Electronic properties and intrinsic transport of
black phosphorus monolayers
Alexander Rudenko
Institute for Molecules and Materials
Radboud University Nijmegen
September 24th , 2016
AMM-2016 Symposium, Ekaterinburg
Introduction Electronic properties Intrinsic transport Conclusion
Outline
− Introduction
− Electronic properties
− Quasiparticle spectrum
− Tight-binding model
− Extrinsic transport
− Phonon scattering and intrinsic transport
− Scattering mechanisms
− Intrinsic mobility in BP
− Conclusion
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Black phosphorus
Orthorhombic symmetry (D2h)
c
[Ne] 3s23p3
15
P
b
a
Phosphorus
1ML BP
(phosphorene)
sp3-hybridization
Weak interlayer
bonding (vdW)
+
−
+
−
Alexander Rudenko
+
−
+
−
z
y
x
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Black phosphorus: motivation
Publications per year
First reports on exfoliated BP appeared Jan 2014
Ultrathin samples
became available
• Thickness dependent direct gap (∼0.3–2.0 eV)
• Strong intrinsic anisotropy (mx∗ /my∗ > 10)
• Suitability for electronic applications (high µ, high Ion /Ioff )
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Energy (eV)
Quasiparticle GW bands in monolayer BP
6
4
2
0
-2
-4
-6
X
S
Γ
Y
G = G0 + G0 ΣG (Dyson eq.)
Σ = i G W (GW appr.)
W = ε−1 v (screening)
Γ
Alexander Rudenko
X
S
Y
24 Sep 2016 (AMM-2016)
Γ
Transport properties of black phosphorus
5 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Energy (eV)
Quasiparticle GW bands in monolayer BP
6
4
2
0
-2
-4
-6
X
S
Γ
Y
G = G0 + G0 ΣG (Dyson eq.)
Σ = i G W (GW appr.)
W = ε−1 v (screening)
Γ
X
S
Y
Γ
• Valence and conduction band edges are isolated
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
5 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Energy (eV)
Quasiparticle GW bands in monolayer BP
pz (%)
>80
6
4
2
0
-2
-4
-6
X
G = G0 + G0 ΣG (Dyson eq.)
S
60
Γ
Y
40
Σ = i G W (GW appr.)
W = ε−1 v (screening)
20
0
Γ
X
S
Y
Γ
• Valence and conduction band edges are isolated
• ...and have predominantly pz character
Alexander Rudenko
k
HGW
R
HTB
multiorbital
Hamiltonian
single-orbital
Hamiltonian
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Minimal model for monolayer BP
H=
X
||
tij ci† cj
i 6=j
z
t
||
2
y
x
||
t1
t
≈ −1.5 eV ;
||
t2
||
(1)
||
∆g (Γ) ≈ 2t2 + 4t1
| {z }
||
1
Band gap
≈ +3.7 eV
6
Band gap appearance
||
||
criterion: t2 > 2|t1 |
Energy (eV)
4
2
X
S
Γ
Y
(1)
∆g
0
-2
-4
-6
Γ
X
S
Y
Γ
A. Rudenko, M. Katsnelson, PRB 89, 201408 (2014)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
6 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Minimal model for monolayer BP
H=
X
||
tij ci† cj
i 6=j
z
t
||
2
y
x
||
(1)
t
||
1
||
t1 ≈ −1.5 eV ; t2 ≈ +3.7 eV
||
||
∆g (Γ) ≈ 2t2 + 4t1
| {z }
Band gap
Why do the hoppings have different signs?
teff ≈ t + U (kinetic + Coulomb energies)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
7 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Minimal model for monolayer BP
H=
X
||
tij ci† cj
i 6=j
z
t
||
2
y
x
||
(1)
t
||
1
||
t1 ≈ −1.5 eV ; t2 ≈ +3.7 eV
||
||
∆g (Γ) ≈ 2t2 + 4t1
| {z }
Band gap
Why do the hoppings have different signs?
teff ≈ t + U (kinetic + Coulomb energies)
888 88 888 88
8
8
Overlap (t1 << 0)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
No overlap (t2 ≤ 0)
Transport properties of black phosphorus
7 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Minimal model for bilayer BP
H=
t||2
t1
i 6=j
t1||
|
┴
z
X
||
tij ci† cj +
{z
i 6=j
}
intralayer term
||
X
|
tij⊥ ci† cj
{z
}
interlayer term
||
t1 ≈ −1.5eV ; t2 ≈ +3.7eV ; t1⊥ ≈ 0.5eV
y
x
6
accounts for the
narrowing of a gap in
multilayer BP
Energy (eV)
4
t1⊥
2
X
S
Γ
Y
(2)
∆g
0
-2
-4
-6
Γ
X
S
Y
Γ
A. Rudenko, M. Katsnelson, PRB 89, 201408 (2014)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Optimal TB model vs. GW
t9||
t8||
t2
t1
t1||
┴
t10||
z
t5||
y
||
tij ci† cj +
monolayer
2
X
S
0
Γ
Y
arbitrary number of layers
X
S
(c)
6
Y
X
S
Γ
Y
−2
Γ
X
Γ
trilayer
S
(d)
6
Y
Γ
bulk
4
4
Energy (eV)
0
−6
Γ
• Applicability to BP with
2
−4
−6
GW in low-energy region
bilayer
4
−2
• Perfect agreement with
tij⊥ ci† cj
(b)
6
−4
||
4
X
i 6=j
4
t
x
(a)
6
t6||
Energy (eV)
┴
X
i 6=j
t||2
┴
t3 t4
┴
H=
Energy (eV)
t3||
2
0
X
S
Γ
Y
Energy (eV)
t7||
−2
2
0
Z
X
S
Γ
Y
−2
−4
−4
−6
−6
Γ
X
S
Y
Γ
Γ
X
S
Y
Γ Z
A. Rudenko, S. Yuan, M. Katsnelson, PRB 92, 085419 (2015)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
9 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Band gap, Eg (eV)
Band gap scaling with the number of layers
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
GW0 calc.
TB model
A exp(−Bn)/nC+D
A=1.71 eV B=0.17
D=0.40 eV C=0.73
0
5
10 15 20 25 30 35
∞
Number of layers, n
• Band gap decays exponentially
• Exp. gap of bulk BP (∼0.35 eV) is reproduced
A. Rudenko, S. Yuan, M. Katsnelson, PRB 92, 085419 (2015)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
10 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Optical properties from large-scale simulations
Two approaches for σxx (ω):
• First-principles (GW )
4.4 Å
3.3 Å
primitive cell
(four atoms)
• Tight-binding
0.7 µm
0.7 µm
large supercell
(∼107 atoms)
A. Rudenko, S. Yuan, M. Katsnelson, PRB 92, 085419 (2015)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
11 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Optical properties from large-scale simulations
Two approaches for σxx (ω):
3.5
3.3 Å
primitive cell
(four atoms)
σxx/(σ0 n)
4.4 Å
n=1
n=2
n=3
n=∞
(a)
3
• First-principles (GW )
2.5
2
GW0
1.5
1
0.5
0
0
0.5
1
1.5
2
Frequency, ω (eV)
2.5
3
3.5
large supercell
(∼107 atoms)
σxx/(σ0 n)
0.7 µm
0.7 µm
(c)
3
• Tight-binding
2.5
2
n=1
n=2
n=3
n=100
TB model
1.5
1
0.5
0
Excellent agreement up to ω ∼ 2.5 eV
0
0.5
1
1.5
2
Frequency, ω (eV)
2.5
3
A. Rudenko, S. Yuan, M. Katsnelson, PRB 92, 085419 (2015)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
11 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Point defects in black phosphorus
Carrier mobility in monolayer and bilayer BP
1250
yy
Single-layer
yy
1000
Point Defects
xx
x
yy
Bilayer
n =0.1%
x
yy
Point Defects
n =0.02%
1000
x
xx
n =0.1%
xx
-1 -1
(cm V s )
x
n =0.02%
x
n =0.1%
x
n =0.02%
x
n =0.1%
x
750
2
750
2
-1 -1
(cm V s )
xx
1250
n =0.02%
500
250
0
-50
500
250
-40
-30
-20
-10
n
e
0
(10
10
12
20
30
40
2
cm
)
50
0
-50
-40
-30
-20
-10
n
e
0
(10
10
12
20
30
40
50
2
cm
)
T = 300 K, ne = 1013 cm−2 , nx = 0.1%: µ ≈ 500 cm2 V−1 s−1
S. Yuan, A. Rudenko, M. Katsnelson, PRB 91, 115436 (2015)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
12 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Point defects in black phosphorus
Carrier mobility in monolayer and bilayer BP
1250
yy
Single-layer
yy
1000
Point Defects
xx
x
yy
Bilayer
n =0.1%
x
yy
Point Defects
n =0.02%
1000
x
xx
n =0.1%
xx
-1 -1
(cm V s )
x
n =0.02%
x
n =0.1%
x
n =0.02%
x
n =0.1%
x
750
2
750
2
-1 -1
(cm V s )
xx
1250
n =0.02%
500
250
0
-50
500
250
-40
-30
-20
-10
n
e
0
(10
10
12
20
30
40
2
cm
)
50
0
-50
-40
-30
-20
-10
n
e
0
(10
10
12
20
30
40
50
2
cm
)
T = 300 K, ne = 1013 cm−2 , nx = 0.1%: µ ≈ 500 cm2 V−1 s−1
T = 300 K, ne = 1013 cm−2 , nx = 0.01%: µ ≈ 1000 cm2 V−1 s−1
S. Yuan, A. Rudenko, M. Katsnelson, PRB 91, 115436 (2015)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
12 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Phonon-limited transport: overview
Phonon scattering
1-phonon
processes
in-plane
modes
Alexander Rudenko
flexural
modes
24 Sep 2016 (AMM-2016)
2-phonon
processes
in-plane
modes
flexural
modes
Transport properties of black phosphorus
13 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Phonon-limited transport: overview
Phonon scattering
1-phonon
processes
in-plane
modes
flexural
modes
τ −1 ∼ T /n0
(e.g., 3D Si,Ge)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
2-phonon
processes
in-plane
modes
flexural
modes
τ −1 ∼ T 2 /n1
(e.g., graphene)
Transport properties of black phosphorus
13 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility in black phosphorus: theory
x-component of the conductivity tensor:
e2 X
∂f
x2
σxx =
τxx vk −
2S
∂εk
k
Scattering rate for carriers propagating in x-direction:
P
∂f
eff 2
x 2
x
′
)
−
δ(ε
−
ε
′
k
k
∂εk (vk − vk′ ) h|Vkk′ | i
π kk
1
=
P x2 ∂f τxx
~
−
v
k k
∂εk
Main ingridients:
− Carrier dispersion εk and band velocities vk
eff
− Scattering potentials of in-plane and flex. phonons Vkk
′
− Material parameters: elastic const. and deform. potential
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Carrier dispersion in black phosphorus
0
ε(kx,0)
ε(0,ky)
−0.2
0 0.05 0.1 0.15 0.2 0.25 0.3
0
Wave vector, k (1/Å)
0.1 0.2
DOS (1/eV)
0.12
0
0.1
εF =
0.0
εF =
0.04
0.02
0.2
εF =
0.06
(c)
holes
εF =
0.1
0.08
0.0
q*
0
2e
7e
0.3
0.25
0.2
0.15
0.1
0.05
0
Band energy, ε (eV)
−0.1
−0.15
(a)
Wave vector, kx (1/Å)
holes
−0.05
5e
V
5e
V
V
V
0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wave vector, ky (1/Å)
electrons
• Band dispersion is
not quadratic
ε(kx,0)
ε(0,ky)
0 0.05 0.1 0.15 0.2 0.25 0.3 0
(b)
Wave vector, kx (1/Å)
Band energy, ε (eV)
Low-energy features of the BP electronic spectrum (GW )
Wave vector, k (1/Å)
DOS (1/eV)
0.12
0
0.1
εF =
0.0
εF =
0.04
0.02
0.2
εF =
0.06
(d)
electrons
εF =
0.1
0.08
0.0
q*
0
0.05
2e
7e
0.1
5e
V
5e
V
V
not elliptical
• Density of states is
not a constant
V
0.1
0.15
0.2
Wave vector, ky (1/Å)
• Fermi surface is
0.25
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
15 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Carrier dispersion in black phosphorus
ε(kx,0)
ε(0,ky)
−0.2
0 0.05 0.1 0.15 0.2 0.25 0.3
0
Wave vector, k (1/Å)
0.1 0.2
DOS (1/eV)
0.12
0
0.1
εF =
0.0
εF =
0.04
0.02
0.2
εF =
0.06
(c)
holes
εF =
0.1
0.08
0.0
q*
0
2e
7e
0.3
0.25
0.2
0.15
0.1
0.05
0
Band energy, ε (eV)
−0.1
−0.15
(a)
Wave vector, kx (1/Å)
holes
5e
V
5e
V
V
V
0.05 0.1 0.15 0.2 0.25 0.3 0.35
Wave vector, ky (1/Å)
electrons
• Band dispersion is
not quadratic
ε(kx,0)
ε(0,ky)
0 0.05 0.1 0.15 0.2 0.25 0.3 0
(b)
Wave vector, k (1/Å)
DOS (1/eV)
0.12
0
0.1
εF =
0.0
εF =
0.04
0.02
0.2
εF =
0.06
(d)
electrons
εF =
0.1
0.08
0.0
q*
0
0.05
7e
2e
0.1
5e
V
5e
V
V
• Density of states is
0.25
Approximation for low-energy dispersion
~2 ky2
~2 kx2
+
εk =
2mxE (ε) 2myE (ε)
not elliptical
not a constant
V
0.1
0.15
0.2
Wave vector, ky (1/Å)
• Fermi surface is
Effective mass, m
0
−0.05
Wave vector, kx (1/Å)
Band energy, ε (eV)
Low-energy features of the BP electronic spectrum (GW )
10
9
8
7
6
5
4
3
2
1
holes
mEy
mV
y
0
0.05
0.1 0.15 0.2
Energy, ε (eV)
0.25
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Phonon-limited transport: scattering matrices
Scattering probability (in-plane phonons)
Isotropic case (C11 = C22 )
2
h|V̄qeff |2 i = T Cḡ11
No q-dependence!
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
16 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Phonon-limited transport: scattering matrices
Scattering probability (in-plane phonons)
Isotropic case (C11 = C22 )
2
h|V̄qeff |2 i = T Cḡ11
No q-dependence!
Anisotropic case (C11 6= C22 )
h|V̄qeff |2 i = T
C66 (ḡx2 qx4 +ḡy2 qy4 −2ḡx ḡy qx2 qy2 )+(C22 ḡx2 +C11 ḡy2 −2C12 ḡx ḡy )qx2 qy2
2 )q 2 q 2
C66 (C11 qx4 +C22 qy4 −2C12 qx2 qy2 )+(C11 C22 −C12
x y
Explicit q-dependence!
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Phonon-limited transport: material constants
Determination of relevant parameters (DFT)
(a) P armchair
∆Ē (εij )
|{z}
2h
l
→
elastic
constants
strain
(b) P zigzag
∆Ẽ (Rij )
|{z}
2h
l
→
0.1
Alexander Rudenko
2
u xy
fit
2G xy
2
(a)
0
Y xxu xx
0.2
0.4
0.6
Strain, u (%)
0.8
24 Sep 2016 (AMM-2016)
1
3
DFT
fit
2
1
0
(b)
0
4
κxx=1.29 eV
κyy=5.62 eV
4
flexural
rigidity
yy h 2
q4
y /
5
κij
|{z}
κ
2
DFT
0
Elastic energy density (meV/Å )
0.2
yy u 2
yy
0.3
Yxx=21 J/m2
2
Yyy=91 J/m
2
Gxy=22 J/m
0.4
Y
2
Elastic energy density (meV/Å )
curvature
0.5
Cij
|{z}
4
2
/4
κ xxh q x
0.01 0.02 0.03 0.04 0.05 0.06
2
hq (1/Å)
Transport properties of black phosphorus
17 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Phonon-limited transport: material constants
Determination of relevant parameters (DFT)
(a) P armchair
2h
l
∆Ē (εij )
|{z}
strain
(b) P zigzag
2h
l
∆Ẽ (Rij )
|{z}
curvature
Relaxed
0
EVBM
Alexander Rudenko
→
g̃ij
|{z}
flexural deformation
potential
Evac
WF 0
E
ḡij
|{z}
in-plane deformation
potential
Deformed
Evac
0
CBM
→
EA0
cond. band
val. band
WF1
∆E
CB <
(g e>M 0
0)
∆E VBM>0
h
(g >0)
24 Sep 2016 (AMM-2016)
EA1
cond. band
val. band
1
ECBM
1
EVBM
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility in black phosphorus
Theory:
• Single- and two-phonon processes on equal footing
• Proper account of different anisotropy sources
• Relevant parameters from first-principles
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
19 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility in black phosphorus
Theory:
• Single- and two-phonon processes on equal footing
• Proper account of different anisotropy sources
• Relevant parameters from first-principles
Single-phonon (in-plane)
−1
τ̄xx
=
T ḡx2 m∗
~3 C11
Āxx
|{z}
Anisotropy
factor
Two-phonon (flexural)
−1
τ̃xx
=
√
T 2 g̃x2
ln( eγ) Ãxx
2
|{z}
8π~εF κx
Anisotropy
factor
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
19 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility in black phosphorus
Theory:
• Single- and two-phonon processes on equal footing
• Proper account of different anisotropy sources
• Relevant parameters from first-principles
Single-phonon (in-plane)
−1
τ̄xx
=
h+
e−
T ḡx2 m∗
~3 C11
Āxx
0.48
15.2
Āxx
|{z}
Anisotropy
factor
Āyy
2.93
0.98
Two-phonon (flexural)
−1
τ̃xx
=
√
T 2 g̃x2
ln( eγ) Ãxx
2
|{z}
8π~εF κx
Anisotropy
factor
h+
e−
Ãxx
0.14
0.26
Ãyy
12.6
2.60
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
19 / 24
Introduction Electronic properties Intrinsic transport Conclusion
400
320
T=200K
1&2−phonon
1−phonon
T=300K
240
160
(a)
T=400K
10
20
30
40
50
60
70
Carrier density, n ⋅ 1012(cm−2)
1400
electrons
1200
T=200K
1000 1&2−phonon
1−phonon
800
T=300K
T=400K
600
(b)
Anisotropy, µxx / µyy
holes
480
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
holes
2.2
T=200K
2
T=300K
1.8
T=400K
1.6
(c)
Anisotropy, µxx / µyy
Mobility, µxx (cm2V−1s−1)
Mobility, µxx (cm2V−1s−1)
Intrinsic mobility in black phosphorus
10
20
30
40
6.8
60
70
electrons
0K
6.6
T=20
6.4
0K
T=30
6.2
0K
6
(d)
50
Carrier density, n ⋅ 1012(cm−2)
T=40
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
400
320
T=200K
1&2−phonon
1−phonon
T=300K
240
160
(a)
T=400K
10
20
30
40
50
60
70
Carrier density, n ⋅ 1012(cm−2)
1400
electrons
1200
T=200K
1000 1&2−phonon
1−phonon
800
T=300K
T=400K
600
(b)
Anisotropy, µxx / µyy
holes
480
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
holes
2.2
T=200K
2
• In-plane phonons
T=300K
1.8
dominate at densities
higher than 1013 cm−2
T=400K
1.6
(c)
Anisotropy, µxx / µyy
Mobility, µxx (cm2V−1s−1)
Mobility, µxx (cm2V−1s−1)
Intrinsic mobility in black phosphorus
10
20
30
40
6.8
60
70
electrons
0K
6.6
T=20
6.4
0K
T=30
6.2
0K
6
(d)
50
Carrier density, n ⋅ 1012(cm−2)
T=40
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
400
320
T=200K
1&2−phonon
1−phonon
T=300K
240
160
(a)
T=400K
10
20
30
40
50
60
70
Carrier density, n ⋅ 1012(cm−2)
1400
electrons
1200
T=200K
1000 1&2−phonon
1−phonon
800
T=300K
T=400K
600
(b)
Anisotropy, µxx / µyy
holes
480
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
holes
2.2
T=200K
2
• In-plane phonons
T=300K
1.8
dominate at densities
higher than 1013 cm−2
T=400K
1.6
(c)
Anisotropy, µxx / µyy
Mobility, µxx (cm2V−1s−1)
Mobility, µxx (cm2V−1s−1)
Intrinsic mobility in black phosphorus
10
20
30
40
6.8
60
electrons
0K
T=20
6.6
6.4
70
• Electron and hole
anisotropies are
significantly different
0K
T=30
6.2
0K
6
(d)
50
Carrier density, n ⋅ 1012(cm−2)
T=40
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
20 / 24
Introduction Electronic properties Intrinsic transport Conclusion
400
320
T=200K
1&2−phonon
1−phonon
T=300K
240
160
(a)
T=400K
10
20
30
40
50
60
70
Carrier density, n ⋅ 1012(cm−2)
1400
electrons
1200
T=200K
1000 1&2−phonon
1−phonon
800
T=300K
T=400K
600
(b)
Anisotropy, µxx / µyy
holes
480
10
20
30
40
50
Carrier density, n ⋅ 1012(cm−2)
holes
2.2
T=200K
2
• In-plane phonons
T=300K
1.8
dominate at densities
higher than 1013 cm−2
T=400K
1.6
(c)
Anisotropy, µxx / µyy
Mobility, µxx (cm2V−1s−1)
Mobility, µxx (cm2V−1s−1)
Intrinsic mobility in black phosphorus
10
20
30
40
6.8
60
electrons
0K
T=20
6.6
6.4
0K
T=30
6.2
6
(d)
50
Carrier density, n ⋅ 1012(cm−2)
T
10
• Electron and hole
anisotropies are
significantly different
• At T = 300K upper
limit for mobilities:
250 (700) cm2 V−1 s−1
K
=400
20
70
30
40
50
Carrier density, n ⋅ 1012(cm−2)
A. Rudenko, S. Brener, M. Katsnelson, PRL 116, 246401 (2016)
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility: phosphorene vs. graphene
Why does the scattering in BP different from graphene?
At εF ≫ T :
µxx
T
=α·
µ
exx
εF
Alexander Rudenko
with
24 Sep 2016 (AMM-2016)
α∼
C11
2
κx N(εF )
Transport properties of black phosphorus
21 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility: phosphorene vs. graphene
Why does the scattering in BP different from graphene?
At εF ≫ T :
µxx
T
=α·
µ
exx
εF
α
Alexander Rudenko
with
holes (BP)
armchair
zigzag
0.89
0.37
24 Sep 2016 (AMM-2016)
α∼
C11
2
κx N(εF )
electrons (BP)
armchair
zigzag
0.99
0.17
Transport properties of black phosphorus
21 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility: phosphorene vs. graphene
Why does the scattering in BP different from graphene?
At εF ≫ T :
µxx
T
=α·
µ
exx
εF
α
with
holes (BP)
armchair
zigzag
0.89
0.37
α∼
C11
2
κx N(εF )
electrons (BP)
armchair
zigzag
0.99
0.17
In graphene α ≈ 26 at n = 5· 1013 cm−2
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
21 / 24
Introduction Electronic properties Intrinsic transport Conclusion
Intrinsic mobility: phosphorene vs. graphene
Why does the scattering in BP different from graphene?
At εF ≫ T :
µxx
T
=α·
µ
exx
εF
α
with
holes (BP)
armchair
zigzag
0.89
0.37
α∼
C11
2
κx N(εF )
electrons (BP)
armchair
zigzag
0.99
0.17
In graphene α ≈ 26 at n = 5· 1013 cm−2
Unlike phosphorene, graphene has superior flexibility (small κ)
and extreme in-plane stiffness (large C11 )
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Conclusion
Methodology
• TB model for multilayer BP is constructed
(GW quality at low computational cost)
• Theory of phonon charge carrier scattering
in anisotropic materials is developed
Applications
• Mechanism of phonon scattering in BP is
determined (in-plane phonons)
• Limits on intrinsic and extrinsic carrier
mobility in monolayer BP are estimated
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Acknowledgments
Coworkers
Radboud University (The Netherlands)
Mikhail Katsnelson
Shengjun Yuan
Sergey Brener
Funding
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
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Introduction Electronic properties Intrinsic transport Conclusion
Related publications:
PRB 89, 201408 (2014)
PCCP 17, 15209 (2015)
PRB 91, 115436 (2015)
PRB 92, 085419 (2015)
PRB 93, 085417 (2016)
PRL 116, 246401 (2016)
Thank you!
Alexander Rudenko
24 Sep 2016 (AMM-2016)
Transport properties of black phosphorus
24 / 24