2019 defects in 2d semiconductors - rensselaer polytechnic …homepages.rpi.edu/~zhangs9/files/2019...
TRANSCRIPT
• Charged defects in monolayer (ML)
materials using jellium approximation, e.g.,
BN & MoS2
• Defect-bound ionization of deep levels &
carrier transport, e.g., in ML MoS2
• Charged defects in multi-ML materials, e.g.,
black P.1
Outline
D. Wang, et al., PRL 114, 196801 (2015)
D. Wang, et al., PRB 96, 155424 (2017)
D. Wang, et al., npj Comp. Mater. 5, 8 (2019)
2
Questions for defects in 2D materials
Are the defects shallow or deep?
Can the defects be used in some way
for electronic applications?
How is the physics different from 3D?
…
3
2D:
Jellium background
mostly outside the
layer; results are
meaningful only when
background charge
goes to zero.
3D:
Jellium
background
inside solid
Problem with calculating charged defects in 2D
Consider electron ionization from a point defect
4
The electrostatic catastrophe
As Lz increases,
Coulomb energy
of a single cell
diverges linearly,
due to attraction
between charged plane and its jellium background charge.
5
Defect ionization energy (𝐼𝐸) also diverges
∆𝐻𝑓 𝑞 = ±, 𝛼 = ∆𝐻𝑓 0, 𝛼 ± [𝜀𝐹 − 𝜀( ± 0)]:
∆𝐻𝑓 𝑞 = ±, 𝛼 → ∞ implies 𝜀( ± 0) → ∞
Defect ionization energy (𝐼𝐸) is defined as
𝐼𝐸(donor)= 𝜀𝐶𝐵𝑀 − 𝜀 + 0 ;
𝐼𝐸(acceptor)= 𝜀 −/0 − 𝜀𝑉𝐵𝑀.
Hence, they diverge too
Physically, defect ionization energy cannot diverge, as
it determines the electrical behavior of 2D systems.
Wang, et al., PRL 114, 196801 (2015)
6
The physically-correct asymptotic behavior of 𝐼𝐸
Consider 2D as a special case of 3D with 𝐿𝑥 = 𝐿𝑦 = 𝐿𝑠
The true ionization energy 𝐼𝐸0 is given by the “dilute limit” provided
that 𝐿𝑧 = 𝑐𝑜𝑛𝑠𝑡 ∙ 𝐿𝑠. Namely, 𝐼𝐸 𝐿𝑠 𝐿𝑠→∞ = 𝐼𝐸0
If 𝐿𝑧 ≫ 𝐿𝑠 and 𝐿𝑧/𝐿𝑠≠ 𝑐𝑜𝑛𝑠𝑡, the problem may be treated as a
uniform charged plane in a compensating background, 𝐼𝐸 𝐿𝑠, 𝐿𝑧 =
𝐼𝐸0 + 𝑐−2,1𝐿𝑧
𝐿𝑠2, where 𝑐−2,1 =
𝑞2
12 3𝜖0
If the charge is not uniform, however, there will be a correction
term, 𝐼𝐸 𝐿𝑠, 𝐿𝑧 =𝑐−1,0
𝐿𝑠+ 𝐼𝐸0 + 𝑐−2,1
𝐿𝑧
𝐿𝑠2. Phys. Rev. E 86, 046702 (2012)
Numerical calculations show an accuracy up to the third decimal point for 𝒄−𝟐,𝟏
More rigorous solution: expansion of 𝐼𝐸 𝐿𝑠, 𝐿𝑧
Strategy: determine non-zero 𝑐𝑖,𝑗 by taking three physical
limits for which 𝐼𝐸(𝐿𝑠, 𝐿𝑧) is known analytically: (1) fix 𝐿𝑠,
let 𝐿𝑧 → ∞; (2) fix 𝐿𝑧, let 𝐿𝑆 → ∞; and (3) let 𝐿𝑠 = 𝐿𝑧 → ∞7
Wang, et al., PRL 114, 196801 (2015)
8
Obtaining 𝐼𝐸0 using asymptotic expression
1𝐿𝑠,12
𝐼𝐸𝑆,𝐿𝑧
𝐿𝑧0
Incr
easi
ng 𝐿𝑠
1𝐿𝑠,22
1𝐿𝑠,32
Calculate 𝐼𝐸 𝐿𝑠 , 𝐿𝑧 using large enough 𝑳𝒔 and 𝑳𝒛
Define 𝐼𝐸 = 𝐼𝐸 𝐿𝑠 , 𝐿𝑧 −𝑐−1,0
𝐿𝑠= 𝑰𝑬𝟎 + 𝒄−𝟐,𝟏
𝑳𝒛
𝑳𝒔𝟐 and fit the
results for different 𝐿𝑠
𝐼𝐸0 1 ∞2= 0
Intercept of
the lines with
different 𝐿𝑠’s
yields 𝑰𝑬𝟎.
Wang, et al., PRL 114, 196801 (2015)
9
Application to ML BN
Standard results
depend on the
supercell size
Using 𝐼𝐸 = 𝐼𝐸0 +
𝑐−2,1𝐿𝑧
𝐿𝑠2, easily get
𝐼𝐸0 from 𝐼𝐸
intercept with 𝑦
axis
Defect levels are exceptionally deep, but consistent with Komsa*.
* PRL110, 095505, (2013); PRX4, 031044 (2014)
𝑆 = 𝐿𝑠2
10
Alternate way to view IE in ML BN: the CB case
12 points from first-
principles calculations
Fit to 𝐼𝐸 𝐿𝑠, 𝐿𝑧 =
𝑐−1,0
𝐿𝑠+ 𝐼𝐸0 + 𝑐−2,1
𝐿𝑧
𝐿𝑠2
with only two
parameters 𝐼𝐸0 and
𝑐−1,0
Wang, et al., PRL 114, 196801 (2015)
IE0 here is within 100 meV of brute-force extrapolation using
cubic supercells up to 19x19x19 (inset).
11
Monolayer MoS2
Similar to BN, good linear fits are also obtained.
Wang, et al., npjComp. Mater. 5, 8 (2019)
12
Among native
defects, 𝑉𝑆 is most
easy to form
Extrinsic dopants
Nb and Re also have
low formation
energies
DOS show shallow
donor/acceptor
levels for Re/Nb,
which are, however,
at variance with
calculated defect
transition energies.
Density of states and formation energy of defects
* red lines are impurity DOS x 10
13
What do we know from experiments?
Measured carrier mobilities are noticeably smaller than theoretical
band edge mobilities, which calls for a new theory to explain.
https://pubs.rsc.org/en/content/articlelanding/2019/nh/c8nh00150b#!divAbstract
(theory)
Defect-bound ionization: 𝐼𝐸1 versus 𝐼𝐸∞
When a donor is fully ionized (𝐼𝐸∞ = 𝐼𝐸0), its electron is fully delocalized.
In the extreme case, the electron becomes a plane wave → jellium model
When the donor is not fully ionized, the electron is bounded. To calculate
the lowest-energy defect-bound state (𝐼𝐸1), we use the constraint DFT,
which is identical to the approach recently proposed by Pantelides et al.
We define 𝐸𝑑𝑏 the ionization
energy of the defect-bound band
edge state (𝐼𝐸1) to vacuum.
Then 𝐼𝐸∞ = 𝐼𝐸1 + 𝐸𝑑𝑏
In 3D, 𝐸𝑑𝑏 is negligible. In 2D,
however, 𝐸𝑑𝑏 can be quite
substantial.
14Note: hydrogenic model is not expected to work; used here only for illustration purpose
15
𝐸𝑑𝑏 in defect transition energy scheme
𝐼𝐸1𝐼𝐸∞
𝐸𝑑𝑏
𝐼𝐸∞
Acceptors
Donors
(0/-1)
(+1/0)
Similar to hBN, full ionization of any of the defect is difficult
𝐼𝐸1 is noticeably smaller than 𝐼𝐸∞, which can be attributed to the sizable
𝐸𝑑𝑏, which can also be viewed as exciton binding energy at the defect.
Wang, et al., npj Comp. Mater. 5, 8 (2019)
16
Localization of defect states
Small cell = 7x7 147-atom cell; large cell = 13x13 507-atom cell
Density contour normalized to hole density: large cell = 0.3 × small cell
A larger 𝑬𝒅𝒃 correlates with a higher degree of hole localization.
𝐸𝑑𝑏
Wang, et al., npj Comp. Mater. 5, 8 (2019)
17
Re donor band & wavefunction overlap
Re density is at
5x1012 cm-2
Noticeable
dispersion of donor
band with larger 𝜇
Noticeable overlap
between defect
wavefunctions→
transport within
defect band
For real defects, one needs evaluate the critical defect density at
percolation threshold. Wang, et al., npj Comp. Mater. 5, 8 (2019)
18
More general situation when 𝑑0 is finite & 𝐿𝑥 ≠ 𝐿𝑦
Yellow star = defect
(a) – the 𝐿𝑧 ≫ 𝐿𝑥 𝐿𝑦 limit
(b) – the 𝐿𝑥 𝐿𝑦 ≫ 𝐿𝑧 limit.
Wang, et al., PRB 96, 155424 (2017)
19
For 𝐿𝑦 = 𝐿𝑥 = 𝐿𝑠, we had
𝐼𝐸 𝐿𝑠 , 𝐿𝑧 =𝑐−1,0𝐿𝑠
+ 𝐼𝐸0 + 𝑐−2,1𝐿𝑧
𝐿𝑠2
For 𝐿𝑦 = 𝛾𝐿𝑥, we have, instead
𝐼𝐸 𝐿𝑥 , 𝐿𝑧 =𝒄−𝟐,𝟎
𝑳𝒙𝟐+𝑐−1,0𝐿𝑥
+ 𝐼𝐸0 + 𝑐−2,1𝐿𝑧
𝐿𝑥2
= 𝐼𝐸0 +𝑐−1,0𝐿𝑥
+ 𝑐−2,1𝐿𝑧′
𝐿𝑥2
where 𝐿𝑧′ = 𝐿𝑧 − 4𝑑0 − 2𝑑0 1 − 𝜀⊥
−1 with 𝜀⊥ = out-of-
plane relative dielectric constant.
The math
Wang, et al., PRB 96, 155424 (2017)
20
Defects in ML-BP: VP, Pi, & XP (X = O, S, Se, and Te)
𝐸𝑔𝑃𝐵𝐸: 0.91 eV
All defects are deep with donor levels near VBM, instead of CBM.
At 𝑉𝑃0, 3 second-
neighbor P atoms
form (trimer)
bonds
Substitutional
impurities have
noticeably small
formation energies
(negative for 𝑂𝑃)
21
Dependence on the layer thickness: TeP in BPs
Critical difference can already be seen at bilayer, which represents
a 36% reduction in 𝐼𝐸0.
Technical: 𝐿𝑧𝑇
stands for 𝐿𝑧 for
which 𝐿𝑧−1 and
𝐿𝑧−2 terms in the
expansion can be
ignored
𝐼𝐸0 monotonically
decreases with
layer thickness
Wang, et al., PRB 96, 155424 (2017)
22
Charge localization & depth of defect level: TeP
Charge
localization in
the 𝑧-direction
strongly
correlates with
the depth of the
defect levels
… but to a lesser
degree in either
𝑥- or 𝑦-
direction.
• Challenges: 2D appears to have universally
very-deep defect transition energies
• Opportunities: utilizing the defect-bound
1st excited state or few-layer 2D structures
for carrier transport
• The true difference between 2D and 3D is
in the reduced dielectric screening.23
Challenges & opportunities