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Local electronic structure and inhomogeneity in heavy fermion systems
Jian-Xin Zhu Los Alamos National Laboratory
QDM 2015, Santa Fe, Mar. 9, 2015
Collaborators: Jean-Pierre Julien (UJF) Joseph Wang (LANL) Ivar Martin (ANL) Alan R. Bishop (LANL) Yoni Dubi (Ben Gurion) Sasha Balatsky (LANL)
Supported by DOE BES Program LANL-LDRD
• Emergent phenomena and electronic inhomogeneity in correlated electron systems
• Staggered Kondo Phase
• Local electronic structure in Kondo hole
• Local electronic structure around a single impurity in topological Kondo insulators
• Summary
Outline
Quantum materials exhibits novel electronic phases
• These phases characterize the states of electrons. • Superconductivity is in close proximity to AF magnetism. • Ground state in different phases can be tuned by control parameters.
• Several brands
• Several competing interactions (kinetic energy, Coulomb energy, electron-lattice coupling)
• Rich selection of competing orders and transitions between them. Consequently, immense tunability, especially near the transition.
• Electronic inhomgeneity important consequnce of competion.
Spin?
Charge?
Lattice? Orbital?
e
They emerge from compe9ng interac9ons!
H. C. Manoharan et al., Nature 403, 512 (2000)
dI/dV maps out eigenmodes!
Quantum mirages formed by coherent project of electronic structure
Stipe, Rezaei & Ho, Science 280, 1732 (1998)
C2H2 C2D2
C2H2: acetylene C2D2: deuterated acetylene
d2I/dV2 maps out the vibration mode!
IETS-‐STM observa9on of local inelas9c scaBering model
S. H. Pan et al., Nature 413, 282 (2001)
1. Spectra gap varies dramatically on the length scale of 50 Angstrom.
2. “Coherence peak” positions are symmetric about zero bias.
3. Coherent peak intensity is suppressed in the wide gap regions.
4. Low energy part of the tunneling spectra are extremely homogeneous.
McElroy et al., Science 309, 1048 (‘05)
1. Large gap regions are positively correlated with the oxygen dopants.
2. A charge density variations are significantly weak.
3. Oxygen dopants are interstitial
Applications of STM to heavy fermion systems
Perfect correlated electron lattice
Kondo hole system
§ Fano interferene, Kondo lattice formation, and ‘hidden order’ transition in URu2Si2 o Schmidt et al., Nature 465, 570 (2010) o Aynajian et al., PNAS 107, 10383
(2010) § Emerging local Kondo screening and
spatial coherence in the heavy-fermion metal YbRh2Si2 o Ernst et al., Nature 474, 362 (2011)
§ Kondo holes in U1-xThxRu2Si2 o Hamidian et al., PNAS 108, 18233
(2011)
Evidence of inhomogeneity in CeRh(In1-xCdx)5
§ Puddle size shrinks with increased pressure. Seo et al., Nat. Phys. 10, 120 (2014)
Elastic tunneling
Metal 1 Metal 2
EF1 E1 E2
Insulating barrier
EF2
E1=E2
Inelastic tunneling
E1=E2+w0
Metal 1 Metal 2
EF1 E1
E2
w0
Insulating barrier
EF2
dIdV(r,V )
d 2IdV 2 (r,V )
ρ(r,E)dρdE(r,E)
STM tunneling conductance
Local density of states
Connec9on between theory and experiment
Kondo stripe in Anderson-Heisenberg model
Perfect correlated electron lattice
H = − tijc + µδ ij( )
ij ,σ∑ ciσ
† cjσ + Vcf ciσ† fiσ +H.c.⎡⎣ ⎤⎦
i,σ∑
+ ε f − µ( )i,σ∑ fiσ
† fiσ + U fi∑ ni↑
f ni↓f + JH
2Si i Sj −
nif nj
f
4⎛
⎝⎜⎞
⎠⎟ij∑
Slave-boson approach for Ufà∞ fiσ = f iσbi†
fiσ†f iσ + bi
†bi = 1MFA: bi treated as a c-number
Heff = − tij
c + µδ ij( )ij ,σ∑ ciσ
† cjσ + Vcfbiciσ† f iσ +H.c.⎡⎣ ⎤⎦
i,σ∑ − χ ij − ε f + λiσ − µ( )δ ij( )
ij ,σ∑ f iσ
†f iσ
Anderson-Bogoliubov-de Gennes Equations
hijc Δij
Δ ji* hij
f
⎛
⎝⎜⎜
⎞
⎠⎟⎟j
∑ujσn
vjσn
⎛
⎝⎜⎜
⎞
⎠⎟⎟= En
uiσn
viσn
⎛
⎝⎜⎜
⎞
⎠⎟⎟
hijc = −tij
c − µδ ij ,Δij =Vcfbiδ ij
hijf = −χ ij + ε f + λiσ − µ⎡⎣ ⎤⎦δ ij
Numerical details: Supercell technique [JXZ et al., PRB 59, 3353 (1999)] T=0.01, χ=0.1 (input parameter) N×N=24×24
JXZ, Martin, Bishop, PRL 100, 236403 (2008)
Staggered Kondo Phase in the PM regime
bi = b0bi = b0 (−1)
(ix+ iy)
Q=(0,0) Q=(π,π)
(a) (b)
(a) Normal Kondo Phase
(b) Staggered Kondo Phase
c
f
In an infinite-U Anderson-Heisenberg model, SKP is lower in energy than NKP for a uniform RVB OP χ>0 and vice versa.
JXZ, Martin, Bishop, PRL 100, 236403 (2008)
Spinon spectral density for NKP and SKP phases for χ>0
c
f
In the presence of strong mismatch between c- and f-patch of fermi topology, even a Kondo stripe phase is possible!
NKP SKP
b-field spinon density
μ=0 V=1
μ=-0.5 V=0.8
JXZ, Martin, Bishop, PRL 100, 236403 (2008)
H = H0 + Himp
H0 = − tijc + µδ ij( )
ij ,σ∑ ciσ
† cjσ + Vcf ciσ† fiσ +H.c.⎡⎣ ⎤⎦
i,σ∑ + ε f − µ( )
i,σ∑ fiσ
† fiσ + U fi∑ ni↑
f ni↓f
Himp = εcI
σ∑ cIσ
† cIσ + ε fI − ε f( )
σ∑ fIσ
† fIσ + VcfI −Vcf( )cIσ† fIσ + H .c.⎡⎣ ⎤⎦
σ∑ −U fnI↑
f nI↓f
Kondo hole: Level at εfI unoccupied
Kondo Hole Model
Density-matrix Gutzwiller approx.
Heff = H0 + Himp
H 0 = − tijc + µδ ij( )
ij ,σ∑ ciσ
† cjσ + Vcf giσciσ† f iσ +H.c.⎡⎣ ⎤⎦
i,σ∑ + ε f + λiσ − µ( )
i,σ∑ f iσ
†f iσ + U f
i∑ di +Cont
Himp = εcI
σ∑ cIσ
† cIσ + ε fI − ε f( )
σ∑ f Iσ
†f Iσ + Vcf
I −Vcf( )cIσ† f Iσ + H .c.⎡⎣ ⎤⎦σ∑ −U f
λiσ: Lagrange multiplier di: double occupation
giσ (~√qi a): Local renormalization parameter
giσ =niσ
f − di( ) 1− nif + di( )niσ
f 1− niσf( )
⎡
⎣⎢⎢
⎤
⎦⎥⎥
1/2
+di niσ
f − di( )niσ
f 1− niσf( )
⎡
⎣⎢⎢
⎤
⎦⎥⎥
1/2
subject to constraint
λiσ =Vcf∂giσ '∂niσ
fσ '∑ ciσ '
† f iσ ' + c.c.( )−U f =Vcf
∂giσ '∂diσ '
∑ ciσ '† f iσ ' + c.c.( )
aJ.-P. Julien and J. Bouchet, Prog. Theor. Chem. Phys. 15, 509 (2006)
U→∞ :giσ = 1− ni
f
1− niσf
⎡
⎣⎢
⎤
⎦⎥
1/2
Again Anderson-Bogoliubov-de Gennes Equations
hijc Δij
Δ ji* hij
f
⎛
⎝⎜⎜
⎞
⎠⎟⎟j
∑ujσn
vjσn
⎛
⎝⎜⎜
⎞
⎠⎟⎟= En
uiσn
viσn
⎛
⎝⎜⎜
⎞
⎠⎟⎟
hijc = −tij
c − µδ ij + ε Icδ iIδ ij ,
Δij =Vcf giσ 1−δ iI( ) +δ iI⎡⎣ ⎤⎦δ ij
hijf = ε f + λiσ( ) 1−δ iI( ) + ε Ifδ iI − µ⎡⎣ ⎤⎦δ ij
LDOS (Gutzwiller QP):
ρiσc E( ),ρiσc f
E( ),ρiσf E( )( ) = − uiσ
n 2,uiσ
n viσn , viσ
n 2( )n∑ ∂ fFD E − En( )
∂E
Fano interference
dIdV eV=E =
2e2πN0Vtc2
ρiσc (E)+ 2rgiσ ρiσ
c f (E)+ r2giσ2 ρiσ
f (E)⎡⎣ ⎤⎦σ∑
Tunneling conductance measured by STM
r =Vtf /Vtc
c f • When f-electrons become itinerant,
additional transport channel is open
Technical details:
Supercell technique [JXZ et al., PRB 59, 3353 (1999)] U=2, tc=1,Vcf=1.0, εf=-1.0, μ=-0.4, T=0.01, N×N=32×32
• gi participates in the Fano interference. Should be any auxiliary field theory
(a)
−15 0 16 −15
0
16
0.95
0.96
0.97
0.98
0.99
1
(b)
−15 0 16 −15
0
16
0.2
0.4
0.6
0.8
(c)
−15 0 16 −15
0
16
0
0.05
0.1
(d)
−15 0 16 −15
0
16
0.75
0.8
0.85
0.9
0.95
(e)
−15 0 16 −15
0
16
0.2
0.4
0.6
0.8
(f)
−15 0 16 −15
0
16
−0.6
−0.4
−0.2
Solution to the Kondo hole problem
gi λi
di nic
nif ni
c f
JXZ, Julien, Dubi, Balatsky, PRL 108, 186401 (2012)
In the presence of strong mismatch between c- and f-patch of fermi topology, even a Kondo stripe phase is possible!
0
2
4
6
8
10
(a1)
(× 10)
0
0.1
0.2
0.3
0.4(a2)
0
1
2
3
4
(b1)
0
5
10
15
(b2)
−0.25 0.0 0.25 0.5 0.75 −6
−4
−2
0
2
(c1)
E−0.25 0.0 0.25 0.5 0.75
−1.5
−1
−0.5
0
0.5
1
(c2)
E
Gutzwiller QP LDOS
On the Kondo hole site NN to Kondo hole site
ρic(E)
ρif (E)
ρic f (E)
ε If = 1.0, ε I
c = 0.0ε If = 100.0, ε I
c = 0.0ε If = 100.0, ε I
c = −1.0
JXZ, Julien, Dubi, Balatsky,PRL 108, 186401 (2012)
§ Fano interference induces asymmetric dI/dV.
§ Impurity bound states robust against Fano interference.
−0.25 0.0 0.25 0.5 0.75 0
2
4
6
8
10
(a)
(× 10)
E−0.25 0.0 0.25 0.5 0.75
0
0.2
0.4(b)
E
(c)
−15 0 16 −15
0
16
0.05
0.1
0.15
(d)
−15 0 16 −15
0
16
0
2
4
6
8
(e)
−15 0 16 −15
0
16
0.2
0.3
0.4
0.5
dI/dV characteristic and imaging
JXZ, Julien, Dubi, Balatsky, PRL 108, 186401 (2012)
r=0.2
§ Single-particle surface band dispersion in TKI.
§ Continuous tunability of TKI between WTKI and STKI.
§ Number of Dirac cones signifies the STKI and WTKI.
!2
!1.5
!1
!0.5
0
0.5
1
1.5
2
E(k
)
!2
!1.5
!1
!0.5
0
0.5
1
1.5
2
E(k
)
!2
!1.5
!1
!0.5
0
0.5
1
1.5
2
E(k
)
!2
!1.5
!1
!0.5
0
0.5
1
1.5
2
E(k
)
!5
!4
!3
!2
!1
0
1
!̄
E(k
)
!5
!4
!3
!2
!1
0
1
!̄
E(k
)
X̄ X̄ M̄ M̄
(b)
(c) (d)
(e) (f)
(a)
Impurity states on the surface of TKI
Dzero et al, PRL (2010)
Wang, Julien, JXZ, Nature Commun. (submitted)
Hhyb = ciσ† Vcf ,ij ,σα f jα + H .c.⎡⎣ ⎤⎦
ij ,σα∑
Vcf ,ij = dij iσ ,dij = (Vxdijx +Vydij
y +Vzdijz )
§ DOS highly asymmetric w.r.t. Fermi energy for STKI state and dual nature tunable by f-electron energy level
§ DOS is much less asymetric w.r.t. Fermi energy for the WTKI state.
0
0.2
0.4
0.6
0.8
LD
OS
0
0.1
0.2
0.3L
DO
S
!4 !3 !2 !1 0 1 2 3 40
0.2
0.4
0.6
E
LD
OS
Layer 1 (s-band)
Layer 2 (s-band)
Layer 3 (s-band)
Layer 1 (f-band)
Layer 2 (f-band)
Layer 3 (f-band)
(a)
(b)
(c)
Impurity states on the surface of TKI (II)
Wang, Julien, JXZ, Nat. Commun. (submitted)
• In STKI, the impurity induced resonance peak shifts monotonically through the hybridization gap with impurity strength.
• In WTKI, the impurity induced resonance peak is moving into the hybridization gap.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
LD
OS
!2 !1.5 !1 !0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
E
LD
OS
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
LD
OS
Uimp
= 100
Uimp
= 10
Uimp
= 5
Uimp
= 2
Uimp
= 1.5
Uimp
= 1
Uimp
= !1
Uimp
= !1.5
Uimp
= !2
Uimp
= !5
Uimp
= 10
Uimp
= !100
(b)
(c)
(a)
Wang, Julien, JXZ, Nat. Commun. (submitted)
Impurity states on the surface of TKI (III)
5 10 15 20 25 30
30
25
20
15
10
5
!6
!5
!4
!3
!2
!1
05
1015
2025
3035
0
10
20
30
0
1
2
3
4
5 10 15 20 25 30
30
25
20
15
10
5
!6
!5
!4
!3
!2
!1
05
1015
2025
3035
0
10
20
30
0
0.2
0.4
0.6
0.8
5 10 15 20 25 30
30
25
20
15
10
5
!6
!5
!4
!3
!2
!1
05
1015
2025
3035
0
10
20
30
0
0.5
1
1.5
2
2.5
3
3.5
(a)
(b)
(c)
Impurity states on the surface of TKI (IV)
Wang, Julien, JXZ, Nat. Commun. (submitted)
§ LDOS imaging at resonance energy of STKI and WTKI.
§ Spatial pattern also reveals the nature of TKI.
• The Gutzwiller and related methods are powerful methods to address electronic inhomogeneity and local electronic structure in heavy fermion.
• We showcase their applications --- Ø to predict an alternative phase, Kondo stripe, as a
consequence of Fermi surface mismatch; Ø to study the local electronic structure around a Kondo-
hole in heavy fermion systems; Ø to address the nature of impurity induced states in
relation to the topological properties of Kondo insulators.
Summary