1 Nanomaterials for Photovoltaics...

1 Nanomaterials for Photovoltaics (v11)

10. Bulk-Heterojunction Solar Cells

Nanostructured solar cells We can imagine a spectrum of nanostructure solar cells ranging from an organic, bulk heterojunction type just discussed to an inorganic, 3-D organized type.

Example: CuSCN/TiO2 DSH An example of a dye-sensitized solar cell is shown below [B. O’Regan and F. Lenzmann, J. Phys. Chem. B 108, 4342 (2004)]. The image is an SEM cross section.


The cell was synthesized by Spray pyrolysis/spin coating of TiO2, followed by incorporation of Ru-535 dye. A CuSCN solution incorporated into and above film. The cell works by absorption of light by dye, with electron injection into the CB of the TiO2. The dye is regenerated by capture of electrons from the VB of CuSCN. This type of cell has been called an "interpenetrating network heterojunction". The J-V characteristics are shown below

Example: TiO2/PbS QDs Quantum dots can also provide control of bandgap and band offsets useful for heterojunctions solar cells [P. Hoyer and R. Konenkamo, Appl. Phys. Lett. 66, 349 (1995)].

Nanostructured solar cells Consider a nanostructured, bulk heterojunction device:


Bulk heterojunction (I) We can idealize the structure as periodic, vertical columns of the two materials. Ignoring band bending, the band diagram is determined by the band offsets and band gaps.

2 1 , CE , 2 1g g gE E E , V gE E

Bulk heterojunction (II) Let's say a period of the superlattice is 1 2D w w . Assume a uniform charge density in each region

1 1

2 2

, 0

, 0

qN w xx

qN x w

The electric field satisfies

d x

dx

We have the boundary condition

1 20 0x x


Bulk heterojunction (III) Assume the two regions have equal thickness.

Then in region 1 ( 1 0w x )

1

1 1 1

21 1 2

x

x w

qN qN wx dx x

while in region 2 ( 20 x w )

2 2

02 2

2 2 22 2

2 2

2 2

2

0 0

0 0 02 2 2

2

x

x

qN qNx dx x

w qN qNw w

qN wx x

Notice that 0 at both 1 2x w and 2 2x w


Bulk heterojunction (IV) The potential in region 1 ( 1 0w x ) is

1

1

01 1 1 1 1 2

01 1 1 2

2 21 1 1

1

2 2

2 2 2

wx

wx x x x x

qN w qN w qNV x x dx x dx x dx

qN w wx

In region 2 ( 20 x w )

2

2

2 2 2 2

02 2 2

2 22 2 2

2

2

2 2 2

wx

wx x x

qN w qNV x x dx x dx

qN w wx

Notice that 0V at both 0, , 2 ,...x D D .

1 11

1

2 22

2

, 02

, 02

qN wx w x

qN wx x w

2 21 1 1

11

2 22 2 2

22

, 02 2 2

, 02 2 2

qN w wx w x

VqN w w

x x w

Flatband solar-cell model The voltage drop across one period is

2 2 2 22 1 2 2 1 1 1 1 2 2

2 1 1 22 2 2 2 2 2 2 2 2

w w qN w qN w q N w N wV V V

We impose charge neutrality on each superlattice period

1 1 2 2N w N w So


21

1 2

Nw D

N N

, 12

1 2

Nw D

N N

Then

2

1 1 2 2

1 2

1 11 12 2

q DV

N NN N

Notice that if 1 2 , then

2

1 2

11 12 2

q DV

N N

which is the sam as a p/h homojunction with 2w D .

If 1 2 2w w D , then 1 2N N N and

2

1 2

1 1

32

q N DV

Flatband solar-cell model (II) The voltage drop across a single SL period increases as 2D . For a nanoscale SL, this remains small (less than kT q . This small voltage variation (band bending) indicates that the flatband approximation is valid

in the nanoscale regime.


Flatband solar-cell model (III)

Flatband solar-cell model (IV) Within the flatband approximation, we can determine the charge density in each region. The quasi-Fermi levels Fn

E and FpE will be roughly constant over the length of a period. The carrier concentrations in

region 1 are

F 1 F

1 F F

21 1 1

21 1 1

e e e

e e e

i i in n

i ip p i

E E kT E E kT E kTi i


n n n

p n n

Those in region 2 are

F 2 F

2 F F

22 2 2

22 2 2

e e e

e e e

i i in n

i ip p i



n n n

p n n

We define an intrinsic carrier concentration

1 2i i iN n n and a heterojunction carrier concentration

2 coshh i iN N E kT

Some other abbreviations are useful

Fe inE E kTx , Fe i pE E kTy

The product gives

F F2 e n pE E kTx y z , 2 2e eF Fn pE E kT qV kTz x y

Also

2e iE kTr These allow us to write


11

in xn

r

, 1 1ip n r y

and

2 2in n x r , 22

in yp

r

Flatband solar-cell model (V) Assume acceptor doping only on side 1 and donor doping only on side 2. Then

1 A 1 1N N n p ,

2 D 2 2N N p n

If we have full ionization of dopants (

A AN N ,

D DN N ), then

1 A 1 1N N n p ,

2 D 2 2N N p n

Let's assume 1 2w w , so charge neutrality ( 1 2N N N ) gives

A 1 1 D 2 2N p n N p n

This becomes

1 22 1 D A

i ii i

n nn r x n r y N N

r r

Notice y z z x , so

1 2 D A2 1

i ii i

n x n z N Nn r n r

r z r x z

Flatband solar-cell model (VI) A few more definitiosn are useful. Define

12

ii

nu n r

r , 2

1i

in

v n rr

, and D Aw N N

We have

x z wu v

x x z

It is useful to write e x z . Then

sinh coshw

u v u vz

The prefactors can be parameterized as

coshu v R , sinhu v R

So

2 2 2R u v u v uv , atanhu v

u v

Using the identity cosh sinh sinh cosh sinh , we have

2 sinhw

uvz


1 0asinh atanh2

w v u

v uuv z

This allows us to write

1 0e ex z Flatband solar-cell model (VII) We have

1 asinh2

w

uv z

so

1e exp asinh2

w

uv z

Also

0 0

0 00

e etanh

e e

v u

v u

so

0 020

1e e ln

2

v v v

u u u

We arrive at a general result

exp asinh2

v wx z

u uv z

:

All other quantities can be determined once x is found.

Flatband solar-cell examples Bandgap differences and band offsets often exceed kT . Assume

1

2

1i

i

n

n and 1 2

2

1 i

i

nr

n

Then 1r , so

22

1h i iN N r N r

r

Also, this assures

12

ii

nn r

r and 2

1i

in

n rr

Notice that

2iu n r and 1iv n r so

1

2

i

i

v n

u n

We also have


2 2 21 2

1

1 2 2

i i h

i

i i i

uv n n N

uv n

n n n

2

1

i

i

n

n2

2

1r

r

which fives

h iuv N N r Then

1

2

exp asinh2

i

i h

n wx z

n N z

Flatband solar-cell examples: case 1 (no doping) (I) With no doping ( 0a dN N ), we can say that 0w and asinh 0 0 , so

12

2

e iqV kT

i

v nx z

u n and

222

1

e iqV kT

i

z ny

x n

Then in region 1

21 1 2

1 ei i qV kT

h

n x nn

r N

and 2

1 1 eqV kTi hp n r y N

In region 2

22 2 eqV kT

i hn n x r N and 2

2 2 22 ei i qV kT

h

n y np

r N

Flatband solar-cell examples: case 1 (no doping) (II) We can say that

21i

hh

nN

N ,

22i

hh

nN

N

so 1 1n p and 2 2p n . Thus

1 1N n 21 1 eqV kT

hp p N , 2 2N p 22 eqV kT

hn N

Apparently,

21 2 eqV kT

hN N N The charge density iss

2

2

e , 0

e , 0

qV kTh

qV kTh

q N w xx

q N x w

Flatband solar-cell examples: case 2 (asymmetric doping) (I) Consider the case D A hN N N . Let's say

D A 2e 122

qV kT

h

w N N

Nuv z

This implies small quasi-Fermi-level splitting, i.e.


D A2ln

2 h

kT N NV

q N

Let's say D AN N , so 0w . Then

1 11

1sinh e e

22

w

uv z

1

1e

2

so

1ew

uv z

Now we have

x zv

w

u u v

z

w

u

wx

u

and

2 2z z uy

x w

Flatband solar-cell examples: case 2 (asymmetric doping) (II) Now

21

1

2

D AD A

e

i

h qV kT

n r z up

w

NN N

N N

,

11

21 D A

D A2

i

i

h

n wn

r u

n N NN N

N

and

22

2

22

D AD A

e

i

i qV kT

n z up

r w

nN N

N N

,

22

2

i

i

n w rn

u

n

D AN N r

2in rD AN N

Therefore

1 A 1 1 AN N n p N and

2 D 2 2

D

N N p n

N

DN A AN N

We see that

1 2 AN N N The charge density is then

A

A

, 0

, 0

q N w xx

q N x w

With small quasi-Fermi level splitting, the charge density is generally equal to the doping level in the more lightly doped side.


Flatband solar-cell model: p/n+ (I) Consider a bulk heterojunction with A DN N .

We have

D 2

2

e iE kT

i

Nx

n

On the p side

21 2

1 AD

e ei ii iE kT E kTn Np N

x N

1 D21 1 A

2

e ei iiE kT E kTi

i

n Nn n x N

n

On the n side

22 2 De iE kT

in n x N The more lightly doped side determines the charge density, so A 1 1 AN N p n N . the quasi-Fermi

level slope is small, so we have AN ionized acceptors on the p-side and AN free holes on the n+ side.

Flatband solar-cell model: p/n+ (II) The solar cell operates in the non-equilibrium condition

F F F F-side -siden p n pn pqV E E E E


We only need to consider electrons on the n-side and holes on the p-side. Consider photogeneration. On the n-side, the photogenerated carrier density is Dgn N , so 2 Dn N . On the p side, g pp G , so

1 gp p . Now

1 F1 e i pE E kT

g ip n

and

F 2D 2 e n iE E kT

iN n The voltage may be much larger than the built-in potential difference we calculated.

D

2ln g

ii

p NqV kT E

N

Notice that holes are the minority carrier on the p-side for the p/n+ junction.

Flatband solar-cell model: p/n+ (III) Let's compare this to a bulk, p/n homojunction, for which our model gave

photoOC

0

ln 1J

qV kTJ

photo n pJ q G w L L

20

A D

pni

n p

DDJ q n

L N L N

In the p/n+ case: D AN N , n pL L , photo nJ q G L , and

20

A

ni

n

DJ q n

L N

So

2photo AA

2 20

gn

n i i

J N nG L N

J D n n

Now

A

2ln g

oci

N nqV kT

n

In this case, electron generation on the p-side dominates.

Nanostructured solar cell: equivalent circuit An equivalent circuit for a nanostructured solar cell is shown below.


The diode characteristics arise from interfacial recombination Many small subcells connected in parallel. Vertical charge percolation/hopping represented by series resistors. Charge separation may involve exciton dissociation

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