The Luttinger–Kohn model is a flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors . The method is a generalization of the single band k· p theory.
In this model, the influence of all other bands is taken into account by using Löwdin 's perturbation method.[1]
Background
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All bands can be subdivided into two classes:
Class A : six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands.
Class B : all other bands.The method concentrates on the bands in Class A , and takes into account Class B bands perturbatively.
We can write the perturbed solution, ϕ {\displaystyle \phi _{}^{}} , as a linear combination of the unperturbed eigenstates ϕ i ( 0 ) {\displaystyle \phi _{i}^{(0)}} :
ϕ = ∑ n A , B a n ϕ n ( 0 ) {\displaystyle \phi =\sum _{n}^{A,B}a_{n}\phi _{n}^{(0)}} Assuming the unperturbed eigenstates are orthonormalized, the eigenequations are:
( E − H m m ) a m = ∑ n ≠ m A H m n a n + ∑ α ≠ m B H m α a α {\displaystyle (E-H_{mm})a_{m}=\sum _{n\neq m}^{A}H_{mn}a_{n}+\sum _{\alpha \neq m}^{B}H_{m\alpha }a_{\alpha }} ,where
H m n = ∫ ϕ m ( 0 ) † H ϕ n ( 0 ) d 3 r = E n ( 0 ) δ m n + H m n ′ {\displaystyle H_{mn}=\int \phi _{m}^{(0)\dagger }H\phi _{n}^{(0)}d^{3}\mathbf {r} =E_{n}^{(0)}\delta _{mn}+H_{mn}^{'}} .From this expression, we can write:
a m = ∑ n ≠ m A H m n E − H m m a n + ∑ α ≠ m B H m α E − H m m a α {\displaystyle a_{m}=\sum _{n\neq m}^{A}{\frac {H_{mn}}{E-H_{mm}}}a_{n}+\sum _{\alpha \neq m}^{B}{\frac {H_{m\alpha }}{E-H_{mm}}}a_{\alpha }} ,where the first sum on the right-hand side is over the states in class A only, while the second sum is over the states on class B. Since we are interested in the coefficients a m {\displaystyle a_{m}} for m in class A, we may eliminate those in class B by an iteration procedure to obtain:
a m = ∑ n A U m n A − δ m n H m n E − H m m a n {\displaystyle a_{m}=\sum _{n}^{A}{\frac {U_{mn}^{A}-\delta _{mn}H_{mn}}{E-H_{mm}}}a_{n}} ,U m n A = H m n + ∑ α ≠ m B H m α H α n E − H α α + ∑ α , β ≠ m , n ; α ≠ β H m α H α β H β n ( E − H α α ) ( E − H β β ) + … {\displaystyle U_{mn}^{A}=H_{mn}+\sum _{\alpha \neq m}^{B}{\frac {H_{m\alpha }H_{\alpha n}}{E-H_{\alpha \alpha }}}+\sum _{\alpha ,\beta \neq m,n;\alpha \neq \beta }{\frac {H_{m\alpha }H_{\alpha \beta }H_{\beta n}}{(E-H_{\alpha \alpha })(E-H_{\beta \beta })}}+\ldots } Equivalently, for a n {\displaystyle a_{n}} (n ∈ A {\displaystyle n\in A} ):
a n = ∑ n A ( U m n A − E δ m n ) a n = 0 , m ∈ A {\displaystyle a_{n}=\sum _{n}^{A}(U_{mn}^{A}-E\delta _{mn})a_{n}=0,m\in A} and
a γ = ∑ n A U γ n A − H γ n δ γ n E − H γ γ a n = 0 , γ ∈ B {\displaystyle a_{\gamma }=\sum _{n}^{A}{\frac {U_{\gamma n}^{A}-H_{\gamma n}\delta _{\gamma n}}{E-H_{\gamma \gamma }}}a_{n}=0,\gamma \in B} .When the coefficients a n {\displaystyle a_{n}} belonging to Class A are determined, so are a γ {\displaystyle a_{\gamma }} .
Schrödinger equation and basis functions
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The Hamiltonian including the spin-orbit interaction can be written as:
H = H 0 + ℏ 4 m 0 2 c 2 σ ¯ ⋅ ∇ V × p {\displaystyle H=H_{0}+{\frac {\hbar }{4m_{0}^{2}c^{2}}}{\bar {\sigma }}\cdot \nabla V\times \mathbf {p} } ,where σ ¯ {\displaystyle {\bar {\sigma }}} is the Pauli spin matrix vector. Substituting into the Schrödinger equation in Bloch approximation we obtain
H u n k ( r ) = ( H 0 + ℏ m 0 k ⋅ Π + ℏ 2 k 2 4 m 0 2 c 2 ∇ V × p ⋅ σ ¯ ) u n k ( r ) = E n ( k ) u n k ( r ) {\displaystyle Hu_{n\mathbf {k} }(\mathbf {r} )=\left(H_{0}+{\frac {\hbar }{m_{0}}}\mathbf {k} \cdot \mathbf {\Pi } +{\frac {\hbar ^{2}k^{2}}{4m_{0}^{2}c^{2}}}\nabla V\times \mathbf {p} \cdot {\bar {\sigma }}\right)u_{n\mathbf {k} }(\mathbf {r} )=E_{n}(\mathbf {k} )u_{n\mathbf {k} }(\mathbf {r} )} ,where
Π = p + ℏ 4 m 0 2 c 2 σ ¯ × ∇ V {\displaystyle \mathbf {\Pi } =\mathbf {p} +{\frac {\hbar }{4m_{0}^{2}c^{2}}}{\bar {\sigma }}\times \nabla V} and the perturbation Hamiltonian can be defined as
H ′ = ℏ m 0 k ⋅ Π . {\displaystyle H'={\frac {\hbar }{m_{0}}}\mathbf {k} \cdot \mathbf {\Pi } .} The unperturbed Hamiltonian refers to the band-edge spin-orbit system (for k =0). At the band edge, the conduction band Bloch waves exhibits s-like symmetry, while the valence band states are p-like (3-fold degenerate without spin). Let us denote these states as | S ⟩ {\displaystyle |S\rangle } , and | X ⟩ {\displaystyle |X\rangle } , | Y ⟩ {\displaystyle |Y\rangle } and | Z ⟩ {\displaystyle |Z\rangle } respectively. These Bloch functions can be pictured as periodic repetition of atomic orbitals, repeated at intervals corresponding to the lattice spacing. The Bloch function can be expanded in the following manner:
u n k ( r ) = ∑ j ′ A a j ′ ( k ) u j ′ 0 ( r ) + ∑ γ B a γ ( k ) u γ 0 ( r ) {\displaystyle u_{n\mathbf {k} }(\mathbf {r} )=\sum _{j'}^{A}a_{j'}(\mathbf {k} )u_{j'0}(\mathbf {r} )+\sum _{\gamma }^{B}a_{\gamma }(\mathbf {k} )u_{\gamma 0}(\mathbf {r} )} ,where j' is in Class A and γ {\displaystyle \gamma } is in Class B. The basis functions can be chosen to be
u 10 ( r ) = u e l ( r ) = | S 1 2 , 1 2 ⟩ = | S ↑ ⟩ {\displaystyle u_{10}(\mathbf {r} )=u_{el}(\mathbf {r} )=\left|S{\frac {1}{2}},{\frac {1}{2}}\right\rangle =\left|S\uparrow \right\rangle }
u 20 ( r ) = u S O ( r ) = | 1 2 , 1 2 ⟩ = 1 3 | ( X + i Y ) ↓ ⟩ + 1 3 | Z ↑ ⟩ {\displaystyle u_{20}(\mathbf {r} )=u_{SO}(\mathbf {r} )=\left|{\frac {1}{2}},{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {3}}}|(X+iY)\downarrow \rangle +{\frac {1}{\sqrt {3}}}|Z\uparrow \rangle }
u 30 ( r ) = u l h ( r ) = | 3 2 , 1 2 ⟩ = − 1 6 | ( X + i Y ) ↓ ⟩ + 2 3 | Z ↑ ⟩ {\displaystyle u_{30}(\mathbf {r} )=u_{lh}(\mathbf {r} )=\left|{\frac {3}{2}},{\frac {1}{2}}\right\rangle =-{\frac {1}{\sqrt {6}}}|(X+iY)\downarrow \rangle +{\sqrt {\frac {2}{3}}}|Z\uparrow \rangle }
u 40 ( r ) = u h h ( r ) = | 3 2 , 3 2 ⟩ = − 1 2 | ( X + i Y ) ↑ ⟩ {\displaystyle u_{40}(\mathbf {r} )=u_{hh}(\mathbf {r} )=\left|{\frac {3}{2}},{\frac {3}{2}}\right\rangle =-{\frac {1}{\sqrt {2}}}|(X+iY)\uparrow \rangle }
u 50 ( r ) = u ¯ e l ( r ) = | S 1 2 , − 1 2 ⟩ = − | S ↓ ⟩ {\displaystyle u_{50}(\mathbf {r} )={\bar {u}}_{el}(\mathbf {r} )=\left|S{\frac {1}{2}},-{\frac {1}{2}}\right\rangle =-|S\downarrow \rangle }
u 60 ( r ) = u ¯ S O ( r ) = | 1 2 , − 1 2 ⟩ = 1 3 | ( X − i Y ) ↑ ⟩ − 1 3 | Z ↓ ⟩ {\displaystyle u_{60}(\mathbf {r} )={\bar {u}}_{SO}(\mathbf {r} )=\left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {3}}}|(X-iY)\uparrow \rangle -{\frac {1}{\sqrt {3}}}|Z\downarrow \rangle }
u 70 ( r ) = u ¯ l h ( r ) = | 3 2 , − 1 2 ⟩ = 1 6 | ( X − i Y ) ↑ ⟩ + 2 3 | Z ↓ ⟩ {\displaystyle u_{70}(\mathbf {r} )={\bar {u}}_{lh}(\mathbf {r} )=\left|{\frac {3}{2}},-{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {6}}}|(X-iY)\uparrow \rangle +{\sqrt {\frac {2}{3}}}|Z\downarrow \rangle }
u 80 ( r ) = u ¯ h h ( r ) = | 3 2 , − 3 2 ⟩ = − 1 2 | ( X − i Y ) ↓ ⟩ {\displaystyle u_{80}(\mathbf {r} )={\bar {u}}_{hh}(\mathbf {r} )=\left|{\frac {3}{2}},-{\frac {3}{2}}\right\rangle =-{\frac {1}{\sqrt {2}}}|(X-iY)\downarrow \rangle } .Using Löwdin's method, only the following eigenvalue problem needs to be solved
∑ j ′ A ( U j j ′ A − E δ j j ′ ) a j ′ ( k ) = 0 , {\displaystyle \sum _{j'}^{A}(U_{jj'}^{A}-E\delta _{jj'})a_{j'}(\mathbf {k} )=0,} where
U j j ′ A = H j j ′ + ∑ γ ≠ j , j ′ B H j γ H γ j ′ E 0 − E γ = H j j ′ + ∑ γ ≠ j , j ′ B H j γ ′ H γ j ′ ′ E 0 − E γ {\displaystyle U_{jj'}^{A}=H_{jj'}+\sum _{\gamma \neq j,j'}^{B}{\frac {H_{j\gamma }H_{\gamma j'}}{E_{0}-E_{\gamma }}}=H_{jj'}+\sum _{\gamma \neq j,j'}^{B}{\frac {H_{j\gamma }^{'}H_{\gamma j'}^{'}}{E_{0}-E_{\gamma }}}} ,H j γ ′ = ⟨ u j 0 | ℏ m 0 k ⋅ ( p + ℏ 4 m 0 c 2 σ ¯ × ∇ V ) | u γ 0 ⟩ ≈ ∑ α ℏ k α m 0 p j γ α . {\displaystyle H_{j\gamma }^{'}=\left\langle u_{j0}\right|{\frac {\hbar }{m_{0}}}\mathbf {k} \cdot \left(\mathbf {p} +{\frac {\hbar }{4m_{0}c^{2}}}{\bar {\sigma }}\times \nabla V\right)\left|u_{\gamma 0}\right\rangle \approx \sum _{\alpha }{\frac {\hbar k_{\alpha }}{m_{0}}}p_{j\gamma }^{\alpha }.} The second term of Π {\displaystyle \Pi } can be neglected compared to the similar term with p instead of k . Similarly to the single band case, we can write for U j j ′ A {\displaystyle U_{jj'}^{A}}
D j j ′ ≡ U j j ′ A = E j ( 0 ) δ j j ′ + ∑ α β D j j ′ α β k α k β , {\displaystyle D_{jj'}\equiv U_{jj'}^{A}=E_{j}(0)\delta _{jj'}+\sum _{\alpha \beta }D_{jj'}^{\alpha \beta }k_{\alpha }k_{\beta },} D j j ′ α β = ℏ 2 2 m 0 [ δ j j ′ δ α β + ∑ γ B p j γ α p γ j ′ β + p j γ β p γ j ′ α m 0 ( E 0 − E γ ) ] . {\displaystyle D_{jj'}^{\alpha \beta }={\frac {\hbar ^{2}}{2m_{0}}}\left[\delta _{jj'}\delta _{\alpha \beta }+\sum _{\gamma }^{B}{\frac {p_{j\gamma }^{\alpha }p_{\gamma j'}^{\beta }+p_{j\gamma }^{\beta }p_{\gamma j'}^{\alpha }}{m_{0}(E_{0}-E_{\gamma })}}\right].} We now define the following parameters
A 0 = ℏ 2 2 m 0 + ℏ 2 m 0 2 ∑ γ B p x γ x p γ x x E 0 − E γ , {\displaystyle A_{0}={\frac {\hbar ^{2}}{2m_{0}}}+{\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{\gamma }^{B}{\frac {p_{x\gamma }^{x}p_{\gamma x}^{x}}{E_{0}-E_{\gamma }}},} B 0 = ℏ 2 2 m 0 + ℏ 2 m 0 2 ∑ γ B p x γ y p γ x y E 0 − E γ , {\displaystyle B_{0}={\frac {\hbar ^{2}}{2m_{0}}}+{\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{\gamma }^{B}{\frac {p_{x\gamma }^{y}p_{\gamma x}^{y}}{E_{0}-E_{\gamma }}},} C 0 = ℏ 2 m 0 2 ∑ γ B p x γ x p γ y y + p x γ y p γ y x E 0 − E γ , {\displaystyle C_{0}={\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{\gamma }^{B}{\frac {p_{x\gamma }^{x}p_{\gamma y}^{y}+p_{x\gamma }^{y}p_{\gamma y}^{x}}{E_{0}-E_{\gamma }}},} and the band structure parameters (or the Luttinger parameters ) can be defined to be
γ 1 = − 1 3 2 m 0 ℏ 2 ( A 0 + 2 B 0 ) , {\displaystyle \gamma _{1}=-{\frac {1}{3}}{\frac {2m_{0}}{\hbar ^{2}}}(A_{0}+2B_{0}),} γ 2 = − 1 6 2 m 0 ℏ 2 ( A 0 − B 0 ) , {\displaystyle \gamma _{2}=-{\frac {1}{6}}{\frac {2m_{0}}{\hbar ^{2}}}(A_{0}-B_{0}),} γ 3 = − 1 6 2 m 0 ℏ 2 C 0 , {\displaystyle \gamma _{3}=-{\frac {1}{6}}{\frac {2m_{0}}{\hbar ^{2}}}C_{0},} These parameters are very closely related to the effective masses of the holes in various valence bands. γ 1 {\displaystyle \gamma _{1}} and γ 2 {\displaystyle \gamma _{2}} describe the coupling of the | X ⟩ {\displaystyle |X\rangle } , | Y ⟩ {\displaystyle |Y\rangle } and | Z ⟩ {\displaystyle |Z\rangle } states to the other states. The third parameter γ 3 {\displaystyle \gamma _{3}} relates to the anisotropy of the energy band structure around the Γ {\displaystyle \Gamma } point when γ 2 ≠ γ 3 {\displaystyle \gamma _{2}\neq \gamma _{3}} .
Explicit Hamiltonian matrix
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The Luttinger-Kohn Hamiltonian D j j ′ {\displaystyle \mathbf {D_{jj'}} } can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off)
H = ( E e l P z 2 P z − 3 P + 0 2 P − P − 0 P z † P + Δ 2 Q † − S † / 2 − 2 P + † 0 − 3 / 2 S − 2 R E e l P z 2 P z − 3 P + 0 2 P − P − 0 E e l P z 2 P z − 3 P + 0 2 P − P − 0 E e l P z 2 P z − 3 P + 0 2 P − P − 0 E e l P z 2 P z − 3 P + 0 2 P − P − 0 E e l P z 2 P z − 3 P + 0 2 P − P − 0 E e l P z 2 P z − 3 P + 0 2 P − P − 0 ) {\displaystyle \mathbf {H} =\left({\begin{array}{cccccccc}E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\P_{z}^{\dagger }&P+\Delta &{\sqrt {2}}Q^{\dagger }&-S^{\dagger }/{\sqrt {2}}&-{\sqrt {2}}P_{+}^{\dagger }&0&-{\sqrt {3/2}}S&-{\sqrt {2}}R\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\E_{el}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{+}&0&{\sqrt {2}}P_{-}&P_{-}&0\\\end{array}}\right)}
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(July 2010 )
References
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