Spin-polarized transmission through correlated heterostructures
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Spin-polarized transmission through correlated
heterostructures
Ulrich Eckern
with L. Chioncel, and W. H. Appelt, A. Droghetti, C. Morari,
A. Östlin, A. V. Postnikov, A. Prinz-Zwick, M. M. Radonjic,
I. Rungger, U. Schwingenschlögl, L. Vitos, A. Weh
Institute of Physics, University of Augsburg
Frank Hekking Memorial Workshop, Les Houches, 30 Jan 2018
Goals, References
• Study correlation eects on transport properties of
heterostructures
• ... by combining density functional and many-body dynamical
mean eld theory
• ... within the non-equilibrium Green's function approach
• ... as implemented, e.g., within the SMEAGOL package
Spin and Molecular Electronics on Atomically Generated Orbital Landscape
L.054431
Chioncel et al., Transmission through correlated Cu-Co-Cu heterostructures, Phys. Rev. B 92,
(2015)
C.heterostructures,
Morari et al., Spin-polarized
Phys. Rev. B ballistic conduction
96, 205137 (2017) through correlated Au-NiMnSb-Au
W. H. Appelt, PhD
A.A. Prinz-Zwick, PhDthesis,
thesis,April 2016
December
Weh, MSc thesis, January 2018 2017
2/15
Outline
Basic concepts
Scattering / NEGF approach to transport
Towards a more realistic description
Electronic correlations: DMFT
Test case
Cu-Co-Cu heterostructure
The half-Heusler compound NiMnSb
Correlations in a model half-metal
Correlations in bulk NiMnSb
Transport through Au-NiMnSb-Au
3/15
Scattering / NEGF approach to transport
• developed by Landauer & Büttiker, Meir & Wingreen, ...
• initial conditions → stationary currents
• t → −∞: ∆V 6= 0, ∆T 6= 0
• lead molecule lead: non-interacting leads, simple coupling
• express currents In , I through the leads' (equ.) distribution
functions and local properties of the molecules
4/15
Scattering / NEGF approach to transport
> consider total currents
> coupling: ∼ (VLC c+ L dC + h.c.), ∼ (VRC cR dC + h.c.)
+
L = left, C = center, R = right
> rst step: In ∼ tr VLC G< <
LC ∼ tr VCR GCR
> general result:
i
d tr [fL ΓL − fR ΓR ](GR − GA ) + [ΓL − ΓR ]G<
R
In = 2h
• special case: non-interacting or ΓL = ΓR ; then:
1
R
In = h d [fL () − fR ()]T (, ...)
• T (, ...) = transmission probability = function of energy, ...
1
R
• energy current: I = h d [fL () − fR ()]T (, ...)
• including correlations via GR , GA
5/15Towards a more realistic description
• combining DFT (GGA) studies (SMEAGOL) with DMFT
• model system / test case: Cu(111) | Co | Cu(111)
• include correlations within the Co plane
• determine spin-resolved transmission
DMFT supercell
Cu3 Co
Cu2
Cu1 Cu4
Lead Scattering region Lead
L. Chioncel et al., 2015
6/15This Hamiltonian describes electrons with spin directions body problem. The parameters that determine the proper-
s ⊂ R or A moving between localized states at lattice sites ties described by the Hubbard model are the ratio of the
i and j. The electrons interact only when they meet on the Coulomb interaction U and the bandwidth W (W is deter-
same lattice site i. (The Pauli principle requires them to Electronic
mined correlations:
by the hopping, tij), the temperature T, and the dop- DMFT
have opposite spin.) The kinetic energy and the interac- ing or number of electrons.
tion energy are characterized by the hopping term tij and The lattice structure and hopping terms influence the
the local Coulomb repulsion U, respectively. These two ability of the electrons to order magnetically, especially in
Time
+0 ¬ +R¬ +RA¬
–
+
DMFT
Vn Vn
e R– eA–
Electron reservoir
Figure 1. Dynamical mean-field theory (DMFT) of correlated-electron solids replaces the full lattice of atoms and electrons
with a single impurity atom imagined to exist in a bath of electrons. The approximation captures the dynamics of electrons
on a central atom (in orange) as it fluctuates among different atomic configurations, shown here as snapshots in time. In the
simplest case of an s orbital occupying an atom, fluctuations could vary among +0¬, +R¬, +A¬, or +RA¬, which refer to an unoc-
cupied state, a state with a single electron of spin-up, one with spin-down, and a doubly occupied state with opposite spins.
In this illustration of one possible sequence involving two transitions, an atom in an empty state absorbs an electron from the
G.
surrounding reservoir in each transition. The hybridization Vn is the quantum mechanical amplitude that specifies how likely
a state flips between two different configurations.
54 March 2004 Physics Today http://www.physicstoday.org
Kotliar & D. Vollhardt, Physics Today 57(3), 53 (2004)
7/15Cu-Co-Cu heterostructure
0.8
1 Cu4 Co Cu4
GGA
Spin-up DMFT, T=80 K
DMFT, T=200 K
T
0.5
0.6 GGA
p (EF)= 0.18
U=3eV, J=0.9eV, T=80K
0 DMFT
p (EF, 80) = 0.33
Spin-down
p(E,T)
1 GGA DMFT
DMFT
0.4 p (EF,200)= 0.32
T
0.5
0
0.2
1.5
T +T
1
0.5
(a) Total (b)
0
0 -2 -1 0 1 2
-2 -1 0 1 2 E-EF (eV)
E-EF (eV)
Left: Spin-resolved transmission: majority spins (upper panel), minority spins (middle
panel), and total (lower panel). Black dashed/red solid lines: GGA/GGA+DMFT
results. Right: Transmission spin polarization obtained within the GGA (black dashed)
and at T = 80 K (red dot dashed), T = 200 K (blue solid). Coulomb and exchange
parameters: U = 3 eV, J = 0.9 eV. Further results: transmission vs. U and J at
xed T
L. Chioncel et al., 2015
8/15Correlations in a model half-metal
0.8 0.0
DMFT Σ
Bethe lattice → semi-circular DOS
EF
Im Σ
0.4 Σ
-0.1
X X † †
EF -2 -1 0 1 2
H = −t ciσ cjσ + cjσ ciσ
DOS(eV )
E(eV)
-1
0.0 σ
0.3 X
+ U ni↑ ni↓
Im χ loc
+-
i
-0.4
U = 2 eV, W = 2 eV
∆ = 0.5 eV, T = 0.25 eV
0.0
0 1 E(eV) 2 HF
-0.8
-2 -1 0 1 2 3 4
E(eV)
• DMFT self-consistency:
Z β Z β Z β
S eff = − dτ dτ 0 c†σ (τ )Gσ−1 (τ − τ 0 )cσ (τ 0 ) + U dτ n↑ (τ )n↓ (τ )
0 0 0
1
S eff → QMC → Gσ → Gσ = iω + µ − t2 Gσ − σ∆
2
L. Chioncel et al., Phys. Rev. B 68, 144425 (2003)
9/15Correlations in bulk NiMnSb
NiMnSb: prototypical half-metal, almost fully polarized; transport?
1.5
DMFT
1.0 LSDA
EF LDA+DMFT self-consistency:
0.5 U = 3 eV, J = 0.9 eV (on Mn)
DOS(eV )
-1
T = 300 K; M
µB ! = 3.96
0.0
X 0 0
-0.5 Σσ (iω) = W σσ · Gσ
-1.0 σ0
-1.5 0.0
-8 -6 -4 -2 0 2 Σ t2g
E(eV)
-0.4 Σe
g
Im Σ (eV)
Im Σ↑ (E) ∼ (E − EF )2 -0.8
Σt
2g
↑-electrons: Fermi liquid EF
-1.2 Σe
Im Σ↓ (E) ∼ (E − EF )x , x < 2 g
↓-electrons: Non-Quasi-Particle states -1.6
-8 -6 -4 -2 0 2
E(eV)
L. Chioncel et al., Phys. Rev. B 68, 144425 (2003)
10/15Transport through Au-NiMnSb-Au
Transport
Interacting region
Lead Interface Bulk Interface Lead
Favorable con guration: Ni terminated (001) interface
11/15States near interfaces: DOS
MnSbNi/Au NiMnSb/Au
0.05 0.05
(a) (b)
DOS (states/eV)
DOS (states/eV)
maj. spin maj. spin
0 min. spin min. spin 0
Total DOS for the Au-(NiMnSb)2 -Au
slab. (a) Ni termination, (b) MnSb termi-
-0.05 LSDA bulk LSDA bulk
-0.05 nation. Solid-blue line: LSDA; solid-red
LSDA LSDA
DMFT DMFT line: LSDA+DMFT (U = 3 eV, J = 0.6
eV)
-2 -1 0 1 2 -2 -1 0 1 2
E-EF(eV) E-EF(eV)
♣ DOS polarization not signi cant, due to strong hybridization of states near
interfaces !
12/15Spin-polarized transmission
Au/(NiMnSb)nNi/Au n=4 Au/(MnSbNi)nMnSb/Au n=4
1 (a) (b)
0.5
0 DMFT maj
DMFT maj
GGA maj
n=3 n=3
1 GGA min
0.5
T(E)
T(E)
0
n=2 n=2
1
0.5
0
1 n=1 n=1
0.5
0
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
E-EF (eV) E-EF (eV)
Spin-resolved transmissions. The dashed lines represent the GGA and the solid lines
the GGA+DMFT results. Black lines: majority spin transmission; red lines: minority
spin transmission. (a) Ni-termination, (b) MnSb-termination
13/15DOS versus transmission: minority spin
0.025 0.025
DOS min(states/eV)
DOSmin(states/eV)
Au/(NiMnSb)nNi/Au Au/(MnSbNi)nMnSb/Au
n=4 n=4
0.0125 0.0125
0 0
n=4 n=4
0.08 DMFT DMFT
0.08
GGA GGA
Tmin(E)
Tmin(E)
U=3eV J=0.6eV U=3eV J=0.6eV
0.04 (a) (b) 0.04
0 0
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
E-E F (eV) E-E F (eV)
Comparison of the minority spin density of states (upper panel), and the minority spin
transmission (lower panel): (a) Ni-terminated structure, (b) MnSb-terminated
structure
♣ Signi cant spin polarization in transmission despite electronic correlations !
14/15Acknowledgements
Thank you for your attention !
Financial support: German Science Foundation (DFG), TRR 80: From
Electronic Correlations to Functionality, University of Augsburg / TU
Munich, 20102021
15/15You can also read
