figure 4**). The Fermi level, located at 0.04 eV below the conduction band minimum, in case of Zr2MnAl, falls into a typical spin gapless semiconducting band gap of 0.41 eV in spin-up channel, according to Ref [25]. In the spin-down channel, a zero band gap is reported around the Fermi level. In both spin channels, the significant contribution to density of states between −4.5 and −1.5 eV comes from the 3d electrons of Mn, while the 4d electrons from Zr atoms have contribu-

*Calculated lattice parameters, partial, total magnetic moments, and energy band gap in Zr2CrZ (Z = Al, Ga, In).*

*The positions of the highest occupied states from the valence band (solid rhombs) and of the lowest unoccupied states from the conduction band (solid stars) of total DOSs (spin-up channel) for Zr2CrAl as function of the* 

Zr2CrAl 6.59b −0.955b −0.768b 2.835b −0.110b 1.000b 0.452b Zr2CrGa 6.635a 0.849a 0.702a −2.591a 0.049a −1.000a 0.629a

Zr2CrIn 6.875a 1.016a 0.859a −2.930a 0.034a −1.000a 0.673a

6.622b −1.011b −0.914b 2.994b −0.068b 1.000b 0.512b

6.812b −1.213b −1.080b 3.343b −0.049b 1.000b 0.615b

**(μB/atom)**

**μt (μB/f.u.) Eg**

**(eV)**

**Alloy a (Å) μZr(4a) (μB/atom) μZr(4c) (μB/atom) μY(4b) (μB/atom) μZ(4d)**

**Figure 5** presents the contribution of double and triple degenerated states (deg and dt2g, respectively) of Zr and Mn atoms, calculated around the Fermi level, at optimized lattice parameters. In Zr2MnAl compound, the highest bonding states from valence band, below the EF, belong to triple degenerated states of manganese

**94**

*Partial and total density of states (PDOS and TDOS) of spin gapless semiconductor Zr2MnAl at equilibrium lattice parameter.*

#### **Figure 5.**

*The densities of states of double and triple degenerated states of Zr and Mn atoms, around the Fermi level, calculated at optimized lattice parameters for Zr2MnAl. The Fermi level, deg and dt2g, are illustrated with black dotted, red dashed, and blue solid line, respectively.*

dt2g, while the lowest anti-bonding states from conduction band come from the triple degenerated states dt2g of Zr1, Zr2, and Mn. As a result, the energy gap from spin-up channel results due to Zr-Mn hybridization. The Zr2MnAl alloy presents

an indirect band gap of 0.41 eV, in the spin up channel with the higher bonding states from valence band located in the Δ point and the lowest anti-bonding states from the conduction band, distributed in the Δ and W high symmetry points of Brillouin zone.

It is obvious that the change in the lattice parameter affects the presence of the zero band gap from spin channel and the width of the semiconducting band gap. In Zr2MnAl alloy, the band gap increases initially by increasing the lattice parameter. The largest band gap is obtained for a lattice parameter of 6.6 Å, which corresponds to a volume increase of 2%. Above the lattice parameter of 6.6 Å, the spin gapless semiconducting properties of Zr2MnAl compound disappear, due to the shifts of the Fermi level in to the conduction band. The width of the energy band gap from spin-up channel decreases as illustrated in **Figure 6**.

The spin gapless semiconductors may present a finite total magnetic moment; however in the particular case, when a perfectly compensated ferrimagnetism appears the total magnetic moment of compound equals zero and the alloy becomes a spin gapless completely compensated ferrimagnet, like Zr2MnAl (see **Figure 7**).

Surprisingly, the zirconium element which does not exhibit natively magnetic properties shows magnetic behavior. A ferrimagnetic interaction occurs between the magnetic moments of Zr and Mn atoms, whereas the zirconium atoms, located in different Wyckoff positions, are coupled ferromagnetically. The magnetic moments of manganese increase with the lattice parameter, in all compounds. The magnetic moments of zirconium atoms coupled ferromagnetically decrease with the lattice parameter increase and compensate the magnetic moment of Mn atoms. The main element Al does not carry significant magnetic moments, but non-negligible contribution to the magnetic moment comes from conduction electrons.

Antiferromagnetic "ab initio" results were reported for Zr2MnZ (Z = A, Ga) [26– 28] and were gathered in **Table 2**. For Zr2MnAl, the band gap is slightly increased from 0.41 eV for ferromagnetic calculation to 0.48 eV to antiferromagnetic results. However, the semiconducting band gap from spin-down channel decreases when the atomic radius of the main element increases (when Ga replaces Al). Due to the different magnetic ordering structures, having the spin moments of manganese antiparallel (antiferromagnetic configuration) or parallel (ferromagnetic configuration) oriented, the sign of partial magnetic moments from **Table 2**, differs. However, the opposite spin orientation is clearly explaining the ferrimagnetic

#### **Figure 6.**

*The positions of the highest occupied states from the valence band (solid rhombs) and of the lowest unoccupied states from the conduction band (solid stars) of total DOSs (spin-down channel) for Zr2MnAl as function of the lattice parameter.*

**97**

for half-metals.

**Figure 7.**

*\**

*a Ref. [25]. b Ref. [26]. c Ref. [27]. d Ref. [28]. e Ref. [29].*

**Table 2.**

**4. Half-metallic ferromagnetic materials**

are classified as half-metallic ferromagnets.

*Zr-Based Heusler Compounds for Biomedical Spintronic Applications*

interaction between the Mn and Zr atoms and the ferromagnetic coupling between the Zr atoms located in the two distinct sublattice. The total magnetic moment per f.u. calculated in both magnetic configurations is fully compensated by partial magnetic moments of constituents and follow the Slater Pauling curve for typical

*Calculated lattice parameters, partial, total magnetic moments, and energy band gap in Zr2MnZ (Z = Al, Ga).*

**Alloy a (A) μZr(4a) (μB/atom) μZr(4c) (μB/atom) μY(4b) (μB/atom) μZ(4d) (μB/atom) μt**

Zr2MnGa 6.59c,\* 1.66c,\* 1.52c,\* −3.08c,\* −0.10c,\* 0.00c,\*

Zr2MnAl 6.56a −0.77a −0.69a 2.44a −0.04a 0.00a 0.41a 6.64b,\* 1.74b,\* 1.50b,\* −3.26b,\* 0.02b,\* 0.00b,\* 6.968d,\* 1.47d,\* 1.35d,\* −4.33d,\* 0.088d,\* 0.00d,\* 0.48d

6.935e −1.48e −1.42e 4.34e −0.030e 0.00e 0.31e

**(μB/f.u.)**

**Eg (eV)**

Ferromagnetic zirconium-based half-metallic Heusler compounds represent another category of materials of specific interest in biomedical spintronic applications where a good response to an external magnetic moment is required and that is mainly related to their large total magnetic moment. From theoretical point of view, the materials which exhibit a metallic character in the majority density of states and a band gap in the minority one, around the Fermi level and the metallic total density of states resulted from summation of partial density of states of all elements

A typical example of density of states for a half-metallic ferromagnet is exemplified in case of Zr2CoAl [30–34] (see **Figure 8**). In the majority channel, the significant contribution to density of states comes from the zirconium, located in the origin of unit cell and the cobalt atom. The band gap from minority channel (**Figure 9**) is formed between the 3d t2g electrons of Co and the 4d t2g unoccupied

*DOI: http://dx.doi.org/10.5772/intechopen.93372*

*Total magnetic moments in spin gapless semiconductor Zr2MnAl.*

*Antiferromagnetic calculation was performed.*

*Zr-Based Heusler Compounds for Biomedical Spintronic Applications DOI: http://dx.doi.org/10.5772/intechopen.93372*

**Figure 7.** *Total magnetic moments in spin gapless semiconductor Zr2MnAl.*


*\* Antiferromagnetic calculation was performed.*

*a Ref. [25].*

*Magnetic Materials and Magnetic Levitation*

spin-up channel decreases as illustrated in **Figure 6**.

Brillouin zone.

conduction electrons.

an indirect band gap of 0.41 eV, in the spin up channel with the higher bonding states from valence band located in the Δ point and the lowest anti-bonding states from the conduction band, distributed in the Δ and W high symmetry points of

It is obvious that the change in the lattice parameter affects the presence of the zero band gap from spin channel and the width of the semiconducting band gap. In Zr2MnAl alloy, the band gap increases initially by increasing the lattice parameter. The largest band gap is obtained for a lattice parameter of 6.6 Å, which corresponds to a volume increase of 2%. Above the lattice parameter of 6.6 Å, the spin gapless semiconducting properties of Zr2MnAl compound disappear, due to the shifts of the Fermi level in to the conduction band. The width of the energy band gap from

The spin gapless semiconductors may present a finite total magnetic moment; however in the particular case, when a perfectly compensated ferrimagnetism appears the total magnetic moment of compound equals zero and the alloy becomes a spin gapless completely compensated ferrimagnet, like Zr2MnAl (see **Figure 7**). Surprisingly, the zirconium element which does not exhibit natively magnetic properties shows magnetic behavior. A ferrimagnetic interaction occurs between the magnetic moments of Zr and Mn atoms, whereas the zirconium atoms, located in different Wyckoff positions, are coupled ferromagnetically. The magnetic moments of manganese increase with the lattice parameter, in all compounds. The magnetic moments of zirconium atoms coupled ferromagnetically decrease with the lattice parameter increase and compensate the magnetic moment of Mn atoms. The main element Al does not carry significant magnetic moments, but non-negligible contribution to the magnetic moment comes from

Antiferromagnetic "ab initio" results were reported for Zr2MnZ (Z = A, Ga) [26– 28] and were gathered in **Table 2**. For Zr2MnAl, the band gap is slightly increased from 0.41 eV for ferromagnetic calculation to 0.48 eV to antiferromagnetic results. However, the semiconducting band gap from spin-down channel decreases when the atomic radius of the main element increases (when Ga replaces Al). Due to the different magnetic ordering structures, having the spin moments of manganese antiparallel (antiferromagnetic configuration) or parallel (ferromagnetic configuration) oriented, the sign of partial magnetic moments from **Table 2**, differs. However, the opposite spin orientation is clearly explaining the ferrimagnetic

*The positions of the highest occupied states from the valence band (solid rhombs) and of the lowest unoccupied states from the conduction band (solid stars) of total DOSs (spin-down channel) for Zr2MnAl as function of* 

**96**

**Figure 6.**

*the lattice parameter.*

*b Ref. [26]. c*

*Ref. [27]. d Ref. [28].*

*e Ref. [29].*

#### **Table 2.**

*Calculated lattice parameters, partial, total magnetic moments, and energy band gap in Zr2MnZ (Z = Al, Ga).*

interaction between the Mn and Zr atoms and the ferromagnetic coupling between the Zr atoms located in the two distinct sublattice. The total magnetic moment per f.u. calculated in both magnetic configurations is fully compensated by partial magnetic moments of constituents and follow the Slater Pauling curve for typical for half-metals.
