Unadulterated spectral function of low-energy quasiparticles
Evtushinsky Daniil
Institute of Metal Physics,Kiev, Ukraine
Moscow Institute of Physics and Technology
Low-Energy QuasiParticlespectrum Σ=Σ'+iΣ"
• Is quasiparticle approach valid ?
• ARPES vs. Transport ?
• Nature of interactions in HTSCs ?
• Offset at ω → 0 and at T → 0 ?
• Behavior near TC ?
• Evolution with doping?
Questions to behavior of Σ"∼1/τ
Problems
• How to remove resolution effectsaccurately?
R-?
• How to disentangle impurity scatteringfrom quasiparticle interaction?
Σ"int=Σ"-Σ"imp
Total response function
• RA – Analyzer• Remains constant• Easy to measure
• RS – Sample Surface• Varies with space and time• Difficult to measure
R=RA ⊗ RS
Spectral Function Extraction from ARPES Data
I(k, ω) ∝ A(k, ω)⊗R(k, ω)
Inte
nsity
-0.05 0.00 0.05
k - km (Å-1)
L G
- 0.01 eV
- 0.1 eV
We measure I(k, ω)We are interested in A(k, ω)We need to remove R(k, ω)
Momentum Resolution
( )( ) 20
2max
00 ))((
)(1),()(ω
ωπ
ωΣ ′′+−⋅
Σ ′′⋅−==
kkvkAkL
Without resolution effects MDC is perfect lorentzian:
Real MDC is a convolution between pure signal and resolution:
κκκ dRkLRLkMDC ⋅⋅−=⊗= ∫ )()()(
2
2
1)( σ
πσ
k
ekR−
=Where resolution is a gaussian:
Σ(k,ω)=Σ(ω) M(k)=const ε(k) linear
What is Voigt Profile?
Lorentzian ⊗ Gaussian = Voigt profile
Real MDC = Pure MDC ⊗ Resolution(Lorentzian) (Gaussian)
Voigt Profile
κκκ dGkLGLkV ⋅⋅−=⊗= ∫ )()()(
42
22
GLL
V WWWW ++≈
Where L is a lorentzian, and G is gaussian. HWHM WL and WG
)/)2ln(exp()2ln(
)( ,/1
1)( 2222 G
L
WW
xW
xGx
xLG
⋅−=+
=π
HWHM of Voigt profile, WV:
GL
VGL WWWWW +≈⇒<<2
error< 1.2 %
L, G:1. Offset2. Max position3. Amplitude4. Width
Lorentz, Gauss10
5
0
HWHM
Am
plitude
max position
offset
G L
and VoigtL, G:
1. Offset2. Max position3. Amplitude4. Width
Lorentz, Gauss
One more parameter:
5. η=WG/WL
10
5
0
HWHM
Am
plitude
max position
offset
G L
------ V
Voigt →Lorentz
η→∞
η→0
Voigt →Gauss
0.500.400.30k, 1/A
-0.4
-0.2
0.0
ω, eV
Old
Lorentz-FitOverall width
Procedure of Self Energy Extraction
0.500.400.30k, 1/A
-0.4
-0.2
0.0
ω, eV
New
Voigt-Fit
Lorentz-FitOverall width
Intrinsic widthand
resolution
Procedure of Self Energy Extraction
MDC Shape
-0.2
0.0
0.2
Errors
0.500.450.40k, A-1
5
4
3
2Intensity
0.500.450.40
k, A-1
DataFit:
Lorentzian Gaussian Voigt profile
Σ(ω) from Voigt Fit
60
50
40
30
20
10
0
HW
HM
(1/
Å) ×
10−3
-0.3-0.2-0.10.0ω (eV)
Voigt fit:
WV
WG WGf
WL WLf
Lorentz fit:
Wlor
Bi-2212 OP 89K, T = 110–135 Ka
Σ(ω) from Voigt Fit
60
50
40
30
20
10
0
HW
HM
(1/
Å) ×
10−3
-0.3-0.2-0.10.0ω (eV)
Voigt fit:
WV
WG WGf
WL WLf
Lorentz fit:
Wlor
Bi-2212 OP 89K, T = 110–135 Ka
Resolution:WG=const
Σ(ω) from Voigt Fit
60
50
40
30
20
10
0
HW
HM
(1/
Å) ×
10−3
-0.3-0.2-0.10.0ω (eV)
Voigt fit:
WV
WG WGf
WL WLf
Lorentz fit:
Wlor
Bi-2212 OP 89K, T = 110–135 Ka
Overall width,WV
Σ(ω) from Voigt Fit
60
50
40
30
20
10
0
HW
HM
(1/
Å) ×
10−3
-0.3-0.2-0.10.0ω (eV)
Voigt fit:
WV
WG WGf
WL WLf
Lorentz fit:
Wlor
Bi-2212 OP 89K, T = 110–135 Ka
Intrinsic width,WL
Examples of Voigt fit Application to ARPES Data
• BSCCO• x=0.21• x=0.16• x=0.12
• YBCO• Temperature dependence• Comparison to high-resolution data
Examples on BSCCO
0.06
0.04
0.02
0.00
W, 1/A
-0.20-0.100.00ω, эВ
WV WL WG
0.05
0.04
0.03
0.02
0.01
0.00
W, 1/A
-0.3-0.2-0.10.0ω, eV
0.08
0.06
0.04
0.02
0.00
W, 1/A
-0.30-0.20-0.100.00 ω, eV
0.50.40.3k, 1/A
-0.3
-0.2
-0.1
0.0
0.1
ω , eV
0.500.400.30k, 1/A
-0.4
-0.2
0.0
ω, eV
0.60.50.40.3k, 1/A
-0.4
-0.2
0.0
ω, eV
x=0.21, T = 100 K x=0.16, T = 40 K x=0.12, Low T
Voigt Fit in application to YBCO
0.550.500.450.400.350.301/A
-0.4
-0.3
-0.2
-0.1
0.0
0.1
eV
0.550.500.450.400.350.301/A
-0.4
-0.3
-0.2
-0.1
0.0
0.1
eV
0.04
0.02
0.00
HW
HM
, 1/A
-0.050.00
ω, eV
Image1 Image2Resolution: Raw Data: Purified Data:
Image1,24 K
Image2,24 K
Temperature dependence of scattering rate Σ"(ω=0,T)
20
15
10
5
0HW
HM
(1/
Å) ×
10−3
3002001000
T (K)
Spectrum width
width decreases!
Temperature dependence of scattering rate Σ"(ω=0,T)
20
15
10
5
0HW
HM
(1/
Å) ×
10−3
3002001000
T (K)
Voigt Gauss Lorentz
width decreases!Overall Intrinsic width increases
Comparison to the low-energy high-resolution data
0.03
0.02
0.01
0.00
W (
1/Å)
-0.15-0.10-0.050.00ω (eV)
Kordyuk PRL 2004 WL Koralek PRL 2006
a
Bi-2212 OP 89 KT = 30 K
J. D. Koralek et al., PRL 2006
R∼(hν−φ)-1/2
R6eV LASER/R27eV synchrotron=0.25
Justification of Voigt Fit procedureA. MDC Lineshape
B. ω-dependence: WG(ω)=const
C. YBCO: different images give same result while exposed to Voigt Fit procedure
D. T-dependence: while WG exhibits almost random and strange behavior, WL(T) always increases with temperature
E. Purified by Voigt Fit data coincides with high-resolution data (Koralek PRL 2006)
Justification of Voigt Fit procedure
-0.2
0.0
0.2
Errors
0.500.450.40k, A-1
5
4
3
2
Intensity
0.500.450.40
k, A-1
DataFit:
Lorentzian Gaussian Voigt profile
A. MDC Lineshape
B. ω-dependence: WG(ω)=const
C. YBCO: different images give same result while exposed to Voigt Fit procedure
D. T-dependence: while WG exhibits almost random and strange behavior, WL(T) always increases with temperature
E. Purified by Voigt Fit data coincides with high-resolution data (Koralek PRL 2006)
Justification of Voigt Fit procedure
60
50
40
30
20
10
0
HW
HM
(1/
Å) ×
10−3
-0.3-0.2-0.10.0ω (eV)
Voigt fit:
WV
WG WGf
WL WLf
Lorentz fit:
Wlor
Bi-2212 OP 89K, T = 110–135 Ka
A. MDC Lineshape
B. ω-dependence: WG(ω)=const
C. YBCO: different images give same result while exposed to Voigt Fit procedure
D. T-dependence: while WG exhibits almost random and strange behavior, WL(T) always increases with temperature
E. Purified by Voigt Fit data coincides with high-resolution data (Koralek PRL 2006)
Justification of Voigt Fit procedure
0.04
0.02
0.00
HW
HM
, 1/A
-0.050.00
ω, eV
Image1 Image2Resolution: Raw Data: Purified Data:
A. MDC Lineshape
B. ω-dependence: WG(ω)=const
C. YBCO: different images give same result while exposed to Voigt Fit procedure
D. T-dependence: while WG exhibits almost random and strange behavior, WL(T) always increases with temperature
E. Purified by Voigt Fit data coincides with high-resolution data (Koralek PRL 2006)
Justification of Voigt Fit procedure
20
15
10
5
0HW
HM
(1/
Å) ×
10−3
3002001000
T (K)
Voigt Gauss Lorentz
b
A. MDC Lineshape
B. ω-dependence: WG(ω)=const
C. YBCO: different images give same result while exposed to Voigt Fit procedure
D. T-dependence: while WG exhibits almost random and strange behavior, WL(T) always increases with temperature
E. Purified by Voigt Fit data coincides with high-resolution data (Koralek PRL 2006)
Justification of Voigt Fit procedure
0.03
0.02
0.01
0.00
W (
1/Å)
-0.15-0.10-0.050.00ω (eV)
Kordyuk PRL 2004 WL Koralek PRL 2006
a
Bi-2212 OP 89 KT = 30 K
A. MDC Lineshape
B. ω-dependence: WG(ω)=const
C. YBCO: different images give same result while exposed to Voigt Fit procedure
D. T-dependence: while WG exhibits almost random and strange behavior, WL(T) always increases with temperature
E. Purified by Voigt Fit data coincides with high-resolution data (Koralek PRL 2006)
Σ"(ω) behavior in vicinity of
Fermi-level
0.03
0.02
0.01
0.00
WL
(1/Å
)
-0.15-0.10-0.050.00ω (eV)
OP 89 K, T = 30 K OP 89 K, T = 110–135 K OD 75 K, T = 80–130 K
b
ωΣ ′′−ωΣ ′′
=ωσ)0()()(
23
2
const 0 const
ω∝σ→ω∝Σ ′′
ω∝σ→ω∝Σ ′′
=σ→ω∝Σ ′′=σ→=Σ ′′
Reduced self-energy function
Evtushinsky Physical Review B 2006
Quasiparticle limit:Σ’’(ω)/ω<<1
Σ” ∼ ω2
Σ” ∼ ω+ αωn
OP, T<TC
OP, OD T>TC
Temperature dependence of scattering rate
T.Valla Physical Review B, 2006T. Yamasaki et al., cond-mat/0603006
MDC Width: 0.030 vs 0.004 1/Å
Yoshida’s MDC used to calculate resistivity. HWHM=0.03 1/ÅCompare to result, obtainedby Voigt-fit procedure: 0.004 1/Å.
Yoshida Physica B 2004
r
imF
vnek
nem Σ ′′
≈τ
=ρη220
*τ from ARPES and Resistivity
Yoshid
a Physic
a B 2004
10-16
10-15
10-14
10-13
τ (s
ec)
0.250.200.150.100.050.00
Doping level
from resistivity linear fit
from ARPES (Σ''~20 meV) isotropic part (Σ''=2 meV)
20 meV
n ~ 1 − x
)0(Σ ′′=
ητ
Forward and Isotropic Scattering
Forward, τf Isotropic, τI
Transport: τ-1= τI-1+ α·τf-1 , α<<1
ARPES: τ-1= τI-1+τf-1
Quasiparticle Lifetime
from ARPES
1/τf
1/τI
1/τ
Σ"
-0.15-0.10-0.050.00
ω (eV)
Full Width Intrinsic Width
Bi-2212 OP 89K, T = 110–135 K
2 meV
r
imF
vnek
nem Σ ′′
≈τ
=ρη220
*Impurity scattering
Yoshid
a Physic
a B 2004
n(x) ?
inhomogeneity
forward and isotropic (unitary)?
10-16
10-15
10-14
10-13
τ (s
ec)
0.250.200.150.100.050.00
Doping level
from resistivity linear fit
from ARPES (Σ''~20 meV) isotropic part (Σ''=2 meV)
20 meV
2 meV
n ~ 1 − x
Resistivity from ARPES*
2
mne τσ =Simple Drude formula: Note that m* is calculated from
renormalized dispersion.n=(1-x)
How resistivity changes if we take in to account some extra features?
Two kinds of scattering: R↓ARPES: τ-1= τI-1+τf-1
Transport: τ-1= τI-1+ α·τf-1 , α<<1Forward, τf Isotropic, τI
Inhomogeneity: R↑n ~ (1-Vm) nd + Vm nmVm – metallic phase volume
Conclusions
• Quasiparticles are well-defined
• Non-zero Σ"(T=0) ≈ 16 meV• No drop at TC
• Σ“ depends rather on dopants than on x• To compare to transport we need to
disentangle isotropic and forward contributions
Conclusions
• Quasiparticles are well-defined
• Non-zero Σ"(T=0) ≈ 16 meV• No drop at TC
• Σ“ depends rather on dopants than on x• To compare to transport we need to
disentangle isotropic and forward contributions
Conclusions
• Quasiparticles are well-defined
• Non-zero Σ"(T=0) ≈ 16 meV• No drop at TC
• Σ“ depends rather on dopants than on x• To compare to transport we need to
disentangle isotropic and forward contributions
Conclusions
• Quasiparticles are well-defined
• Non-zero Σ"(T=0) ≈ 16 meV• No drop at TC
• Σ“ depends rather on dopants than on x• To compare to transport we need to
disentangle isotropic and forward contributions
Forward
Isotropic
Σ"(ω,T) asymmetryΣ"(ω =0,T)= S(πT)Σ"(ω,T=const)= S(ω)+const
0.10
0.08
0.06
0.04
0.02
0.00
Σ″, e
V
-0.12 -0.08 -0.04 0.00ω and -pi*T, eV
Σ″ (ω , T=95..125 K) (ω -dep for average image)
Σ″ (ω =0, T)
Σ"(ω,T)= S(ω)+S(πT) →0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00Σ″, eV
-0.08 -0.04 0.00 0.04ω and -π*T, eV
Σ″(ω, T=209 K) Σ″(ω=0, T)
OP (KEI) Y-doped (YDLY)
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
Истинная ширина MDC, 1/A
300250200150100500T, K
x w[0] + w[1]*T + w[2]*T^2 0.21 {0.0045, 1.90e-05, 7.2e-08} 0.16 {0.0052, 1.02e-05, 8.9e-08} 0.16(Ni){0.0054,-3.84e-06, 9.4e-08} 0.16(Zn){0.0093, 2.64e-05, 8.7e-09} 0.11 {0.0116,-1.44e-06, 1.7e-07} 0.12(Y) {0.0052, 2.41e-05, 2.1e-07}
Energy resolution in Foigt fit
I(k, ω) ∝ A(k, ω) ⊗R(k, ω)R(k, ω)= Rk⊗ Rω
Energy resolution has almost the same effect on the MDC shape as momentum resolution, so both Rω and Rk are taken into account by Voigt-fit procedure.
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