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Physics in 2D Materials
Taro WAKAMURA (Université Paris-Saclay)
Lecture 3
Today’s Topics
Lecture 3: Transition metal dichalcogenides (TMDCs)1
2.3 Graphene spintronics
3.1 Semiconducting TMDCs
3.2 Superconducting TMDCs
First of all... What is “spintronics”?
Electronics and Spintronics
Electron has: charge e Electronics
Electron has: spin 1/2 Spintronics
Spintronics in our daily lives
Magnetoresistive Random Access Memory
(MRAM)
Hard Disc Drive
(HDD)
How was spintronics born?
Spin polarized current
Ie ( = I↑ + I↓) ≠ 0
IS ( = I↑-I↓) ≠ 0
=
Flow of charge and spinI↑: ↑spin current
I↓: ↓spin current
:Charge :Spin
Spin-dependent transport
Currents
in ferromagnets
Birth of Spintronics
Giant Magneto-Resistance (GMR) effect
Peter Grunberg
Albert Fert
Nobel Prize in Physics in 2007
Nonmagnetic metal
Spin polarized current
Ie ( = I↑ + I↓) ≠ 0
IS ( = I↑-I↓) ≠ 0
=
Flow of charge and spin
Pure spin current
=IS ( = I↑-I↓) ≠ 0
Flow of spin only
I↑: ↑spin current
I↓: ↓spin current
:Charge :Spin
Spin-dependent transport
Currents
in ferromagnets
?
Nonlocal spin injection and detectionEasiest way: lateral spin valves
charge current
+ spin current
spin accumulation
spin current
F side N side
Spin Polarized CurrentPure Spin Current
Spin Current
Lateral Spin Valve (LSV) structure
↑
↓
N F
VP
VAP
DV
-500 0 500
-1
0
Magnetic field [Oe]
DV
/I [
m
]
DR
V
Nonlocal spin injection and detection
10
-500 0 500
-1
0
Magnetic field [Oe]
DV
/I [
m
]
DR
Fitting equation
where
PI: spin polarization of tunneling
junction
lX: spin diffusion length of X
T. Wakamura et al., Appl. Phys. Exp. 4, 063002 (2011).
Nonlocal spin injection and detectionData evaluation
Key points of spin transport
Spin can transfer information
Spin transport in a long distance is preferable
However
Spins (to a certain quantized axis) are not conserved
Charges are conserved on the contrary.
Therefore, it is important to choose materials with long spin relaxation length or
spin relaxation time.
Then how does spin relaxation occur in materials?
Spin relaxation mechanismSpin relaxation mechanism
A: Elliot-Yafet mechanism
Periodic ion scattering containing
phonon contribution
B: D’yakonov-Perel’ mechanism
Spin precesses along an effective
magnetic field during momentum
scattering.J. Fabian and S. Das Sarma, J. Vac. Sci. Technol. B 17, 1708 (1999).
e.g. Metals, Graphene…
e.g. Semiconductors, Graphene…
Two mechanisms show different dependence of ts on tp.
ts: spin relaxation time, tp: momentum relaxation time
13
A: Elliot-Yafet mechanism
Basic idea: impurity or phonon scattering + spin-orbit interaction
B: D’yakonov-Perel mechanism
Spin tilts a little every time the electron
experiences momentum scattering.
ps tt
Basic idea: spin precession by random magnetic fields
The system lack of inversion symmetry:
kkEE
Kramer’s theorem: if Hamiltonian is time-reversal symmetric
J. Fabian and S. Das Sarma, J. Vac. Sci. Technol. B 17, 1708 (1999).
kkEE
Spin relaxation mechanism
14
From two equations
kkEE
This can be regarded as a spin split caused by an effective
k-dependent magnetic field (k):
)(2
1)( kΩk Η
J. Fabian and S. Das Sarma, J. Vac. Sci. Technol. B 17, 1708 (1999).
Electrons change their momentum after
each momentum scattering process
Random magnetic field between
the scattering processes
The smaller tp, the smaller the net magnetic field for spin becomes.
(motional narrowing)
Thusps tt 1
Spin relaxation mechanism
Role of graphene in spintronics
The biggest advantage of graphene for spin transport
Intrinsic spin-orbit interaction(SOI) is small for
graphene due to small atomic number (~24 eV [1]).
[1] M. Gmitra et al., Phys. Rev. B 80, 235431 (2009).
Small spin relaxation = long-range spin transport
is possible!
Theoretically, lsf ~ 100 m, tsf ~ 100 ns are possible.
Much larger than other materials!
Atomic Number
Typical metallic materials with low SO (Cu, Ag, Al) have m spin diffusion
length and less than 1 ns spin diffusion time.
First experimental demonstration
N. Tombros et al., Nature 448, 571 (2008).
Lateral spin valve structure with graphene
Co/Al2O3/graphene lateral spin valve (figure)
Nonlocal spin valve signals are observed!
Spin diffusion length: between 1.5 ~ 2 m
L eB B
S
Larmor precession
NM
a 0 rotation
NM
B
b
B = 0
p/2 rotation
NM
B
c p rotation
V/I
B
0
0
VI
Hanle effectAnother way to estimate tsf and D: the Hanle effect
N
B=0V
Time t
t=0
44 46 48
P (
arb
.)
Time (ps)
DN
= 500 cm2/s
t = 40 ps
0
( )cosPV
dt tI
t
21( ) exp e p
4x
4 sfNN
LP t
D t
t
D t
p t
Diffusion Spin-flip
( )P t
F. J. Jedema et al, Nature 416, 713 (2002).
Hanle effectEstimation of tp and D by the Hanle effect
First experimental demonstration
N. Tombros et al., Nature 448, 571 (2008).
Hanle curves clearly demonstrate that
measured voltages are spin currents origin
However, the observed spin diffusion length
is much shorter than theoretically predicted.
Because of low mobility (2000 cm2V-1s-1)?
Low quality tunneling layer?
Spin injection efficiency: 10 %
Spin diffusion time: less than 200 ps
For efficient spin injection
Impedance mismatch problem
“Spin resistance” of material X is defined as 𝑅𝑋𝑠 = 𝜌𝑋
𝜆𝑠𝑓𝑋
𝐴𝑁
Spin impedance mismatch
Large mismatch of the spin resistance for
two different materials prevents efficient spin
transfer between the two materials
How to overcome?
Inserting spacer (tunneling layer) between
ferromagnet and nonmagnet can reduce the
impedance mismatch.S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409 (2003).
AN: Cross-sectional area of X
For efficient spin injection
In the case of graphene and ferromagnetic metal contacts
𝑅𝑋𝑠 = 𝜌𝑋
𝜆𝑠𝑓𝑋
𝐴𝑁
Contact resistance:
Graphene Large r (a few hundred ~ kilo Ohm/sq), large lsf (more than m)
Ferromagnetic metal
Small r (a few ten microOhm cm), small lsf (less than 10 nm)
Huge impedance mismatch emerges at the interface between
ferromagnets and graphene.
Spin injection through direct contact between F and Gr
Direct contact of Co to graphene
Spin injection efficiency 1.3 %
Spin diffusion length 1.5 m
W. Han et al., Appl. Phys. Lett. 94, 222109 (2009).
Spin currents are not
efficiently injected into
graphene with transparent
contacts!
Spin injection through h-BN into graphene
Hexagonal boron nitride (h-BN)
Two dimensional van der Waals insulator
Atomically flat, small lattice mismatch with graphene
Good candidate for tunnel barrier between
graphene and ferromagnetic metals!
Co/h-BN/graphene tunnel junction
M. V. Kamalakar et al., Sci. Rep. 4, 6146 (2014).
h-BN mono ~ a few layer
Mobility of graphene 2000 cm2V-1s-1
Spin injection through h-BN into graphene
Data from five devices:
h-BN0 Transparent contact between Co and graphene
(no h-BN tunneling layer)
h-BN1~4 Co/h-BN/graphene with different thickness of
h-BN
M. V. Kamalakar et al., Sci. Rep. 4, 6146 (2014).
Spin injection through h-BN into graphene
Nonlocal spin valve signals (DRNL) are clearly enhanced
with h-BN tunneling layer (contact resistance (RA)).
The thicker h-BN is, the larger the value of DRNL.
M. V. Kamalakar et al., Sci. Rep. 4, 6146 (2014).
Efficient spin injection into graphene!
Spin transport in graphene
B. Dlubak et al., Nat. Phys. 8, 557 (2012).
Epitaxial graphene on SiC
Flatter than exfoliated graphene
= less ripples, which induce gauge
fields that enhance spin relaxation
High mobility (17000 cm2V-1s-1)
Co/Al2O3/graphene local spin valve
Longer spin diffusion length is expected!
Spin transport in graphene
B. Dlubak et al., Nat. Phys. 8, 557 (2012).
“Mega-Ohm” magnetoresistance signals are observed.
Local spin valve measurements
Magnetoresistance (MR) ratio of 12 % is observed.
Spin transport in graphene
B. Dlubak et al., Nat. Phys. 8, 557 (2012).
Mega Ohm magnetoresistance can be achieved
when the contact (barrier) resistance (Rb) is
more than 1 M.
Al2O3 tunnel barrier should be high quality
lsf > 150 m for graphene on SiC!
tsf > 100 ns!
Considering the observed magnetoresistance
value and the distance between the contacts (L),
the fitting for the experimental data shows the
spin diffusion length lsf is more than 150 m.
Physics in TMDC
What are TMDCs?
TMDC (Transition Metal DiChalcogenides)
2D materials (like graphene)
Composition is MX2
M: Mo, W, Nb, V, Cr, etc.
X: Te, Se, S, etc.
Semiconductors, Semimetals, Metals, Superconductors,
Ferromagnets...
What are TMDCs?
Structural difference
WSe2, WS2, MoS2, etc.NbSe2, NbS2, TaS2, etc.
X. Qian et al., Science 346, 1344 (2014). J. Ribeiro-Soares et al., Phys. Rev. B 90, 155438 (2014).
What are TMDCs?
X. Qian et al., Science 346, 1344 (2014). J. Ribeiro-Soares et al., Phys. Rev. B 90, 155438 (2014).
Semiconducting TMDC
(Examples: WSe2, WS2, MoSe2, MoS2 etc.)
Thickness-dependent properties
Difference between monolayer and bulk
Bulk (crystal): Indirect band-gap semiconductor
Monolayer: Direct band-gap semiconductor
Band gap is located at the K (K’) point.
Slight difference of the lattice constant
(bulk 3.135 A, monolayer 3.193 A)
H. Terrones et al., Sci. Rep. 3, 1549 (2013).
Transition metal dichalcogenides (TMDs)
Similar to graphene with Dirac cones
at K (K’) points
Thickness-dependent properties
Indirect to direct band gap transition in MoS2
Photoluminescence for 1-6 layers MoS2
Spectrum A: c1 to v1 at K
Spectrum B: c1 to v2 at K
Direct transition between the band edges
Spectrum I: Indirect transition between c1 and v1
Strong suppression of spectra except A
Signature of indirect-to-direct transition
from 2-layer to 1-layer MoS2
K. F. Mak et al., Phys. Rev. Lett. 105, 136805 (2010).
Thickness-dependent properties
Indirect to direct band gap transition in MoS2
Photoluminescence for 1-6 layers MoS2
Spectrum A: c1 to v1 at K
Spectrum B: c1 to v2 at K
Direct transition between the band edges
Strong suppression of spectra except A
Signature of indirect-to-direct transition
from 2-layer to 1-layer MoS2
K. F. Mak et al., Phys. Rev. Lett. 105, 136805 (2010).
Spectrum I: Indirect transition between c1 and v1
Electronic properties
High-mobility FET with monolayer MoS2
Monolayer MoS2 on SiO2: mobility 0.5-3 cm2V-1s-1
Screening of charged impurities by a material with
high dielectric constant may increase the mobility
Device with top & bottom gate
HfO2: k=25
(e.g. SiO2: k=4)
B. Radisavljevic et al., Nat. Nanotech 6, 147 (2011).
Electronic properties
High-mobility FET with monolayer MoS2
Carrier-type:n-type
Mobility: = 217 cm2V-1s-1
B. Radisavljevic et al., Nat. Nanotech 6, 147 (2011).
Electronic properties
High-mobility FET with monolayer MoS2
High on-off ratio (108 current modulation)
4 orders of magnitude larger than conventional
Si-based transistor
Steep increase of current by Vg
B. Radisavljevic et al., Nat. Nanotech 6, 147 (2011).
Valley-Zeeman Coupling
Broken inversion symmetry
Effective inplane electric field
Heavy transition metal (Mo, W)
Strong spin-orbit interaction
𝐵 ∝ 𝑣 × 𝐸Direction of spin is locked by an effective
out-of-plane magnetic field
Time reversal symmetry requires
opposite spin directions at K and – K points
Valley-Zeeman couplingD. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
J. M. Riley et al., Nat. Phys. 10, 835 (2014).
40
Strong anisotropy in induced SOI may be
relevant to “Valley-Zeeman Coupling”
Valley-Zeeman Coupling
Electrical control of VZ coupling
VZ coupling in bulk TMDCs: Due to the inversion
symmetry, the splitting is opposite at K (or K’)
in the adjacent layer
Layer 1 Layer 2
Spin-degenerated bands
Applying strong out-of-plane electric fields, broken out-of-plane symmetry
H. Yuan et al., Nat. Phys. 9, 563 (2013).
Valley-Zeeman Coupling
Electric-double-layer transistor (EDLT)
Large interfacial electric field Extremely high carrier density @ the interface
Rxx vs T curves: From insulating to metallic dependence (more carriers can be
doped in Hole-doped regime)
H. Yuan et al., Nat. Phys. 9, 563 (2013).
Valley-Zeeman Coupling
Electrical control of VZ coupling
Crossover from weak localization (WL) to weak antilocalization (WAL) with the gate
Electrical modulation of SOI
SOI increases as a function of carrier density
H. Yuan et al., Nat. Phys. 9, 563 (2013).
Valley-Zeeman Coupling
Electrical control of VZ coupling
Band structure of bulk WSe2: Valence band maximum at G
(Valence bands at K with slightly lower energy)
Out-of-plane electric field E shifts the valence
band maximum from G to K
Broken inversion symmetry by E induces the
spin-splitting at K
Monolayer WSe2 is not affected by E
H. Yuan et al., Nat. Phys. 9, 563 (2013).
Valley Hall effect in TMDCs
D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
Spin Hall effect: Electrons are deflected depending
on the spin index
Valley Hall effect: Electrons are deflected depending
on the valley index
Origin: Valley-dependent anomalous velocity
Figure: Carriers at the valley K (filled circles)
Carriers at the valley K’ (empty circles)
D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
Electronic structure of graphene
C・ O
General form of the Berry phase (when circulates back to at )
R
:Berry connection
:Berry curvature
Vector potential
Magnetic field
Concept of anomalous velocityAnomalous velocity
Effect of the Berry curvature in the equation of motion
Anomalous velocity (normal to the electric field)
Berry curvature defined as
Defines “how much the band bends” Periodic part of the Bloch wave function
Effect of anomalous velocityAnomalous velocity
Equation of motion including the Berry curvature
When the electron moves, it feels the bending of the band.
Electron Wave packet
The center of the wave packet is determined by the interference of the
waves close to a certain wave number k. When the Berry curvature is not
zero, it affects the interference during the motion of the electron and causes
the shift of the center of the wave packet.
Effect of the Berry curvature
“Berry curvature acts as an effective magnetic field”
Valley Hall effect in TMDCs
Berry curvature in the conduction band
Valley index
In the valence band
Berry curvature is “valley-dependent”
Equation of motion including the Berry curvature
Carriers at different valleys (K or -K) are
deflected to the opposite direction! Figure: Carriers at the valley K (filled circles)
Carriers at the valley K’ (empty circles)
D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
Valley Hall effect in TMDCs
When an electric field is applied, the same number of
electrons (or holes) at K and -K are accumulated
at both edges
Optical excitation of carriers
Electrical detection of the valley Hall effect (VHE)
impossible
Selective excitation of carriers at K (or -K) by
the circularly-polarized light
Coupling strength of carriers at K (or -K) to + (-) lights
Valley index D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
Valley Hall effect in TMDCs
Excitation by +, d & -, u
↓ spin electron and ↑ spin hole both at K and -K
No charge Hall effect, no spin Hall effect, finite VHE
Excitation by linearly polarized light with u
↑ spin electron and ↓ spin hole at K &
↓ spin electron and ↑ spin hole at -K
No charge Hall effect,
finite spin Hall effect
finite VHE
D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
Valley Hall effect in TMDCs
Experimental observation of VHE in monolayer MoS2
Longitudinal conductivity: n-doped behavior as a function of Vg
Carriers are electrons
K. F. Mak et al., Science 344, 1489 (2014).
Valley Hall effect in TMDCs
Longitudinal conductivity is measured with a longitudinal electric field and
shining a circularly polarized light
Rapid increase of conductivity
Longitudinal conductivity owing to
a photocurrent
A
K. F. Mak et al., Science 344, 1489 (2014).
Valley Hall effect in TMDCs
At the resonance (A point), the Hall voltage is measured
R-L: Clockwise, L-R counterclockwise
s-p: Linear polarization
Opposite sign of VH for the opposite polarization Valley Hall effect
K. F. Mak et al., Science 344, 1489 (2014).
Valley Hall effect in TMDCs
No valley Hall effect for bilayer MoS2
Monolayer MoS2
Inversion
asymmetric
Bilayer MoS2
Inversion
symmetric
Photocarriers are created at opposite
valley for each layer, so the valley Hall
effect is canceled out.D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).X. Qian et al., Science 346, 1344 (2014).
J. Ribeiro-Soares et al., Phys. Rev. B 90, 155438 (2014).
K. F. Mak et al., Science 344, 1489 (2014).
Superconducting TMDC
Superconducting TMDCs
Superconducting TMDCs: Note
TMDCs like MoS2, MoSe2, WS2, WSe2.......
Semiconducting, and not superconducting intrinsically
TMDCs like NbS2, NbS2, TaS2...
Metallic, and superconducting intrinsically at low T
TMDCs like WTe2
Metallic or semiconducting, and superconducting intrinsically for monolayers
Superconducting TMDCs
Superconductivity in doped MoS2
Electric-double-layer transistor (EDLT) Strong carrier doping
Superconducting transition @ 9.5K
Y. Saito et al., Nat. Phys. 12, 144 (2016).
Superconducting TMDCs
Magnetic field dependence of superconductivity
Out-of-plane field Inplane field
Superconductivity is robust against inplane fieldY. Saito et al., Nat. Phys. 12, 144 (2016).
Superconducting TMDCs
D. Xiao et al., Phys. Rev. Lett. 108, 196802 (2012).
Superconductivity is broken by
Cooper pair is composed by a ↑ electron at K
and ↓ electron at K’
For out-plane-field: Orbital + spin effect
For inplane-field: Spin effect
Strong valley-Zeeman coupling creates
robust Cooper pairs for inplane field
Orbital effect
Orbital broken by Lorenz force
Spin effect
Spin-singlet broken by the Zeeman effect
Superconducting TMDCs
Superconductivity in monolayer NbSe2
2H-NbSe2: Superconducting TMDC (Bulk Tc~ 7 K)
NbSe2 in monolayer limit Spin-valley locking due to inversion symmetry breaking
(similar to MoS2)
Tc of monolayer NbSe2: 3.0 K
Note: NbSe2 is unstable
in atmosphere
X. Xi et al., Nat. Phys. 12, 139 (2016).
Superconducting TMDCs
Superconductivity is highly robust against inplane magnetic field
X. Xi et al., Nat. Phys. 12, 139 (2016).
Superconducting TMDCs
Temperature vs Magnetic field
X. Xi et al., Nat. Phys. 12, 139 (2016).
MoS2 case
Critical magnetic field goes over “Pauli limit”
HC
2/H
PPauli limit: The magnetic field where the condensation energy is
equal to the Zeeman energy
NbSe2
Superconducting TMDCs
Superconductivity in 1T’-WTe2
1T-WTe2 covered by h-BN
Exhibits a superconductivity in
the electron-doped region (Tc ~ 0.65 K)
V. Fatemi et al., Science 362, 926 (2018).
Superconducting TMDCs
When the back gate voltage (Vbg) < 0.75 V, R increases as T decreases
Metal-insulator transitionV. Fatemi et al., Science 362, 926 (2018).
Superconducting TMDCs
By the gate voltage modulation, a wide range of resistance (from
superconductivity to insulating state) is obtained
V. Fatemi et al., Science 362, 926 (2018).
Superconducting TMDCs
1T-WTe2: Quantum spin Hall insulator (2D topological insulator)
In one sample QSHI phase and superconducting phase
can be both observed
Inplane critical field exceeds the Pauli limit
V. Fatemi et al., Science 362, 926 (2018).
Superconducting TMDCs
Similar measurements from another group
Superconducting transition
h-BN/mono-WTe2/h-BN with top &
bottom gates
E. Sajadi et al., Science 362, 922 (2018).
Superconducting TMDCs
Superconducting transition occurs at highly doped region (~20 x1012 cm-2)
Tc ~ 0.5 K
150 mT perpendicular field is enough to suppress superconductivity, but
superconductivity persists up to 3.5 T when the field is inplane
E. Sajadi et al., Science 362, 922 (2018).
Superconducting TMDCs
Even for non-perfect superconducting state, resistance saturates @LT
Resistance gradually increases as a function of perpendicular magnetic field
By contrast, Resistance rapidly increases with inplane magnetic field
Interesting features
E. Sajadi et al., Science 362, 922 (2018).
Summary for today
TMDCs become superconducting even at a monolayer limit, and Cooper pairs
are robust against inplane field owing to the Ising paring
Semiconducting TMDCs are good candidates for 2D electric-field transistors
In terms of spin properties, valley-Zeeman SOI is specific for semiconducting
TMDCs
Valley-related phenomena, such as valley Hall effect are useful for future valley-
tronics
Graphene is an ideal material for spin transport owing to the small spin-orbit
interaction