Electric Field-Controlled Magnetic Anisotropy …
13
which specifically is related to the internal energy and its derivatives. For instance,
the DOS and the spin-polarized DOS are given by
ρ(E) = −
1
π
ImTr
d k
(2π) 2 G(k, E)
(12)
and
m(E) = −
1
π
ImTr
d k
(2π) 2 σ (k, E)G(k, E),
(13)
respectively, where the traces are over the spin components. The uniform static spin
susceptibility for a fixed number of electrons is determined as
χ
αβ
= −
1
π
ImTr
E F
−∞
d E
dk
d k
(2π) 2
ς=α,β
σ
ς G(k, E) − R αβ
(14)
where R αβ = m
α
(E F )m
β
(E F )/ρ(E F ) comes from ensuring that ∂ N e /∂ B
β
= 0.
Substituting spectral representation of the Green function (11) into (14) results in
the following expression:
χ
αβ
= −
1
π
ImTr
d k
(2π) 2
n,n
E F
−∞
d E
σ
α P n (k)
E − E n (k)
σ
β P n (k)
E − E n (k)
,
(15)
where summation over n = n
corresponds to intraband electron transitions and,
consequently, to the intraband part (χ
αβ
intra ) of the magnetic susceptibility. Summation over n = n
corresponds to interband electron transition and, consequently,
to the interband part (χ
αβ
inter ) of the magnetic susceptibility. Therefore, the total
magnetic susceptibility is sum, χ
αβ
= χ
αβ
intra + χ
αβ
inter . The interband magnetic
susceptibility,χ
αβ
inter , contains the integrand, the partial fraction decompositions of
which, in accordance with the Sokhotsky formula, leads to the relation
−
1
π
ImTr
E F
−∞
d E
1
(E − E n (k))
= f n (k).
(16)
The intraband magnetic susceptibility, χ
αβ
intra , contains the integrand which is the
partial fraction with a second-order pole that results in the relation
13
which specifically is related to the internal energy and its derivatives. For instance,
the DOS and the spin-polarized DOS are given by
ρ(E) = −
1
π
ImTr
d k
(2π) 2 G(k, E)
(12)
and
m(E) = −
1
π
ImTr
d k
(2π) 2 σ (k, E)G(k, E),
(13)
respectively, where the traces are over the spin components. The uniform static spin
susceptibility for a fixed number of electrons is determined as
χ
αβ
= −
1
π
ImTr
E F
−∞
d E
dk
d k
(2π) 2
ς=α,β
σ
ς G(k, E) − R αβ
(14)
where R αβ = m
α
(E F )m
β
(E F )/ρ(E F ) comes from ensuring that ∂ N e /∂ B
β
= 0.
Substituting spectral representation of the Green function (11) into (14) results in
the following expression:
χ
αβ
= −
1
π
ImTr
d k
(2π) 2
n,n
E F
−∞
d E
σ
α P n (k)
E − E n (k)
σ
β P n (k)
E − E n (k)
,
(15)
where summation over n = n
corresponds to intraband electron transitions and,
consequently, to the intraband part (χ
αβ
intra ) of the magnetic susceptibility. Summation over n = n
corresponds to interband electron transition and, consequently,
to the interband part (χ
αβ
inter ) of the magnetic susceptibility. Therefore, the total
magnetic susceptibility is sum, χ
αβ
= χ
αβ
intra + χ
αβ
inter . The interband magnetic
susceptibility,χ
αβ
inter , contains the integrand, the partial fraction decompositions of
which, in accordance with the Sokhotsky formula, leads to the relation
−
1
π
ImTr
E F
−∞
d E
1
(E − E n (k))
= f n (k).
(16)
The intraband magnetic susceptibility, χ
αβ
intra , contains the integrand which is the
partial fraction with a second-order pole that results in the relation