Kondo Effect in the Presence of Spin-Orbit Coupling

0.19 MB·English

by  Takashi Yanagisawa

Kondo Effect in the Presence of Spin-Orbit Coupling
Takashi Yanagisawa
Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science
and Technology (AIST) Tsukuba Central 2, 1-1-1 Umezono, Tsukuba 305-8568, Japan
(Dated: Received April 8, 2012; published online September4, 2012)
Recently, a series of noncentrosymmetric superconductors has been a subject of considerable
interest since the discovery of superconductivityin CePt3Si. In noncentrosymmetric materials, the
degeneracyofbandsisliftedinthepresenceofspin-orbitcoupling. Thiswillbringaboutneweffects
intheKondoeffectsincethebanddegeneracyplaysanimportantroleinthescatteringofelectrons
by localized spins. We investigate the single-impurity Kondo problem in the presence of spin-orbit
3
coupling. Weexaminetheeffectofspin-orbitcouplingonthescatteringofconductionelectrons,by
1
0 usingtheGreen’sfunctionmethod,forthes-dHamiltonian,withemployingadecouplingprocedure.
2 Asaresult,weobtainaclosedsystemofequationsofGreen’sfunctions,fromwhichwecancalculate
physicalquantities. TheKondotemperatureTK isestimatedfromasingularityofGreen’sfunctions.
n
We show that TK is reduced as the spin-orbit coupling constant α is increased. When 2αkF is
a
comparable to or greater than kBTK(α = 0), TK shows an abrupt decrease as a result of the
J
band splitting. This suggests a Kondo collapse accompanied with a sharp decrease of TK. The
7
logT-dependenceof theresistivity will be concealed bythespin-orbit interaction.
2

]
l I. INTRODUCTION rectiontoresistivityarediscussedinsubsequentSections
e V and VI. In Section VII we examine the strong limit of
-
r TheKondoeffecthasattractedmanyresearcherssince thespin-orbitinteractionwherethedetailsofcalcuations
t
s thediscoveryofthesolutionoftheresistanceminimum[1, are shown in Appexdix.
.
t 2]. The effect arises from the interactions between a sin-
a
m gle magnetic atom and the many electrons in a metal.
Metals, when magnetic atoms are added, and rare earth
-
d compoundsexhibitmanyinterestingphenomenathatare
n related to the Kondo effect. The spin-flip scattering of
o a conduction-electron spin by a localized impurity spin II. MODEL HAMILTONIAN
c givesrisetoatermproportionaltolnT intheresistivity.
[
Superconductorswithout inversionsymmetry have at-
1 tracted much attention since the discovery of supercon- The Hamiltonianis H =H0+Hsd =HK+Hso+Hsd
where
v ductivity in CePt3Si[3]. A group of noncentrosymmet-
5
ric rare-earth compounds has been reported to exhibit
5
3 superconductivity: for example, Li2Pt3B[4, 5], CeIrSi3, H = ξ (c† c +c† c ), (1)
6 CeCoGe3, CeIrGe3[6–8], and LaNiC2[9]. The absence of K k k↑ k↑ k↓ k↓
k
. spatial inversion yields the splitting of bands due to a X
01 spin-orbit interaction[10, 11]. Hso = [α(ikx+ky)c†k↑ck↓+α(−ikx+ky)c†k↓ck↑],
3 The influence of the spin-orbit interaction was dis- Xk
1 cussedveryrecentlyin two-dimensionalsystemsstarting (2)
: from the single-impurity Anderson model[12–14]. In the J 1
iv conventional Kondo problem, the conduction-electron Hsd = −2 N [Sz(c†k↑ck′↑−c†k↓ck′↓)+S+c†k↓ck′↑
X stateswith spinupanddownaredegenerate. We expect Xkk′
ar that the band splitting has a large effect on the Kondo +S−c†k↑ck′↓]. (3)
effect, and is closely related to a multi-channel Kondo
problem. The purpose of this paper is to investigatethis
subjectonthebasisofthes-dHamiltonianwiththespin- ξ is defined by ξ = ǫ µ where ǫ is the dispersion
k k k k
−
orbit interaction of Rashba type in three dimensions at relation of the conduction electrons and µ is the chemi-
finite temperature. We calculate Green’s functions and cal potential. c and c† are annihilation and creation
kσ kσ
evaluate the Kondo temperature TK from a singularity operators, respectively. We set H0 = HK +Hso. S+,
of them. We show that TK is reduced as a result of the S− and Sz denote the operators of the localized spin.
band splitting and shows a abrupt decrease when αkF is We consider the spin-orbit interactionof Rashba type in
comparable to kBTK. Hso. α indicates the coupling constant of the spin-orbit
The paper is organized as follows. In Section II we interaction. The term H indicates the s-d interaction
sd
showtheHamiltonian,andinSectionIIIwederiveequa- between the conduction electrons and the localized spin,
tions for Green’s functions. We obtain an approximate with the coupling constant J. J is negative for the anti-
solutionin SectionIV. The Kondotemperature and cor- ferromagnetic interaction.
2
III. GREEN’S FUNCTIONS Similarly, the equation of motion for Fkk′ is
∂
First, we define Green’s functions of the conduction Fkk′(τ) = ξkFkk′ α( ikx+ky)Gkk′↑(τ)
∂τ − − −
electrons
J
Gkk′σ(τ) = −hTτckσ(τ)c†k′σ(0)i, (4) − 2N Xq [hhSzcq↓;c†k′↑iiτ −hhS+cq↑;c†k′↑iiτ].
Fkk′(τ) = −hTτck↓(τ)c†k′↑(0)i, (5) (20)
We define
whereT isthetimeorderingoperator. Wenotethatthe
τ
1
spin operators satisfy the following relations: Γkk′(τ) = e−iωnΓkk′(iωn)
β
n
1 1 X
S±Sz = ∓2Sz, SzS± =±2S±, (6) = hhSzck↑;c†k′↑iiτ +hhS−ck↓;c†k′↑iiτ, (21)
3 1
S+S− = 4 +Sz−Sz2, (7) Φqk(τ) = β e−iωnτΦqk(iωn)
n
3 X
S−S+ = 4 −Sz−Sz2. (8) = hhSzcq↓−S+cq↑;c†k↑ii, (22)
then we obtain
We also define Green’s functions which include the lo-
calized spins as well as the conduction electron opera- (iωn ξk)Gkk′↑(iωn) = δkk′ +α(ikx+ky)Fkk′(iωn)
tors. They are for example, following the notation of −
J
Zubarev[15], Γqk′(iωn), (23)
− 2N
qk′
X
hhSzck↑;c†k′↑iiτ = −hTτSzck↑(τ)c†k′↑(0)i, (9) (iωn−ξk)Fkk′(iωn) = α(−ikx+ky)Gkk′↑(iωn)
hhS−ck↓;c†k′↑iiτ = −hTτS−ck↓(τ)c†k′↑(0)i, (10) + 2JN Φqk′(iωn). (24)
hhSzck↓;c†k′↑iiτ = −hTτSzck↓(τ)c†k′↑(0)i, (11) Xq
S c ;c† = T S c (τ)c† (0) . (12) To obtain the solution to the equations above, we need
hh − k↑ k′↑iiτ −h τ − k↑ k′↑ i Green’s functions in eqs.(9)-(12). The equations of mo-
The Fourier transforms are defined as usual: tion for hhSzck↑;c†k′↑ii and hhS−ck↓;c†k′↑ii are
1 (iω ξ ) S c ;c†
Gkk′σ(τ) = e−iωnτGkk′σ(iωn), (13) − k hh z k↑ k′↑iiiω
β Xn = hSziδkk′ +α(ikx+ky)hhSzck↓;c†k′↑iiiω
1
Fkk′(τ) = β n e−iωnτFkk′(iωn), (14) − 2JN hhSz2cq↑;c†k′↑iiiω + 21hhS−cq↓;c†k′↑iiiω
1 X Xq h i
S c ;c† = e−iωnτ S c ;c† ,(15) J
hh z k↑ k′↑iiτ β n hh z k↑ k′↑iiiωn − 2N hhS+ck↑c†q↓cq′↑;c†k′↑iiiω
X . (16) Xqq′ h
······ − hhS−ck↑c†q↑cq′↓;c†k′↑iiiω ,
From the commutation relations
i (25)
[H0,ck↑] = ξkck↑ α(ikx+ky)ck↓, (17)
[Hsd,ck↑] = −−2JN Xk−′ (−Szck′↑−S−ck′↓), (18) = (αi(ω−−ikξxk+)hhkSy−)hchkS↓−;cc†kk′↑↑;ici†kiω′↑iiiω
J
the equation of motion for Gkk′↑(τ) reads − 4N hhS−cq′↓;c†k′↑iiiω
qq′
X
∂ J
∂τGkk′↑(τ) = −δ(τ)δkk′ −ξkGkk′↑(τ) + 2N hhS−ck↓c†q↑cq′↑;c†k′↑iiiω
α(ikx+ky)Fkk′(τ) Xqq′ h
+ −J [ S c ;c† + S c ;c† ]. −hhS−ck↓c†q↓cq′↓;c†k′↑iiiω −2hhSzck↓c†q↓cq′↑;c†k′↑iiiω
2N hh z q↑ k′↑iiτ hh − q↓ k′↑iiτ J i
Xq −2N hhS+S−cq′↑;c†k′↑iiiω. (26)
(19) q′
X
3
We use the commutationrelationS+S− =3/4+Sz−Sz2 proximation that
to obtain
α( ik +k )
(iω ξk)Γkk′(iω) hhSzck↓;c†k′↑iiiω = −iω x ξk y hhSzck↑;c†k′↑iiiω,
− −
= δkk′hSzi+α(ikx+ky)hhSzck↓;c†k′↑iiiω (30)
α(ik +k )
+ α(−ikx+ky)hhS−ck↑;c†k′↑iiiω hhS−ck↑;c†k′↑iiiω = iωx ξ y hhS−ck↓;c†k′↑iiiω.(31)
k
J 3 −
− 2N 4hhcq↑;c†k′↑iiiω +Γqk′(iω)
Xq h i Thismeansthatwehaveneglectedthetermsoftheorder
J of Jα in the right-hand side. Then we obtain
− 2N hhS+ck↑c†q↓cq′↑;c†k′↑iiiω
Xqq′ h α2(k2+k2)
− hhS−ck↑c†q↑cq′↓;c†k′↑iiiω −hhS−ck↓c†q↑cq′↑;c†k′↑iiiω iω−ξk− iωx ξky !Γkk′(iω)
−
++ 2hhhShS−zcckk↓↓cc†q†q↓↓ccqq′↓′↑;;cc†k†k′↑′↑iiiiiωiω . (27) = −2JN Xq h34hhcq↑;c†k′↑iiiω +Γqk′(iω)i
Here we assume that S i = 0. Now we need − 2JN hhS+ck↑c†q↓cq′↑;c†k′↑iiiω
S c ;c† and S hc zi;c† to obtain a solu- Xqq′ h
hhthhioSn−z ckfko↓↑r;cΓk†k′k′↑↑kii′ii. reTadhehheq−uakt↑ionks′↑fioir hhSzck↓;c†k′↑ii and +− hhhhSS−−cckk↓↑cc†q†q↓↓ccqq′′↓↓;;cc†k†k′′↑↑iiiiiiωω +−2hhhShS−zcckk↓↓cc†q†q↑↓ccqq′↑′↓;;cc†k†k′↑′↑iiiiiiωω
(32i)
(iω ξ ) S c ;c†
− k hh z k↓ k′↑iiiω We use the same approximation in the right-hand side
= α( ik +k ) S c ;c†
− x y hh z k↓ k′↑iiiω of eq.(23), that is, (iω ξk)Fkk′(iω) = α( ikx +
− −
+ 2JN hhSz2cq↓;c†k′↑iiiω − 21hhS+cq↑;c†k′↑iiiω ky)Gkk′↑(iω), and we have
Xq h i J
− 2JN hhS+c†q↓cq′↓ck↓;c†k′↑iiiω (iω−ξk)Gkk′↑(iω) = δkk′ − 2N p Γpk′(iω)
Xqq′ h α2(k2+k2X)
− hhS−c†q↑cq′↓ck↓;c†k′↑iiiω , (28) + iωx ξky Gkk′↑(iω) (33)
−
i
Here, we employ the decoupling approximation proce-
dure for Green’s functions[16, 17]. Many-body Green’s
(iω ξ ) S c ;c† functions are approximated as follows.
− k hh − k↑ k′↑iiiω
= −Jδkk′hS−i+α(ikx+ky)hhS−ck↓;c†k′↑iiiω hhS−ck↑c†q↑cq′↓;c†k′↑iiiω ≈ hck↑c†q↑ihhS−cq′↓;c†k′↑iiiω
+ 4N hhS−cq′↑;c†k′↑iiiω + hS−c†q↑cq′↓ihhck↑;c†k′↑iiiω,
q′
X (34)
J
+ 2N hhS−ck↑c†q↓cq′↓;c†k′↑iiiω hhS+ck↑c†q↓cq′↑;c†k′↑iiiω ≈ hS+c†q↓cq′↑ihhck↑;c†k′↑iiiω
− hhS−Xqcqk′↑hc†q↑cq′↑;c†k′↑iiiω +2hhSzc†q↓cq′↓ck↑;c†k′↑iiiω . − hS+c†q↓ck′↑ihhcq′↑;c†k′↑ii(iω35,)
(29i) hhSzck↓c†q↓cq′↑;c†k′↑iiiω ≈ hck↓c†q↓ihhSzcq′↑;c†k′↑iiiω
− hSzc†q↓ck↓ihhcq′↑;c†k′↑iiiω,
(36)
IV. APPROXIMATE SOLUTION
hhS−ck↓c†q↓cq′↓;c†k′↑iiiω ≈ hcq↓c†q′↓ihhS−ck↓;c†k′↑iiiω
We assumethat the spin-orbitcouplingα is smalland + hck↓c†q↓ihhS−cq′↓;c†k′↑iiiω.
we keep terms up to the order of α. We adopt the ap- (37)
4
We define we obtain
J 3
nk = hc†q↑ck↑i= hc†q↓ck↓i, (38) Γkk′(ω) = 2NG0k(ω)G0k′(ω) mk− 4 (1+JG(ω))
Xq Xq 1 h(cid:18) (cid:19)
mk = 3 hS−c†q↑ck↓i=2 (hSzc†q↑ck↑i+hS−c†q↑ck↓i). − nk− 2 JΓ(ω)
q q (cid:18) (cid:19) i
X X 1
(39) , (49)
× 1+JG(ω)+(J/2)2Γ(ω)F(ω)
We used the relation obtained from the rotational sym-
J2
metry in the spin space, Gkk′(ω) = δkk′G0k(ω)− 4NΓ(ω)G0k(ω)G0k′(ω)
hS−c†q↑cq′↓i=hS+c†q↓cq′↑i=2hSzc†q↑cq′↑i=−2hSzc†q↓cq′↓i. × 1+JG(ω)+(J1/2)2Γ(ω)F(ω)
(40)
hTahveen, after the analytic continuation iω → ω +iδ, we = δkk′G0k(ω)+ NJ G0k(ω)G0k′(ω)t(ω), (50)
where we defined
α2k2 3
ω ξk ⊥ Γkk′(ω)+ mk J Γ(ω)
(cid:18) − − ω−ξk(cid:19) (cid:18)4 − (cid:19) t(ω)=−4 1+JG(ω)+(J/2)2Γ(ω)F(ω). (51)
J 1 J
× 2N Gqk′(ω)+ nk− 2 N Γqk′(ω)=0, mk is given by
q (cid:18) (cid:19) q
X X m∗ = 2 ( S c† c S c† c )=2 Γ (τ = 0)
(41) k h z k↑ q↑ih − k↑ q↓i qk −
q q
X X
2
= eiωnδΓ (iω ), (52)
α2k2 J β qk n
ω−ξk− ω ξ⊥ Gkk′(ω)+ 2N Γqk′(ω)=δkk′, qXωn
(cid:18) − k(cid:19) Xq where δ is an infinitesimal constant. Because mk is real,
(42)
we obtain
wherewesetk2 =k2+k2. Then,weobtainfromeqs.(41)
⊥ x y 1
and (42) that m = 4 eiωnδG0(iω )t(iω ). (53)
k − β k n n
Xωn
3 J
Γkk′(ω) = G0k(ω) mk− 4 2NG0k′(ω) Similarly we have
(cid:18) (cid:19) 1
2 n = G0(iω )(1+JF(iω )t(iω )). (54)
G0(ω) n 1 J + m 3 J F(ω) k β k n n n
− k k− 2 k− 4 2 Xωn
h(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) i
1
Γ (ω),
× N qk V. KONDO TEMPERATURE
q
X
(43)
A singularity of t(ω) determines the characteristic
temperature of the system. We investigate the high-
where
temperature region where m =0. Then,
k
G0k(ω) = 12(cid:18)ω−ξk1+αk⊥ + ω−ξk1−αk⊥(cid:19)(44) Γ(ω)=−43F(ω). (55)
F(ω) = 1 G0(ω), (45) We obtain for k=k′
N Xk k G (ω)−1 = G0(ω)−1 3J2 F(ω) +O(J4).
1 1 kk k − 16N 1+JG(ω)
G(ω) = n G0(ω), (46)
N k− 2 k (56)
k (cid:18) (cid:19)
X
Γ(ω) = 1 m 3 G0(ω). (47) The Kondo temperature TK is determined by the van-
N k− 4 k ishing of the denominator in this equation:
k (cid:18) (cid:19)
X 1 1 1
1 J +
Because of − 2N ω ξ +αk ω ξ αk
k (cid:18) − k ⊥ − k− ⊥(cid:19)
X
N1 Γqk(ω)= 2JNΓ(ω)G0k(ω)1+JG(ω)+(J1/2)2Γ(ω)F(ω),× 14 tanh ξk2−Tαk⊥ +tanh ξk2+Tαk⊥ =0,
q (cid:18) (cid:18) K (cid:19) (cid:18) K (cid:19)(cid:19)
X
(48) (57)
5
where This yields the temperature T as
K
1
ξk = 2m(k⊥2 +kz2)−µ. (58) k T = 2eγD exp 1 7ζ(3) 2αkF 2
B K π − ρ J − 2π2 k T
We have used n = (f(ξ αk )+f(ξ +αk ))/2 by h F| | (cid:18) B K(cid:19)
k k ⊥ k ⊥ 4
neglecting the term of the−order of J2. By using the + 31ζ(5) 2αkF , (65)
expansion, 12π4 (cid:18)kBTK(cid:19) −···
i
z ∞ 1 where we introduced the Boltzmann constant kB and
tanh = , (59) the density of states ρ = mk /(2π2). This is a self-
2 z iπ(2n+1) F F
(cid:16) (cid:17) n=X−∞ − consistency equation for TK, and yields
the equation for T is
K α 2 α 4
r r
x = exp 0.21314 +0.0550
∞ 1 2m K iπT sign(2n+1) − x x
1 = Jn=X−∞8(2π)2r µ Z0 dk⊥k⊥h2ω−Kiπ(2n+1)TK − 0.01h655 αxr 6+(cid:16)0.0(cid:17)05396 αxr (cid:16)8−0(cid:17).001822 αxr 10
iπT sign(2n+1) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+ K +
ω+2αk iπ(2n+1)T ···
⊥ K
− g(xi), (66)
iπTKsign(2n+1) ≡
+ , (60)
ω 2αk iπ(2n+1)T
⊥ K with variables
− − i
whereK isacutoffandweuseanapproximateexpression
x=T /T0, α =2αk /k T0, (67)
K K r F B K
K 1
dkzω k2/(2m)+αk k2/(2m)+µ where
Z−K − ⊥ ⊥− z
1 2eγD 1
k T0 = exp . (68)
× k2/(2m)+k2/(2m)+αk µ iπ(2n+1)T B K π −ρ J
⊥ z ⊥− − K (cid:18) F| |(cid:19)
2m 1
iπsign(2n+1)TK . We expanded g(x) in powers of αr/x up to the tenth
≈ µ ω+2αk iπ(2n+1)T
r ⊥− K order. Thefunctiong(x)isshowninFig.1,wherehigher-
(61) order terms are small and negligible except near x
∼
0. The equation x = g(x) has no finite solution when
We set ancutoff n0 ≡D/(2πTK) in the summationwith αr > 1.045. This indicates that TK vanishes when the
respect to n. By using the formula for the digamma spin-orbit coupling αk is greater than 1.045k T0, and
function, F B K
indicates a Kondo collapse with a sharp decrease of T .
K
n0 1 1 1 This may overestimate the reduction of TK. When α is
n+ 1 +x =ψ 2 +x+n0 −ψ 2 +x , (62) verylarge,ifweusetheasymptoticrelationψ(1/2+z)∼
n=0 2 (cid:18) (cid:19) (cid:18) (cid:19) log(z), we obtain
X
we obtain 2eγD 1 2αk 1
F
1 ρ J log log + . (69)
1 = |J|321π2 2µm Kdk⊥k⊥ 4log 2πeTγD +2ψ 21 ≃ F| |h (cid:18)πkBT(cid:19)− 2 (cid:18)πkBT(cid:19) 4i
r Z0 h (cid:18) K (cid:19) (cid:18) (cid:19) This yields
1 1 ω+2αk
⊥
ψ
− 2 2 − i2πT √e (2eγD)2 2 π√e
(cid:18) K (cid:19) k T exp = k T0,
1ψ 1 + ω+2αk⊥ 1ψ 1 ω−2αk⊥ B K ≃ 2αkF π (cid:18)−ρF|J|(cid:19) αr B (K70)
− 2 2 i2πT − 2 2 − i2πT
(cid:18) K (cid:19) (cid:18) K (cid:19) for α 1. We show T as a function of α in Fig.2.
1ψ 1 + ω−2αk⊥ . (63) Were≫xpect that,in thKe strong coupling limr it αr 1,
− 2 (cid:18)2 i2πTK (cid:19)i TK should approach that of single-band model: ≫
We set µ = kF2/(2m) and K = 2kF, and expand the 2eγD 2
digamma function in terms of αk⊥/(2πTK). For ω = 0, kBTKα = π exp −ρ J . (71)
we have (cid:18) F| |(cid:19)
mk 2eγD 7ζ(3) 2αk 2 We will show this in the section 7. This agrees with
F F
1 = J log eq.(70) for αk D. This is very small compared to
| | 2π2 h (cid:18)π4TK (cid:19)− 2π2 (cid:18) TK (cid:19) the original TKF b∼ecause TKα/TK0 ≃ kBTK0/D. Therefore
+ 31ζ(5) 2αkF ... . (64) TK decreases as αr is increased and shows up a sharp
12π4 (cid:18) TK (cid:19) − decrease at αr ∼1.
i
6
1.5 The inverse of the life time τ (ω) is
k
1
= n NImG (ω)−1
i kk
τ (ω)
k
3n J2 1+JK(ω)
= i πρα(ω) ,
1 a = 0.5 16 (1+JK(ω))2+(JL(ω))2
r (72)
g
wheren istheconcentrationofmagneticimpurities. We
1 i
havedefinedK(ω)=ReG(ω+iδ),L(ω)= ImG(ω+iδ),
and ρα(ω)= (1/π)ImF(ω+iδ). Because−we obtain
1.5
−
0.5
2eγD 7ζ(3) 2αk 2
F
K(0) ρ log , (73)
≃ F πk T − 4π2 k T
h (cid:18) B (cid:19) (cid:18) B (cid:19) i
the formula of the conductivity yields
2e2 ∂f
0 σ = τ (ξ )v2 ρ(ξ )dξ
0 0.5 1 1.5 2 − 3 k k k∂ξ k k
x Z k
2e2 16
v2ρ(0) (1 J K(0))
≃ 3 F 3πn J2ρα(0) −| |
FIG. 1: g(x) = exp(−(7ζ(3)/2π2)(αr/x)2+···) up to the i
tenthorderofαr/xasafunctionofxforαr =0.5,1and1.5. 2e2v2ρ 16 ρF log T
The straight line is a linear function x. ≃ 3 F F3πn J ρα(0) T0
i| | h (cid:18) K(cid:19)
2
7ζ(3) 2αk
F
+ . (74)
4π2 k T
(cid:18) B (cid:19) i
Wehavetheterm(α/T)2 thatcomesfromthespin-orbit
interaction, and this term will conceal the logarithmic
dependence of the resistivity. Then, the electrical resis-
tivity R in the high temperature region T T0 is
≫ K
1
R = R0
0TK 2eγD 7ζ(3) 2αkF 2 −1
T/K × "1−|J|ρF log(cid:18)πkBT(cid:19)+ 4π2 |J|ρF (cid:18) kBT (cid:19) #
(75)
0.5 where R0 is a constant. If the term (α/kBT)2 is larger
than the logarithmic term, the resistivity even shows
R T2. Hence, the spin-orbit coupling may change the
∼
temperature dependence of the resistivity drastically.
0
VII. STRONG SPIN-ORBIT COUPLING CASE
0 0.4 0.8 1.2 1.6
a
r
In this section let us consider the case with strong
spin-orbitcoupling. For this purpose,we diagonalizethe
FIG.2: TK asafunctionofαr. x=g(x)hasnosolutionfor
αr >1.045(Kondocollapse)asindicatedbydottedline. The Hamiltonian H0:
dashed line is an expected line. ξ α(ik +k ) c
H0 = (c†k↑c†k↓) α( ikk+k ) xξ y ck↑
x y k k↓
k (cid:18) − (cid:19)(cid:18) (cid:19)
X
= (ξ αk )α†α +(ξ +αk )β†β , (76)
k− ⊥ k k k ⊥ k k
Xk h i
VI. CORRECTION TO RESISTIVITY
where k = k2+k2, and α and β are defined by
⊥ x y k k
q
The imaginary part of Gkk(ω)−1 gives the scattering αk = ukck↑+vkck↓, (77)
rate of conduction electrons due to the localized spin. β = v∗c +u c . (78)
k − k k↑ k k↓
7
The coefficients uk and vk are VIII. DISCUSSION
1 ik +k
u = , v = x y, (79) We investigated the Kondo effect in the presence of
k k
2 − √2k⊥ spin-orbit coupling. The influence of band splitting was
examinedbyusingtheGreen’sfunctionmethodwherewe
satisfyingu2k+|vk|2 =1. Weconsiderthecasewherethe adoptedthedecouplingschemetoobtainanapproximate
bandsplitissolargethatwecanneglecttheupperband. solution. TheKondotemperatureisreducedbythespin-
This means that we keepterms that containα-operators orbitinteraction,andshowsasuddendecreasewhenαk
F
only. In this approximation the s-d interaction term is is of the order of k T0. We call this the Kondo collapse
B K
due to the spin-orbit coupling. The Kondo effect is sup-
J 1
Hsαd = −2 N [Sz{(ukuk′ −vkvk∗′)α†kαk′ pbreewsseeadkeanneddtahnedlcoognTce-daleepde.nTdehnecreedoufctthioenreosfiTstiviatsyawriel-l
kk′ K
X
sultofthespin-orbitcouplingisconsistentwiththeresult
+ S+vkuk′α†kαk′ +S−ukvk∗′α†kαk′]. (80) forthesingle-impurityAndersonmodelusingthenumer-
ical renormalization group technique[13]. In their work
This is the model of one-channel conduction-electron
the Kondo temperature is a decreasing function of the
band that interacts with the localized spin.
Rashba energy E α2 when the level of the localized
Let us consider the following Green’s function: R ∝
electrons is lowered, that is, in the Kondo region, while
it is a increasing function when the localized level is not
Gαkk′(τ)=−hTταk(τ)α†k′(0)i. (81) deep. ThevariationoftheKondotemperatureisapprox-
imately linear as a function of E , namely, quadratic in
By using the same method in previous sections, Gα is R
kk′ α. Thisisconsistentwithourresultwhichshowsasmall
shown to be
variation of the Kondo temperature with the quadratic
Gαkk′(z) = zδkkξ′kα + 2JN (z ξ12k+α)v(zkvk∗′ξk′α)t(z), cKDozorynraedclootsiothenimnswpkhei-reMantouαrrieyiasissinmintaecrrllae.actsiIeondn.ainrTtehcheeenptDrwezsyoearnlkoc[se1h2oi]n,fsttkhhyee-
− − −
(82) Moriya interaction, however, vanishes in the Kondo re-
gion with particle-hole symmetry ǫ = U/2. Hence the
d
−
for arbitrary complex number z where we defined result in ref.[12] seems consistent with the result for the
s-d model.
3J Fα(z) Asalimitofstrongspin-orbitinteraction,wecaninves-
t(z)= , (83)
16 1+ JG (z) 3 J 2F (z)2 tigate a crossoverto a one-channel Kondo problem. The
2 α − 16 2 α Kondo problem with the spin-orbit coupling is closely
(cid:0) (cid:1) related to a multi-channel Kondo problem. The Kondo
1 1 temperature is reduced considerably because the degen-
F (z) = , (84)
α eracy of the conducting electrons becomes half of the
N z ξ
kα
Xk − conventional Kondo system in this limit. The specific
1 n¯kα 1/2 heatalsoexhibitsalogT-terminthe presentmodelwith
G (z) = − . (85)
α N z ξ one-channel conduction band, and this term appears in
kα
k −
X thefifth-orderofρJ. ThisagreeswiththeoriginalKondo
We derive this formula in Appendix. The Kondo tem- problem.
perature Tα is determined from a singularity of t(z) in The author expresses his sincere thanks to K.Yamaji,
K
the same way as previous sections. We obtain I. Hase and J. Kondo for helpful discussion.
2eγD 2
kBTKα = π exp − J ρ . (86) IX. APPENDIX
(cid:18) | | F(cid:19)
The characteristic energy Tα is reduced significantly In this appendix we derive the equation of motion for
K
comparedtotheconventionalKondotemperaturebyfac- Green’s functions for the single-band s-d model and dis-
tor2intheexponentialfunction. Thisfactorappearsbe- cuss its physical properties.
cause the number of channel of the conduction electrons
inthiscaseisjusthalfofthenormalKondosystem. The
resistivity is also calculated as A. Green’s functions
R=R0 1+ ρF|J|log 2eγD + , (87) H0 was diagonalized by αk and βk:
2 πk T ···
h (cid:18) B (cid:19) i αk = ukck↑+vkck↓, (88)
with a factor 1/2. β = v∗c +u c . (89)
k − k k↑ k k↓
8
The coefficients u and v are
k k
u = 1, v = ikx+ky, (90) iωnhhSzαk;α†k′iiiωn =ξkαhhSzαk;α†k′iiiωn
k 2 k − √2k⊥ + J [ (u u v v∗) S2α ;α†
2N − k q− k q hh z q k′iiiωn
q
satisfying u2 + v 2 = 1. The s-d interaction part be- X
comes k | k| + vku∗qhhS+(nkα− 12)αq;α†k′iiiωn
1
J 1 u v∗ S (n )α ;α† ],
Hsd = −2 N [Sz{(ukuk′ −vkvk∗′)α†kαk′ − k qhh − kα− 2 q k′iiiωn
kk′ (100)
X
− (ukuk′ −vk∗vk′)βk†βk′
− (ukvk′ +vkuk′)α†kβk′ −(ukvk∗′ +vk∗uk′)βk†αk′}
+ S+(vkuk′α†kαk′ −ukvk′βk†βk′ −vkvk′α†kβk′ iωnhhS+αk;α†k′iiiωn =ξkαhhS+αk;α†k′iiiωn
+ ukuk′βk†αk′) + 2JN [−(ukuq−vkvq∗)hhS+(nkα− 12)αq;α†k′iiiωn
+ S−(ukvk∗′α†kαk′ −vk∗uk′βk†βk′ +ukuk′α†kβk′ Xq
− vk∗vk′βk†αk′)]. (91) + 2ukvq∗hhSznkααq;α†k′iiiωn −ukvq∗hhS+S−αq;α†k′iiiωn],
(101)
Thesingle-bands-dmodelcontainsonlythefollowings-d
interaction,
Hsαd = −J2 N1 Xkk′[Sz{(ukuk′ −vkvk∗′)α†kαk′ + i2ωJNnhhS−[(αukk;uαq†k−′iiviωknvq∗=)hhξSk−αh(hnSk+αα−k;21α)†kα′iqi;iαωn†k′iiiωn
+ S+vkuk′α†kαk′ +S−ukvk∗′α†kαk′]. (92) Xq
− 2vkuqhhSznkααq;α†k′iiiωn −vkuqhhS−S+αq;α†k′iiiωn],
We consider the following Green’s functions: (102)
Gαkk′(τ) = −hTταk(τ)α†k′(0)i, (93) whSerne nαkα;α=† αf†koαrka.=zW, +e ahnadve u.nknown functions
S α ;α† = T (S α )(τ)α† (0) , (94) hh a kα q k′ii −
hh z k k′iiτ −h τ z k k′ i
hhS+αk;α†k′iiτ = −hTτ(S+αk)(τ)α†k′(0)i. (95)
B. Approximate Solution
The Fourier transforms are defined similarly:
Toobtainaconsistentsolution,weadoptthefollowing
1 approximation:
Gαkk′(τ) = β eiωnτGαkk′(iωn), (96)
Xωn S n α ;α† = n S α ;α† . (103)
S α ;α† = 1 eiωnτ S α ;α† , (97) hh a kα q k′ii h kαihh a q k′ii
hh z k k′iiτ β hh z k k′iiω
Xωn UsingthisapproximationandtherelationS+S− =3/4+
hhS+αk;α†k′iiτ = β1 eiωnτhhS+αk;α†k′iiω. (98) Sz−Sz2, we obtain
Xωn (iωn−ξkα)Gαkk′(iωn)=δkk′
2
J n¯ 1/2
Tdehreiveeqduaastifoonllsowofs.motion for these Green’s functions are + (cid:18)2N(cid:19) Xq iωqn−−ξq Xq′ hvkuq′hhS+αq′;α†k′iiiωn
iωnGαkk′(iωn) = δkk′ +ξkαGαkk′(iωn) + ukvq∗′hhS−αq′;α†k′iiiωn
J [(u u v v∗) S α ;α† + (ukuq′ −vkvq∗′)hhSzαq′;α†k′iiiωn
− 2N k q− k q hh z q k′iiiωn 2 i
+ vkuqXhqhS+αq;α†k′iiiωn + 83(cid:18)2JN(cid:19) Xq iωn−1 ξqα Xq′ (ukuq′ +vkvq∗′)Gαq′k′(iωn),
+ u v∗ S α ;α† ], (99) (104)
k qhh − q k′iiiωn
9
where n¯q =hnqαi. Here we define Then Gαkk′ and Γkk′ read
Γkk′(iωn) = Xq h(ukuq−vkvq∗)hhSzαq;α†k′iiiωn Gαkk′(z) = z−δkkξ′kα + 2JN (z−ξ21k+α)v(zkv−k∗′ξk′α)t(z),
+ vkuqhhS+αq;α†k′iiiωn +ukvq∗hhS−αq;α†k′iiiωn . 1 +v v∗ (112)
(105i) Γkk′(z) = 2 k k′t(z), (113)
− z ξk′α
−
This quantity reads after substituting the equations for
S α ;α† forarbitrarycomplexnumberz. TheKondotemperature
hh a q k′ii Tα is determined from a singularity of t(z) in the same
K
J 1 way as previous sections. We obtain
Γkk′ =
2N iω xi
Xq n− kα Xq′ h Tα = 2eγD exp 2 . (114)
1 K π − J ρ
− vkuq′ n¯q− 2 hhS+αq′;α†k′iiiωn (cid:18) | | F(cid:19)
(cid:18) (cid:19) The characteristic energy Tα is reduced significantly
1 K
− ukvq∗′ n¯q− 2 hhS−αq′;α†k′iiiωn ctoorm2painretdhteoetxhpeocnoennvteianltifounnacltiKonon:dotemperaturebyfac-
(cid:18) (cid:19)
1
− (ukuq′ −vkvq∗′) n¯q− 2 hhSzαq′;α†k′iiiωn Tα TK0 T0. (115)
(cid:18) (cid:19) K ∼ D K
− 38(ukuq′ +vkvq∗′)Gαq′k′(iωn) (cid:18) (cid:19)
Thisfactorappearsbecausethenumberofchannelofthe
J 3 n¯ 1/2 i J 3 1
k conductionelectronsinthiscaseisjusthalfofthenormal
= − Γkk′
−2N 8 q iωn−ξqα − 2N 8 q iωn−ξqα Kondo system. The resistivity is also calculated as
X X
× Xq′ (ukuq′ +vkvq∗′)Gαq′k′(iωn). (106) R=R0h1+ ρF2|J|log(cid:18)π2ekγBDT(cid:19)+···i, (116)
Then we obtain
with a factor 1/2.
2
Gαkk′(iωn) = iωδkk′ξ + 38 2JN iω 1 ξ
n− kα (cid:18) (cid:19) n− kα C. Entropy and Specific Heat
1 1
× Xq′ iωn−ξq′α1+ 2JN p in¯ωpn−−1ξ/p2α The energy expectation value E =hHi is given by
× Xq (ukuq+vkvq∗)Gαqk′(Piωn). (107) E = ξkhα†kαki− 2JN h{Sz(ukuk′ −vkvk∗′)
k kk′
X X
Wanedhvave2s=et1u/k2,=w1e/h√a2v.eBecause vk satisfies vk =−v−k + S+vkuk′ +S−ukvk∗′}α†kαk′i
| k| = 1 ξ G† (iω ) J 1 Γ (iω )
v∗ 3 J 2 1 β kα kk n − 2 βN kk n
Xk vk∗Gαkk′(iωn) = iωn−k′ξk′αh1− 8(cid:18)2(cid:19) 2Fα(iωn)2 = 1 kXωn ξkα + J 1 Xk iωnt(iωn) .
1 −1 β iωn ξkα 2 βN (iωn ξkα)2
, (108) kXωn − kXωn −
× 1+(J/2)G (iω )
α n (117)
i
where we set
The expectation value of the interaction Hamiltonian is
1 1 denoted as V. V is given by
F (z) = , (109)
α
N z ξ
kα
k − J
1 Xn¯kα 1/2 V = −2N h{Sz(ukuk′ −vkvk∗′)+S+vkuk′
Gα(z) = − . (110) kk′
N z ξkα X
Xk − + S−ukvk∗′}α†kαk′i
We define J 1 J 1 t(iω )
n
= Γ (iω )= .
kk n
t(z)= 3J Fα(z) . (111) −2 βN Xk 2 βN kXωn iωn−ξkα
16 1+ JG (z) 3 J 2F (z)2 (118)
2 α − 16 2 α
(cid:0) (cid:1)
10
This is written as To estimate V, we use the expansion formula for the
Fermi distribution function f(ω):
J D
V = 2ρ(0)ReZ−Ddωf(ω)t(ω−iδ), (119) D dωf(ω)h(ω)= 0 dωh(ω)+ π2(kBT)2h′(0),
6
Z−D Z−D
where we adopted the approximation (129)
for a differentiable function h(ω). Using this, we obtain
F (ω iδ)= πρ(0)i. (120)
α
± ∓ 3π 0 1
V = ρ(0)JIm dω
ρ(ω) is the density of states of conduction electrons. −16 log(Tα/T) g(βω)
Z−D K −
We need t(z) to estimate V. Gα(z), which appears in 3ππ2 ∂ 1
the denominator of t(z), contains a singularity. G (z) is (k T)2ρ(0)JIm .
written as α − 16 6 B ∂ωlog(TKα/T)−g(βω)(cid:12)ω=0
((cid:12)130)
(cid:12)
J 1 1 t(iω )
n
G (z) = R (z)+ ,
α α 2N βN kXωn z−ξkα(iωn−ξkα)2 lWoge(Dar/ekBiTnt)e2reasntdedsoinonloignatrhitehmreigciotner|mlosg(TloKαg(/DT/)|k≫BT1).,
(121) The second term is written as
π4 1
where
V2 =−128kBTρ(0)J(log(Tα/T))2. (131)
K
1 F (iω ) F (z) 1
α n α
R (z)= − F (z). (122)
α α This is expanded as in terms of ρ(0)J:
β z iω − 2
n
Xωn −
π4 ρ(0)J 4 D
Rα(z) is evaluated as[18] V2 = 32 2 kBT log k T
(cid:18) (cid:19) (cid:18) B (cid:19)
1 βD 1 βz 3π4 ρ(0)J 5 D 2
R (ω iδ) ρ(0) ψ + ψ +i , k T log . (132)
α B
− ≈ 2 2π − 2 2π − 64 2 k T
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:18) (cid:19) (cid:18) B (cid:19)
(123)
In the first term of V, the logarithmic corrections never
where ψ is the digamma function and D is the cutoff
emerge from the region where βω is large because we
energy. We use the following relation,
have g(βω) log(βω) for large ω and the T-dependence
ρ(0)J D 1 βω is canceled w∼ith log(TKα/T). When βω is small, g(βω) is
1+ log ψ +i expressed in a power series of βω. A dominant contri-
2 2πk T − 2 2π
ρ(0)J hTα (cid:18) B (cid:19) (cid:18) (cid:19)i bution is of the order of (log(TKα/T))−2. The integral is
= log K g(βω) , (124) restrictedontheinterval( kBT,0)andthefirsttermV1
2 T − is estimated as −
h i
where 3π 1 0 1 βω
V1 ≃ −16ρ(0)J(log(Tα/T))2 dωImψ 2 +i2π
1 x 1 K Z−kBT (cid:18) (cid:19)
g(x)=ψ +i ψ . (125) 3π 1 1
2 2π − 2 = k Tρ(0)J π +0.0052
(cid:18) (cid:19) (cid:18) (cid:19) −16 B (log(Tα/T))2 − 8
K
h
Then the interaction energy is
0.00738+0.000026 . (133)
− ···
3π D 1 i
V = ρ(0)JIm dωf(ω) , As a result, V is given as
−16 log(Tα/T) g(βω)
Z−D K −
(126) A 1
V = k Tρ(0)J , (134)
where we neglected the term of the order of (ρ(0)J)2 in −2 B (log(Tα/T))2
K
the denominator of t(z). V has a logarithmic temper-
ature dependence. Because of the relation between the for a constant A>0.
free energy and V, From the relation TKα =Dexp(2/(ρ(0)J)), we have
dρ(0)J 1
∂F = dlogTα. (135)
V =J , (127) ρ(0)J −log(Tα/D) K
∂J K
Using this formula, the entropy obtained from the inter-
the additional entropy ∆S(T) is
action energy V is
∆S(T)= ∂ (F F0)= J dJ′ ∂ V(J′,T). (128) ∆S = ∂∆F, (136)
−∂T − − 0 J′ ∂T − ∂T
Z