Luttinger Options

Luttinger parameters are dimensionless parameters characterizing the dispersion of the semiconductor valence bands in the framework of the Kohn-Luttinger approach. Introduced by Luttinger in 1956 when recording effective $(\mathbf {k} \cdot \mathbf {p} )$ ${\ displaystyle (\ mathbf {k} \ cdot \ mathbf {p})}$ ${\ displaystyle (\ mathbf {k} \ cdot \ mathbf {p})}$ Hamiltonian for Ge and Si in a magnetic field ^[1] .

Content

1 Definition
2 Relationship with effective mass
3 References
4 Literature
5 notes

Definition

The sixfold degenerate valence band in the semiconductors of the zinc blende structure splits as a result of spin-orbit interaction into a twofold degenerate CO band and a fourfold degenerate band, which generates branches of light and heavy holes . In the effective Hamiltonian $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ recorded for the zone $\Gamma _{15}$ ${\ displaystyle \ Gamma _ {15}}$ , three independent dimensionless parameters are involved $\gamma _{1}$ ${\ displaystyle \ gamma _ {1}}$ , $\gamma _{2}$ ${\ displaystyle \ gamma _ {2}}$ , $\gamma _{3}$ ${\ displaystyle \ gamma _ {3}}$ called the Kohn-Latinger parameters:

\mathbf {D} ={\dfrac {\hbar ^{2}}{2m}}\left[\left(\gamma _{1}+{\frac {5}{2}}\gamma _{2}\right)\mathbf {k} ^{2}-2\gamma _{2}\left(\mathbf {k} \cdot \mathbf {J} \right)^{2}-2\gamma _{3}\left(\sum _{i=1}^{3}\left\{\mathbf {k} _{\alpha _{i}},\mathbf {k} _{\beta _{i}}\right\}\left\{\mathbf {J} _{\alpha _{i}},\mathbf {J} _{\beta _{i}}\right\}\right)+D^{\left(A\right)}\right],

{\ displaystyle \ mathbf {D} = {\ dfrac {\ hbar ^ {2}} {2m}} \ left [\ left (\ gamma _ {1} + {\ frac {5} {2}} \ gamma _ {2} \ right) \ mathbf {k} ^ {2} -2 \ gamma _ {2} \ left (\ mathbf {k} \ cdot \ mathbf {J} \ right) ^ {2} -2 \ gamma _ {3} \ left (\ sum _ {i = 1} ^ {3} \ left \ {\ mathbf {k} _ {\ alpha _ {i}}, \ mathbf {k} _ {\ beta _ {i} } \ right \} \ left \ {\ mathbf {J} _ {\ alpha _ {i}}, \ mathbf {J} _ {\ beta _ {i}} \ right \} \ right) + D ^ {\ left (A \ right)} \ right],}

Where $D^{\left(A\right)}={\frac {e}{c}}\kappa \ \mathbf {J} \cdot \mathbf {H} +{\frac {e}{c}}q\left(\mathbf {J} _{x}^{3}\mathbf {H} _{x}+\mathbf {J} _{y}^{3}\mathbf {H} _{y}+\mathbf {J} _{z}^{3}\mathbf {H} _{z}\right)$ ${\ displaystyle D ^ {\ left (A \ right)} = {\ frac {e} {c}} \ kappa \ \ mathbf {J} \ cdot \ mathbf {H} + {\ frac {e} {c} } q \ left (\ mathbf {J} _ {x} ^ {3} \ mathbf {H} _ {x} + \ mathbf {J} _ {y} ^ {3} \ mathbf {H} _ {y} + \ mathbf {J} _ {z} ^ {3} \ mathbf {H} _ {z} \ right)}$ - relativistic member $\mathbf {J}$ ${\ displaystyle \ mathbf {J}}$ Is the operator of the angular momentum matrix for the state with spin 3/2, $\mathbf {H}$ ${\ displaystyle \ mathbf {H}}$ - a magnetic field, $\kappa$ ${\ displaystyle \ kappa}$ , $q$ ${\ displaystyle q}$ - dimensionless constants. The sum sign means the sum of the cyclic permutations $\alpha _{i}=x,y,z$ ${\ displaystyle \ alpha _ {i} = x, y, z}$ , $\beta _{i}=y,z,x$ ${\ displaystyle \ beta _ {i} = y, z, x}$ .

Dimensionless parameters similar to the Luttinger parameters appear when recording effective Hamiltonians for other bands and symmetries. For example, in the 8-band Kane Hamiltonian, they are called Kane parameters.

Effective Mass Relationship

In the structures of cubic syngony, near the point $\Gamma$ ${\ displaystyle \ Gamma}$ :

mass of heavy holes: $m_{\mathrm {hh} }={\frac {m_{0}}{\gamma _{1}-2\gamma _{2}}}$ ${\ displaystyle m _ {\ mathrm {hh}} = {\ frac {m_ {0}} {\ gamma _ {1} -2 \ gamma _ {2}}}}$
mass of light holes: $m_{\mathrm {lh} }={\frac {m_{0}}{\gamma _{1}+2\gamma _{2}}}$ ${\ displaystyle m _ {\ mathrm {lh}} = {\ frac {m_ {0}} {\ gamma _ {1} +2 \ gamma _ {2}}}}$

References

GaAs : $\gamma _{1}$ ${\ displaystyle \ gamma _ {1}}$ = 6.98; $\gamma _{2}$ ${\ displaystyle \ gamma _ {2}}$ = 2.06; $\gamma _{3}$ ${\ displaystyle \ gamma _ {3}}$ = 2.93 ^[2]
InAs : $\gamma _{1}$ ${\ displaystyle \ gamma _ {1}}$ = 20; $\gamma _{2}$ ${\ displaystyle \ gamma _ {2}}$ = 8.5; $\gamma _{3}$ ${\ displaystyle \ gamma _ {3}}$ = 9.2 ^[3]
InP : $\gamma _{1}$ ${\ displaystyle \ gamma _ {1}}$ = 5.08; $\gamma _{2}$ ${\ displaystyle \ gamma _ {2}}$ = 1.60; $\gamma _{3}$ ${\ displaystyle \ gamma _ {3}}$ = 2.10 ^[2]

Literature

Yu P., Cardona M. Fundamentals of Semiconductor Physics. M. - Fizmatlit, 2002. 87.

Notes

↑ Luttinger, JM (1956), " Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory ", Phys. Rev. (American Physical Society). - T. 102 (4): 1030-1041 , DOI 10.1103 / PhysRev.102.1030
↑ ¹ ² I. Vurgaftmana, JR Meyer, R. Ram-Mohan, J. Appl. Phys. 66 , 11, (2001) p. 5815-5874
↑ See Vurgaftman (2001), meaning $\gamma _{1}$ ${\ displaystyle \ gamma _ {1}}$ questionable

[Luttinger1956-1] Luttinger, JM (1956), " Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory ", Phys. Rev. (American Physical Society). - T. 102 (4): 1030-1041 , DOI 10.1103 / PhysRev.102.1030

[Vurgaftman-2] ¹ ² I. Vurgaftmana, JR Meyer, R. Ram-Mohan, J. Appl. Phys. 66 , 11, (2001) p. 5815-5874

[3] See Vurgaftman (2001), meaning $\gamma _{1}$ ${\ displaystyle \ gamma _ {1}}$ questionable