• 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