Scattering in graphene associated with charged out-of-plane impurities
Yue Liu1,*, Aditi Goswami1, Feilong Liu 1, Darryl L. Smith1,2, and P. Paul Ruden1,2
1University
2Los
of Minnesota, Minneapolis, Minnesota, 55455, USA
Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
Supplementary material
SI. Graphene Hamiltonian
Neglecting spin-orbit interaction, the low-energy graphene Hamiltonian, 𝐻0 , can be written as
0
𝑘𝑥 − 𝑖𝑘𝑦
𝐻0 = ℏ𝑣𝐹
0
0
[
𝑘𝑥 + 𝑖𝑘𝑦
0
0
0
0
0
0
𝑘𝑥 − 𝑖𝑘𝑦
0
0
𝑘𝑥 + 𝑖𝑘𝑦
0
]
(𝑆. 1)
In this 4x4 matrix, rows/columns 1,2 refer to spin up and rows/columns 3,4 refer to spin down.
⃗ ) = ℏ𝑣𝑓 |𝑘
⃗ | can be written
Diagonalizing 𝐻0 , two degenerated states with energy eigenvalue 𝜀𝑐 (𝑘
as:
1
1 𝑒 𝑖𝜃𝑘⃗
⃗ ⟩ = ( 𝑖𝜃 ) 𝑒 𝑖𝑘⃗∙𝑟 ,
|𝑐1, 𝑘
2 𝑖𝑒 𝑘⃗
𝑖𝑒 2𝑖𝜃𝑘⃗
1
𝑖𝜃𝑘
⃗
1
⃗ ⟩ = ( 𝑒 𝑖𝜃 ) 𝑒 𝑖𝑘⃗∙𝑟
|𝑐2, 𝑘
2 −𝑖𝑒 𝑘⃗
−𝑖𝑒 2𝑖𝜃𝑘⃗
⃗ ) = −ℏ𝑣𝑓 |𝑘
⃗ | for holes are:
The other two degenerated states with energy eigenvalue 𝜀𝑣 (𝑘
1
(S. 2a)
1
1 −𝑒 𝑖𝜃𝑘⃗
⃗⟩= (
|𝑣1, 𝑘
) 𝑒 𝑖𝑘⃗∙𝑟 ,
2 −𝑖𝑒 𝑖𝜃𝑘⃗
𝑖𝑒 2𝑖𝜃𝑘⃗
1
1 −𝑒 𝑖𝜃𝑘⃗
⃗ ⟩ = ( 𝑖𝜃 ) 𝑒 𝑖𝑘⃗∙𝑟
|𝑣2, 𝑘
2 𝑖𝑒 𝑘⃗
−𝑖𝑒 2𝑖𝜃𝑘⃗
(S. 2b)
Considering the Rashba interaction term 𝑅0 , the system symmetry is broken, and the total
⃗ ⟩ and |𝑣±, 𝑘
⃗ ⟩ are given in the equations (2)
graphene Hamiltonian 𝐻 and four eigenstates |𝑐±, 𝑘
and (3) of the main text. These eigenstates are linear combinations of the eigenstates of 𝐻0 :
⃗ ⟩, |𝑐2, 𝑘
⃗ ⟩, |𝑣1, 𝑘
⃗ ⟩ and |𝑣2, 𝑘
⃗⟩
|𝑐1, 𝑘
⃗⟩=
|𝑐+, 𝑘
1
√2 ∙ √1 + 𝑏𝑐+
⃗⟩=
|𝑐−, 𝑘
⃗⟩=
|𝑣+, 𝑘
1
(𝑆. 3𝑏)
⃗ ⟩ + (1 − 𝑏𝑣+ )|𝑣1, 𝑘
⃗ ⟩]
[(1 + 𝑏𝑣+ )|𝑐1, 𝑘
(𝑆. 3𝑐)
⃗ ⟩ + (1 − 𝑏𝑣− )|𝑣2, 𝑘
⃗ ⟩]
[(1 + 𝑏𝑣− )|𝑐2, 𝑘
(𝑆. 3𝑑)
2
1
√2 ∙ √1 + 𝑏𝑣−
⃗ ⟩ + (1 − 𝑏𝑐− )|𝑣2, 𝑘
⃗ ⟩]
[(1 + 𝑏𝑐− )|𝑐2, 𝑘
2
√2 ∙ √1 + 𝑏𝑣+
⃗⟩=
|𝑣−, 𝑘
(𝑆. 3𝑎)
2
1
√2 ∙ √1 + 𝑏𝑐−
⃗ ⟩ + (1 − 𝑏𝑐+ )|𝑣1, 𝑘
⃗ ⟩]
[(1 + 𝑏𝑐+ )|𝑐1, 𝑘
2
bc± and bv± are defined in the main text.
SII. Screened potential calculation
Poisson’s equation for the graphene model in FIG.1 can be written as,
𝜕 2𝑈 𝜕 2𝑈 𝜕 2𝑈
+
+
= −4𝜋𝑒 2 [𝛿(𝑥, 𝑦, 𝑧 + 𝑧0 ) − 𝑛(𝑥, 𝑦)𝛿(𝑧)]
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2
2
(𝑆. 4)
Here, 𝑛 is the induced charge density in the graphene layer, calculated in (T = 0) Thomas-Fermi
approximation:
𝐸𝑓 +𝑈(𝑟)
𝑛(𝑟) =
∫
𝐸𝑓
𝑔(𝜀) =
2
𝜋(ℏ𝑣𝑓 )
2
2
1
1
1
𝑔(𝜀) 𝑑𝜀 = 𝑎 (𝐸𝑓 + 𝑈(𝑟)) − 𝑎𝐸𝑓 2 = 𝑎𝐸𝑓 𝑈(𝑟) + 𝑎𝑈 2
2
2
2
|𝜀|, is the graphene density of states function, and 𝑎 =
2
𝜋(ℏ𝑣𝑓 )
2
(𝑆. 5)
.
Taking a 2D- Fourier transform in the xy plane and using the cylindrical symmetry,
̂ (𝑞, 𝑧) = ∫ 𝑑 2 𝑟 𝑒 𝑖𝑞⃗𝑟 𝑈(𝑟, 𝑧)
𝑈
1
̂ (𝑞) + 𝑎 [∫ 𝑑 2 𝑟 𝑒 𝑖𝑞⃗𝑟 𝑈 2 (𝑟)]
𝑛̂(𝑞) = 𝑎𝐸𝑓 𝑈
2
(𝑆. 6𝑎)
(𝑆. 6𝑏)
̂ (𝑞, 0) = 𝑈
̂ (𝑞) we find:
Solving for 𝑈
̂ (𝑞) =
𝑈
2𝜋𝑒 2 𝑒 −𝑞𝑧0
𝑞 + 2𝜋𝑒 2 𝑇̂(𝑞)
(𝑆. 7)
̂ (𝑞) + 1 𝑎[∫ 𝑑 2 𝑟 𝑒 𝑖𝑞⃗𝑟 𝑈 2 (𝑟)]
𝑛̂(𝑞) 𝑎𝐸𝑓 𝑈
2
𝑇̂(𝑞) =
=
̂ (𝑞)
̂ (𝑞)
𝑈
𝑈
1
2
̂
̂
1 (2𝜋)2 ∫ 𝑑 𝑞′ 𝑈(𝑞′)𝑈(|𝑞 − 𝑞 ′|)
= 𝑎𝐸𝑓 + 𝑎
̂ (𝑞)
2
𝑈
(𝑆. 8)
S.7 and S.8 may be solved iteratively.
̂ (𝑞) and U(r)
Taking the inverse Fourier transform of the converged result, 𝑈(𝑟) is obtained. 𝑈
are displayed in FIG.S1 and FIG.S2.
3
FIG. S1.
̂ (𝑞) for different Fermi energies; impurity distances are 𝑧0 = 2.5𝑛𝑚 (solid), 𝑧0 = 5𝑛𝑚 (dashed) and
𝑈
𝑧0 = 10𝑛𝑚 (dotted).
4
FIG. S2. 𝑈(𝑟) for different Fermi energies; impurity distances are 𝑧0 = 2.5𝑛𝑚 (solid), 𝑧0 = 5𝑛𝑚 (dashed) and 𝑧0 =
10𝑛𝑚 (dotted).
SIII. Potential and Rashba scatttering cross-sections
⃗ to 𝑘
⃗′
For the states represented in equation (S.2a) and (S.2b), the scattering matrices from 𝑘
⃗ ′|𝑈(𝑟)|𝑐1, 𝑘
⃗ ⟩ as an
under the action of the potential 𝑈(𝑟) are calculated as follows (taking ⟨𝑐1, 𝑘
example)
5
1
⃗ ′|𝑈(𝑟)|𝑐1, 𝑘
⃗ ⟩ = ⟨(1
⟨𝑐1, 𝑘
4
1
𝑒 −𝑖𝜃𝑘⃗′
−𝑖𝑒 −𝑖𝜃𝑘⃗′
−𝑖𝑒 −2𝑖𝜃𝑘⃗′ )𝑒
⃗ ′∙𝑟
−𝑖𝑘
𝑖𝜃⃗
𝑒 𝑘
𝑖𝜃
𝑖𝑒 ⃗𝑘
2𝑖𝜃⃗
𝑖𝑒 𝑘
|𝑈(𝑟)| (
) 𝑒 𝑖𝑘⃗∙𝑟 ⟩
1
= [⟨𝑒 −𝑖𝑘⃗′∙𝑟 |𝑈(𝑟)|𝑒 𝑖𝑘⃗∙𝑟 ⟩ + ⟨𝑒 −𝑖𝜃𝑘⃗′ ∙ 𝑒 −𝑖𝑘⃗′∙𝑟 |𝑈(𝑟)|𝑒 𝑖𝜃𝑘⃗ ∙ 𝑒 𝑖𝑘⃗∙𝑟 ⟩
4
+ ⟨𝑒 −𝑖𝜃𝑘⃗′ ∙ 𝑒 −𝑖𝑘⃗′∙𝑟 |𝑈(𝑟)|𝑒 𝑖𝜃𝑘⃗ ∙ 𝑒 𝑖𝑘⃗∙𝑟 ⟩ + ⟨𝑒 −2𝑖𝜃𝑘⃗′ ∙ 𝑒 −𝑖𝑘⃗′∙𝑟 |𝑈(𝑟)|𝑒 2𝑖𝜃𝑘⃗ ∙ 𝑒 𝑖𝑘⃗∙𝑟 ⟩]
1
1
2
̂ (𝑞)
= (1 + 2𝑒 −𝑖𝜃 + 𝑒 −2𝑖𝜃 ) ∫ 𝑑 2 𝑟 𝑒 −𝑖𝑞⃗∙𝑟 𝑈(𝑟) = (1 + 𝑒 −𝑖𝜃 ) 𝑈
4
4
(𝑆. 9)
By a similar method, it can be shown that,
1
2
̂ (𝑞)
(1 + 𝑒 −𝑖𝜃 ) 𝑈
4
(𝑆. 10𝑎)
1
⃗ ′|𝑈(𝑟)|𝑐1, 𝑘
⃗ ⟩ = ⟨𝑐1, 𝑘
⃗ ′|𝑈(𝑟)|𝑣1, 𝑘
⃗ ⟩ = (1 − 𝑒 −𝑖𝜃 )2 𝑈
̂ (𝑞)
⟨𝑣1, 𝑘
4
(𝑆. 10𝑏)
⃗ ′|𝑈(𝑟)|𝑣1, 𝑘
⃗ ⟩ = ⟨𝑐1, 𝑘
⃗ ′|𝑈(𝑟)|𝑐1, 𝑘
⃗⟩=
⟨𝑣1, 𝑘
For eigenstates of Hamiltonian 𝐻 represented in equation (3a)-(3b) of the main text, the
scattering matrix elements with respect to potential 𝑈(𝑟) for intra-band scattering can be written
as,
⃗ ′ |𝑈(𝑟)|𝑐+, 𝑘
⃗⟩
⟨𝑐+, 𝑘
=
=
1
2(1 + 𝑏𝑐+ 2 )
1
2(1 + 𝑏𝑐+ 2 )
⃗ ′ | + (1 − 𝑏𝑐+ )⟨𝑣1, 𝑘
⃗ ′ ||𝑈(𝑟)|(1 + 𝑏𝑐+ )|𝑐1, 𝑘
⃗ ⟩+(1 − 𝑏𝑐+ )|𝑣1, 𝑘
⃗ ⟩⟩
⟨(1 + 𝑏𝑐+ )⟨𝑐1, 𝑘
⃗ ′ |𝑈(𝑟)|𝑐1, 𝑘
⃗⟩
[(1 + 𝑏𝑐+ )2 ⟨𝑐1, 𝑘
⃗ ′|𝑈(𝑟)|𝑣1, 𝑘
⃗ ⟩ + ⟨𝑣1, 𝑘
⃗ ′|𝑈(𝑟)|𝑐1, 𝑘
⃗ ⟩)
+ (1 − 𝑏𝑐+ 2 )(⟨𝑐1, 𝑘
⃗ |𝑈(𝑟)|𝑣1, 𝑘
⃗ ⟩]
+ (1 − 𝑏𝑐+ )2 ⟨𝑣1, 𝑘
(𝑆. 11)
6
Using in (𝑆. 10𝑎) and (𝑆. 10𝑏),
⃗ ′ |𝑈(𝑟)|𝑐+, 𝑘
⃗⟩=
⟨𝑐+, 𝑘
̂ (𝑞)
𝑈
2
2
4(1 + 𝑏𝑐+ )
2
[(1 + 𝑒 −𝑖𝜃 ) (1 + 𝑏𝑐+ 2 ) + (1 − 𝑒 −𝑖𝜃 ) (1 − 𝑏𝑐+ 2 )] (𝑆. 12)
Similarly, for inter-band scattering:
̂ (𝑞)
𝑈
⃗ ′ |𝑈(𝑟)|𝑐+, 𝑘
⃗⟩=
⟨𝑐−, 𝑘
2
(1 − 𝑒 −2𝑖𝜃 )
(𝑆. 13)
2
2√(1 + 𝑏𝑐− )(1 + 𝑏𝑐+ )
This leads directly to equations (4a) and (4b) in the main text.
For impurity induced Rashba interaction 𝑅𝑖𝑚𝑝 (𝑟), scattering matrixes such as
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑐1, 𝑘
⃗ ⟩ can be calculated as,
⟨𝑐1, 𝑘
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑐1, 𝑘
⃗⟩
⟨𝑐1, 𝑘
1
= ⟨(1 𝑒 −𝑖𝜃⃗𝑘′
4
=
−𝑖𝑒 −𝑖𝜃⃗𝑘′
0
⃗ ′∙𝑟 0
−𝑖𝑘
|[
−𝑖𝑒 −2𝑖𝜃⃗𝑘′ )𝑒
0
0
0
0
0
0
1
𝑖𝜃⃗
−𝑖𝑅𝑖𝑚𝑝 (𝑟) 0
𝑒 𝑘
] | ( 𝑖𝜃⃗ ) 𝑒 𝑖𝑘⃗∙𝑟 ⟩
𝑖𝑒 𝑘
𝑖𝑅𝑖𝑚𝑝 (𝑟)
0
0
2𝑖𝜃
𝑖𝑒 ⃗𝑘
0
0
0
1
𝑅̂ (𝑞)𝑒 −𝑖𝜃
2 𝑖𝑚𝑝
(𝑆. 14)
Consequently,
1
𝑅̂ (𝑞)𝑒 −𝑖𝜃
2 𝑖𝑚𝑝
(𝑆. 15𝑎)
1
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑐1, 𝑘⟩ = ⟨𝑐1, 𝑘
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑣1, 𝑘⟩ = − 𝑅̂𝑖𝑚𝑝 (𝑞)𝑒 −𝑖𝜃
⟨𝑣1, 𝑘
2
(𝑆. 15𝑏)
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑐1, 𝑘
⃗ ⟩ = ⟨𝑣1, 𝑘
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑣1, 𝑘
⃗⟩=
⟨𝑐1, 𝑘
7
⃗ ′ |𝑅𝑖𝑚𝑝 (𝑟)|𝑐+, 𝑘
⃗ ⟩ can be related to equation (𝑆. 15𝑎) and (𝑆. 15𝑏) by
Similarly, ⟨𝑐+, 𝑘
⃗ ′ |𝑅𝑖𝑚𝑝 (𝑟)|𝑐+, 𝑘
⃗⟩
⟨𝑐+, 𝑘
=
=
1
2(1 + 𝑏𝑐+ 2 )
𝑏𝑐+ 2
1 + 𝑏𝑐+ 2
⟨(1 + 𝑏𝑐+ )⟨𝑐1, 𝑘′| + (1 − 𝑏𝑐+ )⟨𝑣1, 𝑘′||𝑅𝑖𝑚𝑝 (𝑟)|(1 + 𝑏𝑐+ )|𝑐1, 𝑘⟩+(1 − 𝑏𝑐+ )|𝑣1, 𝑘⟩⟩
𝑅̂𝑖𝑚𝑝 (𝑞)𝑒 −𝑖𝜃
(𝑆. 16)
⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑐+, 𝑘
⃗ ⟩ one obtains,
For ⟨𝑐−, 𝑘
⟨𝑐−, 𝑘⃗ ′|𝑅𝑖𝑚𝑝 (𝑟)|𝑐+, 𝑘⃗⟩
1
=
2
⟨(1 + 𝑏𝑐− )⟨𝑐2, 𝑘′| + (1 − 𝑏𝑐− )⟨𝑣2, 𝑘′||𝑅𝑖𝑚𝑝 (𝑟)|(1 + 𝑏𝑐+ )|𝑐1, 𝑘⟩+(1 − 𝑏𝑐+ )|𝑣1, 𝑘⟩⟩
2
2√(1 + 𝑏𝑐− )(1 + 𝑏𝑐+ )
=0
(𝑆. 17)
Hence the Rashba scattering cross-sections in equations (6a) and (6b) of the main text are
obtained.
SIV. Consideration of the optical theorem
Calculations of potential and Rashba scattering cross-sections are to lowest order Born
approximation. It is useful to explore intra-band scattering (c+ to c+) on the basis of the optical
theorem,
𝜎=√
8𝜋
𝐼𝑚[𝑓(0)]
𝑘
(𝑆. 18)
where 𝑓(𝜃) is the scattering amplitude and  is the total scattering cross section. In first order
Born approximation, the scattering amplitude is real. However,  may be calculated from the
8
differential scattering cross section as presented in the main text. 𝐼𝑚[𝑓(0)] may then be
obtained. We may check the accuracy of of the Born approximation by calculating 𝑓 𝑏𝑜𝑟𝑛 (0)/
𝐼𝑚[𝑓(0)].
𝑓 𝑏𝑜𝑟𝑛 (0)/𝐼𝑚[𝑓(0)] for the intra band scattering processes is tabulated in Tables 1 and 2 for
different Fermi energies and impurity distances.
𝐸𝑓 = 25𝑚𝑒𝑉
𝐸𝑓 = 50𝑚𝑒𝑉
𝐸𝑓 = 100𝑚𝑒𝑉
𝐸𝑓 = 200𝑚𝑒𝑉
𝑧0 = 2.5𝑛𝑚
13.85
27.67
31.50
42.60
𝑧0 = 5𝑛𝑚
13.84
31.49
42.61
74.31
𝑧0 = 10𝑛𝑚
15.75
21.30
34.46
63.01
Table. 1. 𝑓 𝑏𝑜𝑟𝑛 (0)/𝐼𝑚[𝑓(0)] for intra-band potential scattering.
𝐸𝑓 = 25𝑚𝑒𝑉
𝐸𝑓 = 50𝑚𝑒𝑉
𝐸𝑓 = 100𝑚𝑒𝑉
𝐸𝑓 = 200𝑚𝑒𝑉
𝑧0 = 2.5𝑛𝑚
7.01 × 105
4.37 × 105
3.23 × 105
2.98 × 105
𝑧0 = 5𝑛𝑚
8.73 × 105
6.46 × 105
5.96 × 105
6.13 × 105
𝑧0 = 10𝑛𝑚
1.29 × 106
1.19 × 106
1.23 × 106
1.20 × 106
Table. 2. 𝑓 𝑏𝑜𝑟𝑛 (0)/𝐼𝑚[𝑓(0)] for intra-band Rashba scattering.
9
𝑓 𝑏𝑜𝑟𝑛 (0)/𝐼𝑚[𝑓(0)] is greater than 10 for potential scattering and greater than 105 for Rashba
scattering.
10