Multiexciton Generation in Seeded Nanorods - The Journal of Physical

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Multixciton Generation in Seeded Nanorods Hagai Eshet, Roi Baer, Daniel Neuhauser, and Eran Rabani J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/jz5010279 • Publication Date (Web): 05 Jul 2014 Downloaded from http://pubs.acs.org on July 15, 2014

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The Journal of Physical Chemistry Letters

Multiexciton Generation in Seeded Nanorods Hagai Eshet,

†

Roi Baer,

‡

Daniel Neuhauser,

¶

and Eran Rabani

School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University,Tel Aviv 69978, Israel, Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel, and Department of Chemistry, University of California at Los Angeles, CA-90095 USA

E-mail:

∗ †

To whom correspondence should be addressed School of Chemistry, The Sackler Faculty of Exact

Sciences, Tel Aviv University,Tel Aviv 69978, Israel

‡

Fritz Haber Center for Molecular Dynamics, Insti-

tute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

¶

Department of Chemistry, University of California

at Los Angeles, CA-90095 USA

ACS Paragon Plus Environment 1


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Abstract The stochastic formulation of multiexciton generation (MEG) rates is extended to provide access to MEG eciencies in nanostructures containing thousands of atoms. The formalism is applied to a series of CdSe/CdS seeded nanorod hetero-structures with dierent core and shell dimensions. At energies above 3Eg (where Eg is the band gap) the MEG yield increases with decreasing core size, as expected for spherical nanocrystals. Surprisingly, this behavior is reversed for energies below this value. This anomalous behavior is explained by the behavior of the density of states near the valence band edge, which increases with the core diameter. Our predictions indicate that the onset of MEG can be shifted to lower energies by manipulating the density of states in complex nanostructure geometries.

Graphical TOC Entry

Multiexciton generation yield in CdSe/CdS seeded nanorods.

Keywords: Semiconductors, heterostructures, electronic structure, solar cells, nanostructures. Multiexciton generation by which more than

of impact ionization.

18,19,21,29

Using this frame-

a single electron-hole pair is generated from

work and a unied Green function approach to

a single photon upon optical excitation is a

MEG it was shown that size scaling of the band

promising direction to collect hot carriers, po-

gap, Coulomb couplings and density of states

tentially increasing photovoltaic power conver-

(DOS) lead to an overall increase in MEG rates

sion eciencies in solar cells to above

and eciencies with decreasing NC size.

45%. 1,2

It is common to dene an onset photon energy

Eon

Several dierent approaches have been pro-

beyond which MEG eciencies are

posed recently to lower the onset energy ra-

noticeable. One of the necessary conditions for MEG to be of technological use is that

Eon

32

tio

be

Eon /Eg .

to decrease

Gabor et al.

Eon /Eg

33

used internal elds

in carbon nanotubes pho-

close to the lower bound of twice the quasi-

todiodes,

particle band gap (2 Eg ).

eld-induced acceleration of the charge carri-

tors

Eon /Eg > 5

1

In bulk semiconduc-

and thus MEG is not use-

ers.

34

a result that was rationalized by

Sandberg et al.

35

studied the role of

ful for photovoltaic applications. This large ra-

shape on MEG eciency and argued for a

tio is a consequence of energy and momentum

60%

conservation constraints which severely limit

tor nanorods (NRs) compared to spherical NCs.

the number of multiexciton states that can be

The eect of NR diameter and length on MEG

formed coupled with the existence of very rapid

rates was analyzed by considering the scaling

non-radiative relaxation pathways.

of the density of trion states (DOTS) and the

3,4

Conned

increase in MEG yields in semiconduc-

36

semiconductor nanocrystals (NCs) on the other

Coulomb couplings.

hand, oer improved MEG eciencies,

with

ing from spherical NCs leading to a surprising

This re-

result where the MEG rates, which are given by

sult was rationalized by several dierent theo-

the product of the two, were roughly indepen-

retical treatments.

dent of the NR length.

an onset energy below

1831

Eon < 3Eg . 717

5,6

Perhaps the most intu-

itively appealing picture of MEG in NCs is that

Both show distinct scal-

36

This prediction was

recently veried by experiments.


37,38

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Two additional handles can be used to control the onset energy ratio

Eon /Eg .

generating an exciton

The rst

Klimov and cowork-

In the above equations,

energy in Type-II core-shell NCs can be re-

dressing the role of phonon emission in nanos-

40,41

a, b, c... for unocr, s, t, u... are general indices, and σ =↑, ↓ is the spin index. µ0S is the transition dipole from |0i to |Si, γ is the phonon emission rate, and ΓS is the MEG rate from state|Si.

This allows for reduction of In order to achieve this, we

study MEG eciency in seeded nanorods heterostructures

42,43

It is interesting to note two limits of Eq. (1):

using a stochastic formulation

of the MEG formation rates.

30

When the MEG rate is much larger than the

We then extend

the formalism to calculate the MEG eciency in a series of seeded nanorods with dierent seed

MEG becomes

highly ecient ( nex

and when the

(nex

the MEG eciency increases with the

of

Eg .

γ

32

approach, which assumes that each exciton gen-

is computation-

).

ΓS

is written as as sum of rates

+ ΓS ≡ Γia = Γ− a + Γi ,

erated upon absorption can either decay to the

γ

(3)

and a stochastic formula for the negative and positive trion formation rates is adopted:

or produce another electron-hole pair with a

30

The master equation approach was

previously applied to study MEG in spherical nanocrystals and predicts MEG eciencies similar to those calculated by a direct absorption Green function formalism within the wide band limit.

Γs

of creating positive and negative trions:

seeded nanorods we adopt the master equation

32

and

from experiments and assume it is energy

independent

In order to calculate the MEG eciencies in

ΓS .

γ

ally expensive for large NCs. We take the value

eciency even when the photon energy is scaled

rate

MEG is suppressed

(ω) → 1).

The calculation of

seed radius, i.e., larger seeds show higher MEG

band edge by phonon emission with a rate

ΓS )

to the MEG rate ( γ

the case of spherical NCs, we nd that below

by

γ), (ω) → 2)

phonon emission rate (ΓS

phonon emission rate is much larger compared

size and dierent NR diameter. In contrast to

Eon /Eg

i, j, k...

cupied (electron) states.

and at the same time increase of the

MEG eciency.

We use indices

for occupied (hole) states and

is based on controlling the density of states near

Eon /Eg

|0i.

Slater determinant,

The other

handle, which forms the crux of our approach, the band edge.

aσ i = a†aσ aiσ |0i |Si ≡ |Siσ

obtained by applying the annihilation ( aiσ ) and † creation (aaσ ) operators on the ground state

Theoretically, ad-

tructures is rather challenging.

(2)

is a shorthand notation for a single exciton state

duced markedly by partially blocking the non-

39

ES :

PS (ω) = P

ers have recently shown that the MEG onset

radiative decay channel.

with energy

|µ0S |2 δ (ES − ~ω) . 2 S 0 |µ0S 0 | δ (ES 0 − ~ω)

is based on suppressing the competing channel of phonon emission.

|Si

2π ~ 2π = ~

= Γ− a

Wa2 ρ− T (εa )

Γ+ i

Wi2 ρ+ T (εi ) ,

Using the master equation approach it

(4)

is straightforward to show that the MEG e-

where the negative and positive trion density of

ciency is related to the number of excitons gen-

state (DOTS) is given by

erated at steady state, which is given by

nex (ω) =

X S

2ΓS + γ , PS (ω) ΓS + γ

21

ρ− T (ε) =

X

ρ+ T

X

δ (ε − (εb + εc − εj ))

bcj

(1)

(ε) =

δ (ε − (εj + εk − εb )) ,

(5)

jkb

nex (ω) represents the number of excitons generated at a photon energy ~ω (the MEG eciency is simply given by nex (ω)−1) and PS (ω) where

and the average square Coulomb coupling for electrons and holes is given by

is the probability for absorbing the photon and



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Page 4 of 10

tic procedure.

X

2 2 Wa = Wa;cbj δ (εa − (εb + εc − εj )) /ρ− T (εa )

3. Filter target electron and hole states with energies sampled from

bcj

X

2 2 Wi = Wi;jkb δ (ε − (εj + εk − εb )) /ρ+ ). T (εi(6) jkb

Γ− a

and

ES .

Γ+ i

an average over the sampled electronP 2ΓS +γ 1 hole pairs: nex (ω) = S PS (ω) ΓS +γ NS where PS (ω) = ρ (εa ) ρ (εi ) PS (ω). The

where

product

is the screened Coulomb interactions and

at energy

summation expression (cf., Eq. (1)) with

2 Wi;jkb = 2 |Vjibk − Vkijb |2 + |Vkijb |2 + |Vjibk(7) |2 ,

ψr (r)ψs (r)ψu (r0 )ψt (r0 ) |r − r0 |

µ0S

to generate the MEG rate − for each electron-hole pair: ΓS≡ia = Γa + + Γi . Obtain nex (ω) by replacing the full

2 = 2 |Vacjb − Vjcab |2 + |Vjcab |2 + |Vacjb |2 Wa;cbj

d3 rd3 r0

(above and

for each electron-hole pair, calculate the

4. Use

couplings are given by

¨

ρ (ε)

below the Fermi energy, respectively) and transition dipole

In the above equations, the squared Coulomb

Vrsut =

30

ρ (εa ) ρ (εi )

is used to correct for

the density of electron and hole pairs at energy

(8)

ES . NS

is the number of excited

ES = ~ω , NS = 200.

states sampled at energy cally taken to be

ψr (r)

typi-

are the single-particle orbitals (see more details

The rst two steps have been developed previ-

below). We assume a static dielectric screening,

ously by Baer and Rabani and applied to both

,

NCs and NRs.

for the Coulomb matrix elements.

30

The last two provide an exten-

To calculate the MEG eciency given by

sion to calculate the MEG eciencies by sam-

Eq. (1), one requires as input the MEG rates

pling the singly excited states rather than cal-

and the transition probabilities for all possiaσ ble single excitons |Si ≡ |Siσ i. For spherical nanocrystals these are readily available for

culating them all.

moderate excitation energies ( ≤

in Ref.

3Eg ). 21

In Fig. 1 we plot the negative and positive trion formations rates (upper panel, shown also

How-

30

) and the MEG eciency (lower panel,

3nm

ever, for nanorods (and also seeded nanorods),

shown here for the rst time) for a

it is practically impossible to generate all possi-

ameter CdSe nanocrystal.

ble singly excited states, even at relatively low

sults are compared with the full summation ap-

excitation energies, due to the large density of

proach, labeled deterministic. The agreement

states.

Therefore, in order to obtain the MEG

between the stochastic and deterministic calcu-

eciency, we propose a stochastic approach,

lations is excellent, albeit the fact that stochas-

where we sample the singly excited states and

tic rates are slight underestimated compared

the corresponding MEG rates rather than cal-

with the average deterministic rates.

culating them all, to perform the full summa-

ing the results near the valence and conduction

36

tion formula for

nex (ω).

The approach can be

Compar-

approach of the energy falls short in describing the trion formation rates for

1. Calculate the single particle density of

the phonon emission rate,

Use convolutions to generate the

density of single excited states ( ρX

ε/Eg → ±1.

How-

ever, since these rates are very low compared to

states (ρ (ε)) using a stochastic trace for-

30

30

band edge, it is clear that the uniform sampling

summarized as follows:

mula.

di-

The stochastic re-

γ , assuming a vanish-

ing value does not aect the MEG eciencies.

(E))

and the negative/positive trion density of − + 30 states (ρT (ε)/ρT (ε)). 2. Generate the negative/positive trion formation rates (Eq. (4)) using the stochas-


Page 5 of 10

only one trajectory was used for each structure.

2

-1

Γ (ps )

10

The single particle orbitals ( ψr (r)) required for

0

10

the calculation of the MEG rates were then

-2

generated for the relaxed congurations using

10

a semi-empirical pseudopotential models on a

-4

10 -2

-1

0

1

2

3

4

real space grid,

ε / Eg

2

function.

Stochastic Deterministic

1.8

n(ω)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


46,47

48

with a Gaussian ltering

1.6

In Fig.2 we show the MEG eciencies (left

1.4

panels) and the trion formation rates (right

1.2

panels) for the two sets of rod diameters. The MEG eciency increases with photon energy

1 2

3

2.5

4

3.5

with an onset close to 3 Eg and

ω / Eg Figure 1:

Drod = 4

The trion formation rate (upper

and

6nm,

respectively.

3.5Eg

for the

This is con-

sistent with experiments for a wide range of

panel) and the MEG eciency (lower panel) for

conned nanostructures.

5

At energies above the

3nm diameter CdSe nanocrystal (Cd 232 S e251 )

onset energy, we nd that the MEG eciency

as a function of scaled energy ( ω is the pho-

at a scaled energy increases as the size of the

a

ton energy while

ε

is the energy of the electron

nanocrystal seed decreases, consistent with cal-

21

(hole), for negative (positive) trions, measured

culations of spherical nanocrystals.

from the bottom of the conduction (top of the

ton energies below the onset (between

valence) band). The full summation approach

3Eg ) the reverse is true and the MEG eciency

labeled deterministic is shown by red circles

is higher for larger seeds.

and the stochastic approach spanning a larger

the inset of Fig. 2 and is qualitatively dierent

energy range is shown by black circles.

from spherical nanocrystals.

For pho-

2Eg

and

This is depicted in

Despite the fact that the MEG eciencies for

In the lower panel of Fig. 1 we compare

2Eg

energies between

ministic approach.

in seeded nanorods, understanding the origin of

The stochastic approach

and

3Eg

the stochastic MEG eciencies to the deter-

are rather small

resulting from

the behavior below the onset is still signicant

the small systematic deviation of the trion for-

as it may prove useful in designing materials

mation rates. However, the overall agreement is

with higher MEG in this important range of

excellent and the stochastic approach is reliable

energies. In the right panels of Fig. 2 we plot − the negative trion formation rate ( Γe (ε)) for all corresponding cases. Since the MEG pro-

slightly underestimates

nex (ω),

at energies much higher than those accessible by the deterministic calculations. Moving

on

nanorods of

to

the

20nm

CdSe/CdS

cess is dominated by the formation of negative

core/shell

trions for CdSe NCs,

length, we calculated the

Drod

core diameters

in the calculations of the MEG eciencies), the − positive channel. For all cases studied, Γe (ε) increases with the electron energy, similar to

4nm and 6nm and for several Dcore in the range 2 − 4.5nm in of

each case. The rods were faceted and the center of the spherical CdSe core was placed at of the length of the rod.

the behavior of spherical

1/3

nanostructures.

The congurations

100ps

recently developed force-eld

at

44

300K

with a

accurately de-

45

29,32

and rod shaped

36

However, there seems to be

an isosbestic point about ε/Eg = 1.75 where Γ− e (ε) changes behavior (this is more clearly observed in the DOTS discussed below). This

of the seeded rods were relaxed using molecular dynamics runs for

we focus on this channel

and ignore, for the sake of the discussion (not

MEG formation rates and eciencies for rod diameters

21

The

isosbestic point is correlated with the crossover

nal congurations were quenched to remove

of the MEG eciency with the nanocrystal seed

structural eects of thermal uctuations, thus,

size.

scribing

CdSe/CdS

hetero-structures.

To delineate the factors governing this be-


The Journal of Physical Chemistry Letters 2

4

10

2

10

-1

Γe (ps )

nex(ω)

1.75 1.5 1.25

2

2.5

3

D=4nm

0

10

-2

10

D=4nm

-4

10

1

2

10 -1

Γe (ps )

1.75

nex(ω)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 10

1.5 1.25

D=6nm

1 2

2.5

3

4

3.5

4.5

0

10

-2

10

D=6nm

-4

10

5

0

1

ω / Eg Figure 2: MEG eciency for

2

3

4

ε / Eg

Drod = 4

6

and

nm diameter nanorods (upper and lower left panels,

respectively) and the corresponding trion formation rates (right panels) for scaled energies. For the

Drod = 4nm rods the blue, black, red and green curves represent seed sizes of Dcore = 0, 2, 2.5 and 3nm, respectively. For the Drod = 6nm rod the black, red and green curves represent seed sizes of Dcore = 2.5, 3.5, and 4.5nm, respectively. Inset: zoom in on the low energy range. havior, we rst note that the trion formation

crease of the band gap was mainly attributed

rates are given in terms of a product of the av-

to the change of the highest hole energy, which

erage square Coulomb coupling and the trion

is localized on the seed.

density of states (cf., Eq. (4)).

47

The former

In the right panels of Fig. 3 we plot the DOTS

shows a monotonic core-size dependence and

for two additional cases. The bottom panel as-

a weak energy dependence (not shown).

sumes that the DOS of the holes for all seed

30,36

Therefore, the isosbestic point must reect a

sizes is identical so the observed change in the

non-monotonic behavior of the DOTS with

DOTS reects the dierences in the DOS of the

Dcore ,

which is indeed the case, as shown by

electrons (i.e., the portion of the DOS above

the left panels of Fig. 3. We nd that both + ρ− T (ε) and ρT (ε) exhibit an isosbestic point, while that for the negative DOTS (circled in

the Fermi energy). Clearly, all three sizes give

black) is correlated with the isosbestic point of Γ− e (ε).

tive DOTS originates from the hole DOS (i.e.,

Since the DOTS is given in terms of a triple

ergy). The upper right panel of Fig. 3 shows the

convolution of the DOS, we also plot the DOS

DOTS for the case where the only dierence in

as insets to each panel in Fig. 3.

the DOS for the three values of

the same DOTS. Therefore, one can conclude that the isosbestic point observed in the negathe portion of the DOS below the Fermi en-

We focus

Dcore is assumed

on the energy regime about the Fermi energy,

to be the aforementioned shoulder near the va-

which is assumed at the top of the valence band

lence band edge.

and is set as the origin of the energy axis. Per-

account for the isosbestic point. Thus, we con-

haps the most notable eect of the seed size

clude that the crossover behavior observed in

on the DOS is the appearance of a shoulder

the MEG eciency can be attributed to the in-

near the valence band edge as

Dcore

Clearly, this is sucient to

increases.

crease in the DOS near the valence band edge

This shoulder reects the increasing DOS on

and at the same time the rather insignicant

the seed due to volume eects. In addition, the

decrease in the band gap energy.

DOS shows a systematic shift in the position

In conclusion, we have extended our stochas-

of the valence and conduction peaks resulting

tic approach to calculate the MEG eciency in

from small changes in the band gap as

Dcore

in-

systems containing thousands of atoms within

creases (note that we scale the energy by the

a master equation formalism. Rather than cal-

band gap

Eg

for each system).

Here, the de-

culating all possible excitations and the cor-


Page 7 of 10

The Journal of Physical Chemistry Letters 12

D=4nm

D=4nm

10

ρDOTS(ε)

2

10

ρ(ε)

ρDOTS(ε)

10

1

8

10

0 -3 -2 -1 0

6

10

1 2 ε / Eg

3

4

10

10

8

10

5 6

10

12

10

D=6nm

10

ρDOTS(ε)

4

10

ρ(ε)

ρDOTS(ε)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2

8

10

0 -3 -2 -1 0

6

10

1 2 ε / Eg

3

4

D=4nm

10

10

8

10

5 6

-3

-2

-1

0

1

2

3

10 -3

-2

-1

0

ε / Eg

1

2

3

ε / Eg

Drod = 4nm and 6nm nanorods (upper and lower panels, respectively) and the articial DOTS for a Drod = 4nm diameter (right panels) as described in the text. Insets show the corresponding DOS for each Drod . For the Drod = 4nm rod the blue, black, red and green curves represent seed sizes of Dcore = 0, 2, 2.5 and 3nm, respectively. For the 6nm rod the black, red and green curves represent seed sizes of Dcore = 2.5, 3.5, and 4.5nm, respectively. Figure 3: The DOTS for

responding MEG rates, we proposed to sam-

Grants No.

ple the electron-hole pairs and for each pair,

spectively.

obtain the MEG rates (sum of the positive

support of the US-Israel Bi-National Science

and negative trion formation rates) using our

Foundation.

recently developed expeditious stochastic ap-

Center for Re-Dening Photovoltaic Eciency

proach.

Application of the approach to quasi-

Through Molecule Scale Control, an Energy

type II CdSe/CdS seeded nanorods uncovered

Frontier Research Center funded by the U.S.

an interesting and unexpected behavior. Above

Department of Energy, Oce of Science, Oce

30

the onset of MEG (around

Dcore

increases as

3Eg )

the eciency

1020/10 and No.

611/11, re-

R. B. and D. N. acknowledge the E. R. would like to thank the

of Basic Energy Sciences under Award Number DE-SC0001085.

decreases, as expected. How-

ever, below this onset, the opposite is true and

References

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