Theory of Impedance Spectroscopy of Ambipolar Solar Cells with Trap

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Theory of Impedance Spectroscopy of Ambipolar Solar Cells with Trap Mediated Recombination Luca Bertoluzzi, Pablo P. Boix, Ivan Mora-Sero, and Juan Bisquert J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 23 Dec 2013 Downloaded from http://pubs.acs.org on December 31, 2013

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The Journal of Physical Chemistry

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Theory of Impedance Spectroscopy of Ambipolar Solar Cells with Trap Mediated Recombination Luca Bertoluzzi1, Pablo P. Boix2, Ivan Mora-Sero1, Juan Bisquert*1 1-Photovoltaics and Optoelectronic Devices Group, Departament de Física, Universitat Jaume I, 12071 Castelló, Spain 2-Energy Research Institute @NTU (ERI@N), Research Techno Plaza, X-Frontier Block, Level 5, 50 Nanyang Drive, Singapore 637553.

Email: [email protected] 20 December 2013

Abstract The analysis of recombination in solar cells suggests in many cases the presence of trap mediated recombination in the absorber. We present a theory of the recombination of electrons and holes, the Shockley-Read-Hall model, using the impedance spectroscopy (IS) technique. We derive the impedance functions and the corresponding equivalent circuit model. After examining some cases of interest, we show that two semicircles can be obtained in the recombination circuit only if the chemical capacitance associated to traps is substantially larger than the chemical capacitances of free electrons and holes in the absorber bands, while in the other cases, the normal behaviour of one recombination arc will be obtained. Keywords: solar cell, impedance spectroscopy, recombination

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Introduction In the fundamental operation of a solar cell the carriers generated in the absorber are transported to distinct selective contacts for electrons and holes.1 Ambipolar transport refers to the situation in which electrons and holes of similar number and mobilities govern the carrier transport in the semiconductor.2 A minority carrier governs the behavior of most classes of solar cells, such as crystalline silicon solar cells and dyesensitized solar cells, while the majority carrier is a spectator whose background density is not altered by illumination or voltage injection except in very extreme circumstances. However, recently some solar cells that display acute ambipolar characteristics have come to the forefront of research. These are the quantum dot film solar cells3 and organometal halide perovskite solar cells.4,5 In both classes of cells it is likely that similar number of electrons and holes are transported simultaneously in the same medium (the absorber) towards macroscopic selective contacts. These mechanisms are currently a matter of investigation. Given the fact that the materials are prepared by low temperature processing methods, it is likely that the semiconductor absorber contains abundant traps that will act as recombination centers for electrons and holes. Impedance spectroscopy (IS) is a method that has become widely popular for the understanding of electronic processes and technical characterization of solar cells.6 The theory of diffusion-recombination impedance7 describes well the ac dynamics of solar cells dominated by single carrier transport. This technique has been extended to include the retarding effects of disordered localized levels in the bandgap as well as nonlinear recombination.8 This model has been widely used for the description of dye-sensitized solar cells and organic solar cells.6 In this paper we develop the theory of impedance spectroscopy of Shockley-ReadHall (SRH) recombination in ambipolar semiconductors. The general small perturbation equivalent circuit model, leading to a transmission line, was first calculated by Shockley9 and perfected by Sah10 but to our knowledge the application to solar cells of the full impedance model has not been derived in the previous literature. Here we provide a schematic formulation of the main results and the application of the general ac model to solar cells, with explicit derivation of the impedance function and a general analysis of spectral characteristics. Theory We formulate a model semiconductor solar cell as shown in Fig. 1. In this paper we are mainly interested in disclosing the recombination dynamics of electrons and holes that include spectral characteristics of the separate carriers. Therefore here we simplify all transport features and we assume that carrier densities are homogeneous in the semiconductor layer of thickness L . If only direct recombination of electrons and holes is considered, the ac equivalent circuit (EC) consists on a single arc with mixed parameters of both species.11 In order to obtain more structured spectra it is important to consider the recombination via inner band gap traps in the SRH model, as shown in Fig.


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1. The circuit obtained in this work will become part of a transmission line in the case in which transport is considered explicitly.

Fig. 1. Ambipolar solar cell model with two selective contacts that allow injection and extraction of electrons and holes at either side, and SRH recombination via a trap state in the bandgap. In the present model, the conduction band lower edge with effective density of states N c is at energy E c . The upper edge of the valence band with effective density of states N v is at energy E v . The recombination center is at the energy E t with density N t . The density of electrons n can be related to the electron Fermi level E Fn as n = N c e ( E Fn − Ec ) / k BT

(1)

= ni e ( EFn − Ei ) / k BT

where k B T is thermal energy and ni and Ei are respectively the density and energy of the intrinsic level. Similarly for the hole density p p = Nve = pi e

( Ev − E Fp ) / k BT

(2)

( Ei − E Fp ) / k BT

For the fractional occupancy of the trap, f , using a trap Fermi level E Ft , we obtain f = =

1 1+ e

( Et − E Ft ) / k BT

(3)

1 1 + e ( qVε + Ei − E Ft ) / k BT

where qVε = E t − Ei , being q the positive elementary charge. It is convenient to define the following voltages:

− qV n = E Fn − Ei

(4)

− qV p = E Fp − E i

(5)



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− qVt = E Ft − E i

(6)

Therefore we can write the carrier densities as

n = ni e − qVn / k BT p = pi e

f =

(7)

+ qV p / k BT

(8)

1

(9)

1 + e q (Vε +Vt ) / k BT

The carriers dynamics are determined by continuity equations for the three species ∂n ∂t

∂p ∂t ∂f ∂t

1 ∂j n

+ G − β n N t (1 − f )n + ε n N t f

(10)

+ G + ε p N t (1 − f ) − β p N t pf

(11)

= β n (1 − f )n − ε n f + ε p (1 − f ) − β p pf

(12)

=+

=−

q ∂x

1 ∂j p q ∂x

Here G is a generation rate, β k and ε k are the capture and release coefficients between the trap and the bands. jn and j p are the local electric current density of electrons and holes respectively. The detailed balance condition12 implies that

εn ni β n

=

pi β p

εp

= e ( Et − Ei ) / k BT

(13)

The boundary conditions associated to ideal selective contacts are

j n ( x = L ) = j p ( x = 0) = 0

(14)

Under the assumption that the active layer is homogeneous, we can integrate Eqs. (10) and (11) as follows:

∂n ∂t

∂p ∂t

=−

jn

+ G − β n N t (1 − f )n + ε n N t f

(15)

+ G + ε p N t (1 − f ) − β p N t fp

(16)

qL

=−

jp qL

where j n = j n ( x = 0) and j p = j p ( x = L) are the injection (or extraction) currents. Since the current at the boundaries is taken by either carrier, we must have

j = jn = j p

(17)

In steady state (indicated by overbar), all the quantities are time independent (i.e. ∂y k / ∂t = 0 ) and we find that the occupation of the trap is given by


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f =

βnn + ε p

(18)

βnn + β p p +ε n +ε p

In particular, at equilibrium:

f0 =

β n n0 β n n0 + ε n

=

εp

(19)

β p p0 + ε p

Thus the current can be stated as

(

j = qL G − U SRH

)

(20)

where

U SRH = N t β n β p

n p − ni pi

(21)

βnn + β p p + ε n + ε p

is the standard SRH recombination rate. By the structure of the solar cell device, the applied voltage or photovoltage is related to the above defined voltages as

Vapp = Vn − V p

(22)

The determination of the separate voltages as a function of the applied voltage, Vapp , requires stating a relationship between the number of electrons and holes. In a homogeneous medium we have the condition of electro-neutrality. Then, with a net + background of doping of positive ionic charge N ion we obtain: + n − p − N ion =0

(23)

Eqs. (18) – (23) allow determining all steady state characteristics such as the currentvoltage curve. In particular, Eqs. (22) and (23) allow one to express the densities of free electrons and holes (eqs. (7) and (8)) as a function of the applied voltage Vapp in the presence of a net density of positively charged dopant:   n 1 +    i n = N ion  1 + 4 + 2   N ion  

2  − qV / k T   e app B     

  n 1 +   p = N ion  1 + 4 +i  N 2   ion  

2  −qV / k T   e app B     

1/ 2

1/ 2

  + 1  

(24)

  − 1  

(25)

Note that for an n-type semiconductor, the previous equations can be simplified, at first order, into:



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+ n ≈ N ion

p≈

ni 2 +

(26) e

− qVapp / k BT

(27)

N ion These last equations are valid as long as n(Vapp ) >> p (Vapp ) . In the calculation of the impedance we use a small sinusoidal perturbation of voltage imposed over a steady state. The small signal is modulated with angular frequency ω so that we can use the substitution ∂ / ∂t → iω to solve the equations in the Laplace domain. We can split all quantities as follows y k = y k + yˆ k (t )

(28)

j k = j k + ik (t )

(29)

Vk = Vk + v k (t )

(30)

The definition of the impedance is the ratio of modulated voltage to current, i.e.

Z=

vn − v p

(31)

itot

where itot = in = i p . The equations for the modulated quantities are obtained by linear expansion of Eqs. (12), (15) and (16), with the following result

in

[

)]

(

+ iω + β n N t 1 − f nˆ − (β n n + ε n )N t fˆ = 0

(32)

qL ip

[

] (

)

+ iω + β p N t f pˆ + β p p + ε p N t fˆ = 0

(33)

qL

[iω + β n n + ε n + β p p + ε p ] fˆ − β n nˆ (1 − f )+ β p pˆ f = 0

(34)

We introduce the equilibrium chemical capacitances13

C µn = − qL

C µp = qL

∂n ∂Vn

∂p ∂V p

C µt = −qL

∂f ∂Vt

=

q 2 Ln

(35)

k BT

=

q 2 Lp

(36)

k BT

Nt =

q2L

(

)

f 1− f N t

k BT

then we can write the following expressions for the modulated densities


(37)

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nˆ = −

C µn

(38)

vn

qL pˆ =

C µp

(39)

vp

qL t

Cµ N t fˆ = − vt qL

(40)

We define the following resistances14-16

Rtn −1 = C µn β n N t (1− f )=

Rtp −1 = C µp β p N t f = n Rrec

−1

q2L N t β n n (1− f ) kT

(41)

q2L N t β p pf kT

= Rtn −1 − C µt [β n n + ε n ] =

(42)

q2L (1− f )U SRH kT

q2L fU SRH kT It is also convenient to introduce admittances and conductances as follows p Rrec

−1

[

]

= Rtp −1 − C µt β p p + ε p =

(43)

(44)

Yµk = iωC µk

(45)

G k = R k −1

(46)

Then Eqs. (32) – (34) can be expressed in the form n in = Yµn v n + Gtn (v n − vt )+ Grec vt

(

(47)

)

p − i p = Yµp v p + Gtp v p − vt + Grec vt

(

)

(

(48)

)

n p 0 = Gtn (v n − vt )+ Gtp v p − vt + Grec + Grec −Yµt vt

(49)

These three equations state the conservation of current and can be represented as an EC, shown in Fig. 2.



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Fig. 2. Equivalent circuit for the SRH recombination model. The electrical components of the EC are indicated along with the chemical potentials: vn for electrons in the CB, v p for holes in the VB, vt for traps. vi is the intrinsic chemical potential. This EC was derived by Shockley for the equilibrium situation and by Sah for the general case that we show here.9,10 The left vertical branch is indicated as a current generator because the current depends on the voltage drop at the capacitor associated to the traps. Solving the linear system of equations (47) − (49) we obtain the impedance as follows n p ( Yµt +Yµn +Yµp )(Gtp + Gtn )− (G rec + Grec −Yµt )(Yµn + Yµp ) Z=

det

(50)

where

[ ( ) ( )] + G [G (Yµ +Yµ + Yµ )+ Yµ (Yµ +Yµ − G )]

n n p det = Yµn Gtp Yµp + Yµt − G rec −Yµp Grec + Grec −Yµt tn

tp

t

n

p

p

t

n

p rec

(51)

Discussion Before entering into the details of the practical application of the EC of Fig. 2, it is useful to examine when it can be reduced to well-known reference cases. The first case we discuss here is the impedance of the equilibrium steady state in which electrons and holes remain in dark equilibrium ( n = n0 , p = p 0 ). This situation corresponds to the case depicted by Shockley.9 The corresponding EC is given in Fig. n p 3(a). According to Eqs. (43) – (44) we have Grec = Grec = 0 and the current sources can be removed from the EC. Note that the conductance Gtn and Gtp can be simplified into: Gtn =

q2L N t f 0ε n kT

(52)

q2L N t f 0 β p p0 kT The total impedance is given by: Gtp =


(53)

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Za =

(Yµt +Yµn +Yµp )(Gtp +Gtn )+Yµt (Yµn +Yµp ) Yµn [Yµp Yµt + Gtp (Yµp + Yµt )]+ Gtn [Gtp Yµp + (Yµp + Gtp )(Yµt +Yµn )]

(54)

Fig. 3. Reduction of the equivalent circuits of Fig. 2 at equilibrium (a), for an N type semiconductor (b) and in the absence of traps (c). In the latter case, a band to band recombination resistance R0 has been introduced. The second case that we shall discuss is the presence of a majority carrier. Let us take the example of an n-type semiconductor. In this situation, the electron concentration in + , as indicated by Eq. (26). A the CB does not vary with the applied voltage and n ≈ N ion recombination center situated below the equilibrium Fermi level remains occupied by n electrons ( f ≈ 1 ). In this case Grec ≈ 0 and the total impedance becomes: Zb =

(Yµt +Yµn +Yµp )(Gtp +Gtn )− (Grecp −Yµt )(Yµn +Yµp ) p )]+Gtn [Gtp (Yµt +Yµn +Yµp )+Yµp (Yµt +Yµn −Grecp )] Yµn [Gtp (Yµp +Yµt )+ Yµp (Yµt − Grec (55)

This scenario is depicted by the EC of Fig 3(b) in which the current source associated to electrons in the conduction band has been removed. The upper part of the circuit associated to majority carrier corresponds to the equilibrium situation depicted by



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Shockley. Therefore majority carriers behave as if they were at equilibrium. Obviously this statement is true only if the applied voltage is such that the electron concentration remains much higher than the holes concentration. The last case that we examine hereafter is the absence of traps ( N t = 0 ). In this configuration, all the conductances are zero and only remains the capacitances of electrons ( C µn ) and holes ( C µp ). Since there is no trap assisted recombination, band to band recombination occurs, as discussed by Garcia-Belmonte et. al.11 It is consequently necessary to introduce other conductances associated to band to band electrons and holes recombination as follows n G rec = − qL

p G rec = qL

∂U np

(56)

∂Vn

∂U np

(57)

∂V p

In Eqs. (56) and (57), U np corresponds to the rate of band to band electron hole recombination. The corresponding EC is given in Fig. 3(c) and the associated impedance can be written as follows:

Zc =

Yµn + Yµp n p Yµp Yµn + Yµp Grec + Yµn G rec

=

1 R0 −1 + iωC µeff

(58)

where

R0 =

C µn + C µp n C µ Grec

C µeff =

p

(59)

n

p + C µ G rec

C µp C µn

(60)

C µn + C µp

We shall now examine in further details the practical use of the general equivalent circuit of Fig. 2. Two issues of particular interests will be addressed in the following. The first one is related to the required conditions for the observation of two semicircles impedance spectra. It should be pointed out that in the vast majority of cases of IS measurements of solar cells only one recombination semi-circle is observed. In fact, one single example where both carriers show a distinct spectral footprint has been reported to the best of our knowledge.17 The second issue involves the necessity to take into account the current source of the EC of Fig. 2, or whether it can be omitted to simplify the treatment of experimental data. We first examine when two semi-circles can be observed. To do so we shall distinguish between two extreme cases: small and large trap capacitance. Indeed, the trap capacitor actuates as an interrupter: when the trap capacitance is small, it behaves as


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an infinite resistance (open circuit) while if it is large, the trap capacitance behaves as a wire (short circuit). Note that the trap chemical capacitance reaches its maximum in the intrinsic regime, i.e. when the population of electrons and holes are equal, and saturates at this value, contrary to the case of a unipolar semiconductor where the trap capacitance displays a peak (see Fig. 4(a)). Also note that the chemical capacitances follow the evolution of the corresponding electrochemical potentials, as depicted by Fig. 4(b). At low voltages, traps are filled by electrons and the chemical potential of traps follows the one of electrons. Since we simulated the case of an n-type semiconductor, the electrochemical potential of electrons is constant within a given range of voltage (for Vapp < 1.4 V here). Therefore when voltage is increased, the chemical potential of holes first increases with voltage. When the latter chemical potential is sufficiently high to fill traps with a substantial quantity of holes, the chemical potential of traps starts increasing along with the holes chemical potential (at around Vapp = 0.5 V). Once voltage is sufficiently high ( Vapp > 1.4 V), both, the chemical potential of electrons and holes vary at the same rate (intrinsic regime) and the system becomes ambipolar. In this case, traps are equally filled by electrons and holes and the traps electrochemical potential does not vary with voltage. In the case of a small trap capacitance, the EC of Fig. 2 can be reduced to the one of Fig. 5(a) and the valence and conduction band capacitances are in series. In this configuration, the IS spectra will be featured by one single semicircle, as indicated by the example of Fig 4(c). In the case of a large trap capacitance the equivalent circuit can be reduced to the one of Fig. 5(b), which generates two semicircles associated to the capacitances of electrons and holes as depicted by the example of Fig. 4(d). Both semicircles will be distinguishable if the trapping times associated to electrons and holes are different enough and the trapping resistances similar. It is worth remarking that for an intrinsic semiconductor trapping lifetimes only depend on the position of the traps in the band gap and are usually very similar. As a result, both semi-circles will never be neatly distinguishable for an intrinsic semiconductor, unlike for an extrinsic semiconductor where the chemical capacitances of the majority and minority carriers differ by several orders of magnitude.



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Fig. 4. (a) Evolution with voltage of the chemical capacitances associated to the tree storage modes of an N type semiconductor: conduction band ( C µn ) (green plain line), valence band ( C µp ) (blue plain line), and traps ( C µt ) (red plain line). (b) Evolution of the corresponding electrochemical potentials: Vn (green plain line), V p (blue plain line) and Vt (red plain line). (c) Impedance spectra obtained at 0.4 V, i.e. when C µt > C µn , C µp . Under this condition, the EC of Fig. 4(b) can be applied. Parameters of the simulation: N t = N v = N c =10 20 cm-3, β n = β p = 10 5 cm-3s-1, Et = −0.25 eV, Ev = −1 eV, E c = 1 eV and N ion = 1015 cm-3.


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Fig. 5. Reduction of the EC of Fig. 2 when the trap capacitance is much smaller (a) or larger (b) than that of the conduction and valence bands. In the case of a small trap capacitance, one single semi-circle can be observed in the IS spectra while for a large trap capacitance, the current sources are short-circuited and two semicircles may be observed, as detailed in the text.

Fig. 6. IS spectra simulated at applied voltage Vapp = 0.4 V with the impedance given by Eqs. (51) and (52) (blue plain line) and by taking off the terms corresponding to the n p current sources (i.e. Grec and Grec ) (red dots) in the case of an intrinsic semiconductor



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(a) and an N type semiconductor (b). For the simulation of the intrinsic case (Fig. 6(a)), we used N t = 10 7 cm-3 and β n = β p = 1 cm-3s-1 in order to impose C µt

Theory of Impedance Spectroscopy of Ambipolar Solar Cells with Trap

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