Numerical study of exciton states of core‐shell CdTe/CdS nanotetrapodsby using COMSOL Multiphysics

Yuanzhao YAO，Kazuaki SAKODA

Quantum Dot Research Center, National Institute for Materials Science

Graduate School of Pure and Applied Sciences, University of Tsukuba

Colloidal quantum dots (QDs)

3D(Bulk)

2D(Quantum Well)

1D(Quantum Wire)

0D(Quantum Dot)

Five different QD solutions are shown excited with the same long-wavelength UV lamp; the size of the nanocrystaldetermines the color.(from HP of “invitrogen” )

Colloidal QDs are synthesized from 

precursor compounds 

dissolved in solutions. 

(Chemical processes)

Application of colloidal QD

Colloidal QDlight‐emitting device pixelsP.O.Anikeeva, et al. (Nano Lett.,9,2532,2009)

• Infrared detector , sensor• QD electroluminescence device• solar cell• luminescent marker

CdSe QDs are injected into a mouse, and fluoresce under UV‐light. Mark the location of cancer tumour.(from National Geographic)

Shape control of colloidal QD

SphericalQD

nanorod nanotetrapod

Colloidal QD

Proposed model of a CdTe tetrapod

L. Manna, et al. Nature materials, vol.2, 382 (2003)

L. Choi, et al., Annu. Rev. Phys. Chem. 61, 369, 2010

CdS

t=CdS shell thickness

CdTeband diagram

R. B. Vasiliev, et al. Mendeleev Commun. 19, 128 (2009)

Type I

Type II

CdTe/CdS core-shell tetrapods

CB

VBE

nerg

y

e

hshell

• With continuously grown CdS shell on CdTe tetrapod , features of type II heterostructures were observed in experiment, e.g. featureless absorption tail @ t=1.2 nm

• Study the influence of CdS shell on the exciton state , consequently the optical properties of core‐shell tetrapod  

CdTe(ZB)Core

CdTe(WZ)Arm

CdS shell t

cross-section of one branch

e

h

nice tool for modeling QD 

with complicated geometry

Theoretical model

(1)Single particle Schrodinger equation (Effective‐mass approximation)Solved with finite element method by using COMSOL software

(2)Two‐body Schrodinger equationSolved with configuration interaction method

i= e or h

is the envelope function and       is the atomic wave function

3D model of CdTe/CdScore‐shell tetrapod

Consider the lowest 20 electron and 20 hole states, whose wave functions only have A1 or T2 symmetry

Same method as: K. Sakoda et al., Opt. Mat. Express 1, 379 (2011).

Lowest electron state(e1) and highest hole state(h1) wave function distribution

t=0(nm) t=0.1 t=0.2 t=0.3 t=0.6 t=0.9 t=1.2

(e1)

(h1)

0.0 0.4 0.8 1.20.4

0.6

0.8

1.0

O

verla

p in

tegr

al

Shell thickness (nm)

Single-particle state e1&h1 overlap integral

e‐h NOT totally separated

type II heterostructureNOT apparent

Shell thickness dependence of exciton energy with A1 and T2 symmetry

0.0 0.2 0.4 0.6 0.8 1.0 1.21.61.71.81.92.02.12.22.3

-0.01

0.00

0.01

0.02

0.03

Exci

ton

ener

gy (e

V)

shell thickness (nm)

A1 T2-1 T2-2 T2-3

Energy difference (eV)

Analytical calculation (1)

two‐body matrix element

In which matrix element

Constructed electron wave function, combination of 4 independent wave function on each branch

@ t=1.2 nm, the order of lowest 4 exciton states NOT change.

Safe to choose only lowest 4 pair states for analytical calculation. (e1h1, e2h1, e3h1, e4h1)

(1)

(2)

(3)

(4)

(5)

(6)

direct Coulomb exchange interaction

1 state

3 degenerated states

t=1.2

Analytical calculation (2)

Diagonal matrix element Off-diagonal matrix element(A) Coulomb integral

same value for 4 diagonal elements

(B) exchange interaction integral  (e1h1)

exchange interaction integral  (e2h1, e3h1, e4h1)

diagonal element of e1h1(A1) is larger than other three(T2)

(A)direct Coulomb integral

(B) exchange interaction integral

All off‐diagonal elements for exchange interaction integral are zero

All off‐diagonal elements for direct Coulomb integral are zero

Conclusion of analytical calculation:The symmetry of lowest exciton state (t=1.2 nm) is T2

740 760 780 800 8200.00

0.05

0.10

0.15

0.20

0.25

Osc

illat

or st

reng

th (a

rb. u

nit)

Wavelength (nm)740 750 760 770 780 790

0.000.030.060.090.120.150.18

Osc

illat

or st

reng

th (a

rb. u

nit)

Wavelength (nm)

740 745 750 755 760 765 7700.00

0.05

0.10

0.15

0.20

0.25

Osc

illat

or st

reng

th (a

rb. u

nit)

Wavelength (nm)

t=1.2

Symmetry break in core-shell tetrapod

Modification: one thicker armModification: one Arm with thicker shell

For the imperfect cs‐tetrapod, oscillator strength of the lowest‐energy exciton state is NOT zero

t=1.2 t=1.2

Conclusion• The electronic states of core-shell tetrapod with various shell

thickness are calculated. Lowest 20 electron and hole wavefunctions have A1 or T2 symmetry.

• At t=1.2 nm, the carriers separation is not serious, core-shelltetrapod is not apparent type II heterostructure.

• Exciton states were investigated as a function of t. For large t,the lowest exciton state has T2 symmetry, which implysnonluminescence in emission spectrum.

• Core-shell tetrapod with broken symmetry shows non-zerooscillator strength for lowest exciton state.

13

Thank you for your attention!