Equilibrium theory of space charge layers in conjugated polymers

Synthetic Metals 104 Ž1999. 197–209

Equilibrium theory of space charge layers in conjugated polymers II. The transition to high densities G. Paasch

a,)

, P.H. Nguyen

a,b

, S.-L. Drechsler a , J. Malek ´

c

a Institut fur Werkstofforschung Dresden, D-01171 Dresden, Germany ¨ Festkorper-und ¨ Lehrstuhl fur ¨ Experimentalphysik II, UniÕersitat ¨ Bayreuth, D-95440 Bayreuth, Germany Institute of Physics, Academy of Sciences of the Czech Republic, 18040 Prague 8, Czech Republic b

c

Received 28 December 1998; accepted 28 January 1999

Abstract For conjugated polymers whose charged excitations are electron and hole polarons ŽP. and bipolarons ŽBP. we presented in part I a non-degenerate equilibrium description for space charge layers occurring in devices. In this paper a description is given which considers the BP lattice formation at high densities including the transition between both limits which is important at finite temperatures. The theory is based on a novel approximation for the potential and temperature dependence of the P and BP densities which makes use of the degenerate limit. The latter is treated separately first for the soliton lattice. The puzzle of a high surface electric field at low band bending and a divergency for the differential capacitance are clarified. Numerical ground state energy calculations for long chains yield the dependences of the BP and P formation energies on the density. The results can be described analytically with high accuracy in terms of the Žmore simple. functional dependence of the soliton lattice. The formation energies used in the new approximation must be slightly modified to account for the deviation from the ground state at finite temperatures. Surface electric field and capacitance are calculated and discussed. Both are still extreme large in accumulation Žand inversion.. Although the divergency of the capacitance disappears there remains a large maximum indicating an extremely small extension of the accumulation layer almost at the limit of the theoretical model based on the use of a macropotential. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Conjugated polymers; Space charge layers; Bipolarons; Polarons

1. Introduction In Ref. w1x, hereafter referred to as I, for conjugated polymers with polarons ŽP. and bipolarons ŽBP. as charged states, for the first time a systematic theoretical description for the equilibrium properties of space charge layers in the non-degenerate limit Žin the sense of statistics. has been presented. In the present paper the theory is extended into the degenerate state using a new approximation w2x which describes the transition between both limits. As in I the theory is concentrated on device related quantities. Since the properties of Ps and BPs w3–6x have been discussed in detail in part I they will be summarized here only shortly. P and BP states are split off from the valence and conduction bands, charge and spin are Ž"1, 1r2. or Ž"2, 0. for P or BP, respectively, they are connected with a distortion of the polymer chain, the states do exist only ) Corresponding author. Tel.: q49-351-4659-700; fax: q49-3514659-500; e-mail: [email protected]

when the charges are present on the chain, and they are different for Ps and BPs. Therefore, the spectrum itself depends on its occupation. For this reason in the high density limit a statistical description does not exist. On the other hand, in the low density limit the statistics have been given by Refs. w7,8x. Most important quantities are the formation energies, i.e., the energies needed to add one further P or BP. In the high density limit, e.g., if the BPs are energetically favored, the BP states are broadened into a bipolaron lattice band and the P states are shifted to higher Žlower. values above Žbelow. the upper Žlower. edge of the upper Žlower. BP band Ževentually merging with it for very high densities w9x.. These situations have been described theoretically only in the zero temperature limit. Consequently, only in this limit few publications dealt with the degenerate case of space charge layers in such systems, i.e., with deviations from neutrality and the corresponding band bending near interfaces with metals or insulators. In Refs. w10,11x the soliton lattice model has been used to describe the metal–polymer contact and in

0379-6779r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 9 - 6 7 7 9 Ž 9 9 . 0 0 0 4 8 - X

198

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

Ref. w12x a model describing both the BP lattice and Ps Žour results for the soliton model but including doping have been mentioned briefly in Refs. w13–15x.. In all these papers it has been ignored that for the interesting finite temperature and for semiconductor-characteristic doping the Fermi energy must lie within the gap w1x. However, this requirement cannot be obeyed by these models and therefore their conclusions are of minor relevance for all device related properties. In this paper an approximate description of the transition between the both limits is presented based on the dependence of the B and BP densities on the chemical potential and on the temperature developed in Ref. w2x. The paper is organized as follows. In Section 2 as a model system the Su–Schrieffer–Heeger ŽSSH. continuum model for the soliton lattice w5x is considered for the following reasons. At first, its applications to space charge layers w10,11x were oversimplified Žneglect of doping. and important restrictions have not been considered in the discussion of the results. Further, it will be shown here that this model allows one to understand general limitations in systems with solitonicrpolaronic states including orders of magnitude and moreover, based on numerical calculations for chains with a non-degenerate ground state, the connection between the density and the chemical potential of this model is shown to be well suited to parametrize the actual dependence of polaron and bipolaron densities on the chemical potential needed to describe the transition from non-degeneration to degeneration. Then in Section 3 the new approximation is described. An approximate description for the dependence of the P and BP formation energies on both densities beyond the ground state is presented and finally results are given for the dependence of the surface electric field and of the differential capacity Žhence the extension of the space charge layer. on the band bending. These results are discussed and conclusions are drawn in Section 4.

2. Degenerate model systems 2.1. Soliton lattice in the continuum approximation of the SSH-model Strictly speaking, this model w5,10x describes systems with a two-fold degenerate ground state such as transpolyacetylene. Let the projection of the carbon–carbon ŽC–C. distance on the chain direction be a and the band width of the undimerized p-electron system 4 t. In the actual dimerized state the elementary cell is 2 a, the gap 2 D. With the energy zero at the midgap position the band edges of the valence and conduction bands occur at "D. The monomer density is then n m s N2 drŽ2 a. with N2 d as the areal density of the chains. If charges are added Želectrons, e. or removed Žholes, h. states are split off from the valence and conduction bands and the band edges are

shifted to "´qs "Drr and a soliton band is formed in the region from y´ - to q´ -s D'1 y r 2 rr. Here r F 1 is a parameter which is implicitly determined by the density p of holes Ž n for electrons. by p

2c s

ns

2

rK Ž r .

,

ns s

nm m

,

ms

4t 2D

Ž 1.

where K Ž r 2 . is the complete elliptic integral of second order Ž p s 0 for r s 1. Žin the original model w5,10x one has c s 1, but with c / 1 more complicated systems can be described approximately.. 1 m is the number of monomer units characterizing the extension of the soliton Ž2 m in units of C–C distances.. Thus, n s and m are the same as defined more generally for the non-degenerate limit w8,1,2x of systems with polarons and bipolarons. But in this limit the ‘segment’ density n s is the maximum density Ž r s 0.891 for p s n s .. Formally, here the density can become even larger, actually the description becomes inappropriate for even lower densities and in reality, e.g., commensurate phases may change the properties of the systems already for densities smaller than n s . Adding one further charge Že or h. to the system with a given density Žof e or h. requires the formation energy ´s Ždefined as a positive quantity for both electron and hole solitons., the chemical potential is then given by

´s s ´s0

m s "´s ,

EŽ r 2 .

´s0 s

2

D Ž 2. r p Žnegative sign in Eq. Ž2. for hole soliton.. ´s0 is the formation energy for one soliton in the low-density limit. The expression for ´s0 given in Eq. Ž2. is valid only within the continuum model, but in general one has ´s0 - D whereas addition of one e Žh. without any chain deformation would require just D. The dependence ´sŽ n. given parametrically by Eqs. Ž1. and Ž2. leads to an almost constant value of the formation energy for densities below n Q 0.4 n s and changes then rather abruptly into an increase with a small curvature ŽFig. 1a.. This increase is a consequence of the repulsive, short range interaction between the solitons. Expressing the density in units of the monomer density one sees that the abrupt increase starts at a concentration which depends on the solitonic extension m ŽFig. 1b.. The slope of the inverse dependence pŽ m . diverges for low densities ŽFig. 1c. leading to peculiarities to be discussed below. The dependence of this divergency d prd m s yŽ n sr´s0 .2 EŽ r 2 .rŽ1 y r 2 . K 3 Ž r 2 .4 on either p or m arises Žfor details see Ref. w18x. from the properties of the complete elliptic integrals.

1

,

'

The notations w16,17x EŽ r 2 . sH0p r2 1y r 2 sin2w d w and K Ž r 2 . s

H0p r2

(

2

2

y1

Ž 1y r sin w . d w are used, deviating from those used by Refs. w5,10x. Below the following relations are used: dw EŽ r 2 .r r xrd r s y1rw rK Ž r 2 .x and d1rw rK Ž r 2 .x4rd r sy EŽ r 2 .rw r 2 Ž1y r 2 . K 2 Ž r 2 .x.

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

199

bulk value r b from Eq. Ž1.. In the following examples we use ´s0 s 1.2 eV Žas in Ref. w1x, but as before D s p´s0r2 is used.. Undoped material and bulk values r b s 0.995, p brn s s 0.544 corresponding for m s 10 to a doping of 5.4 mol% are considered. Even for such high doping the bulk value of the chemical potential m b s y1.225 eV deviates only very slightly from the value y´s0 s 1.2 eV for the undoped material. With the energy zero at the bulk potential V b s 0 one has with Eq. Ž2. the macropotential as V s ß y m s y´s0

ž

E Ž r b2 . rb

y

EŽ r 2 . r

/

,

ßsmb .

Ž 3.

The edges of the valence and conduction bands are ´c,v s "´qq V and the edges of the soliton band are "´yq V. Fig. 2 shows the resulting bend bending. In the energy scale of the figure one cannot distinguish between ß and y´s0 . Therefore, the complete depletion of the extremely high doped material leads to a negligible band bending, whereas the narrowing of the soliton band is clearly seen. Accumulation is shown down to a value of r s 0.895 corresponding already to an unrealistic high value prn s f 1. Thus, realistic values of the surface potential are expected to be clearly less than 0.2 eV. Due to the broadening of the soliton band its lower edge is bent downwards for the upwards bent potential. The length scale for the Fig. 1. Dependence of the chemical potential on the density according to Eqs. Ž1. and Ž2.. Ža. and Žb. show the same dependence where the density is given in mole percent of the monomer density for different extensions m of the soliton. Žc. shows the divergency of the derivative d prd m.

It follows from Eqs. Ž1. and Ž2. that Žin equilibrium. there can be either e or h on the chains and in any case one has for the chemical potential < m < G ´s0 . However, the realistic case of finite temperatures and doping such as in usual semiconductors leads to a position of the chemical potential in the gap Ž< m < F ´s0 . w1x which cannot be described within the soliton lattice model. Thus the sketch of the bands given in Ref. w10x is completely misleading. The same situation will occur in a conventional semiconductor for T ™ 0, here the Fermi energy lies either in the valence or in the conduction band, < ´ F < G D, but not within the gap. 2.2. Band bending, surface electric field and differential capacity for the soliton lattice model A deviation from neutrality leads to a macropotential w19x Žfor restrictions see Ref. w1x. i.e., to a band bending. Its spatial dependence follows from the solution of the Poisson equation, but the principal insight is already obtained by representing the band bending as a function of the density or, according to Eq. Ž1. as a function of the parameter r. In equilibrium one has a constant electrochemical potential ß s m q V. Supposing p-doping Ždensity p . with an effective density NAy of negative counterions or ionized dopants bulk neutrality NAys p b gives the

Fig. 2. Dependence of the band bending on the soliton density Žexpressed via the parameter r according to Eq. Ž1.. for a p-doped system: Valence and conduction band and soliton band edges Žfull., macropotential V Ždashed. and the electrochemical potential Ždash-dotted.. Ža. accumulation and Žb. depletion.

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

200

band bending becomes clear below and lies in the region less than 1 nm. Therefore, near an interface only either one or the other of the two situations shown in Fig. 2 and nothing else is realized. As mentioned above a value of the electrochemical potential <ß < - ´s0 is not possible within this model. At finite Žroom. temperature values <ß < - ´s0 occur usually at interfaces and in the depletion region with the smooth transition to inversion and require to consider the transition to the non-degenerate case. Bulk neutrality is used as the boundary condition, i.e., V X Ž`. s 0, and the potential zero is chosen as V Ž`. s V b s 0. Then the electric field at a planar interface Žsurface, denoted by the index s. Es s VsXre follows from the first integral of the one-dimensional Poisson equation Žfor the potential energy V . e 0 e V Y s e r . As already mentioned one can have either e- or h-solitons on the chains. For simplicity only p-doping is considered here. Then the total charge density is with Eq. Ž1. 1

½

r s e Ž p y NAy . s 2en s

1 y

2

rK Ž r .

r b K Ž r b2 .

5

.

Ž 4.

For the surface electric field one obtains E s ysgn Ž r b y r .

=

Esy

LD s

ž

EŽ r 2 .

´s0re

)

y

r

LD

LD E Ž r b2 rb

½

ee 0 ´s0 2

´s0re

2e n s

1 r2 .

1

½ . /5 r

2

1 y

r b2

2 y

r b K Ž r b2 .

1r2

,

Ž 5.

1r2

y1

5

for r b s 1 Ž undoped. ,

Y Fig. 3. Surface electric field Es Ža., differential capacity C Žb., and extension d sc Žc. of the space charge layer in dependence on the band bending for an undoped system Ždashed. and for bulk p-doping Ž NAyr n s s 0.544. Žfull. for the soliton lattice model. Parameters are: ´s0 s1.2 eV, n s s 5=10 20 cmy3 , e s 4. The dash-dotted lines show the dependences for the undoped case with the model Eq. ŽA2., parameters bs8, as 0.5.

Ž 6. Ž 7.

Together with Eq. Ž3. this gives the parametrical connection EsŽ Vs . for E s Es and V s Vs . Here L D is the Debey length appropriate for this system. The differential capacity Žper unit area. follows from the total charge per unit area Q Y s yee 0 Es as dQ Y rs ee 0 Y C se sy ' . Ž 8. dVs Es d sc Here in the last equation d sc is the extension of the space charge layer. From these equations one obtains directly estimates for the possible orders of magnitude. With typical values w1x ´s0 s 1.2 eV, n s s 5 = 10 20 cmy3 , e s 4, one gets L D s 0.515 nm, ´s0re L D s 2.33 = 10 7 Vrcm, and 2e 2 n s L D r´s0 s 6.87 mFrcm2 . One sees already from these numbers that fields of the order of usual breakthrough fields of oxides will occur and that the extension of the space charge layer will be in the order of few atomic spacings. Thus even the concept of using a macropotential becomes somewhat doubtful. More details are displayed in Fig. 3 showing the surface electric field ŽEqs. Ž5. and Ž6.., the differential capacity,

and d sc ŽEq. Ž8.. as functions of the band bending. The dependences are shown for the undoped and for an extremely highly doped material. Evidently, in the undoped case only accumulation is possible, but even the full depletion of the highly doped material requires only a negligibly small band bending of about 25 meV. In accumulation already for only 0.15 eV band bending fields of 3 to 5 = 10 6 Vrcm are reached. The capacitance increases from accumulation towards depletion Žjust opposite as in a usual semiconductor. and diverges when the density tends to zero, and accordingly d sc becomes zero. Since the space charge layer cannot be concentrated in a region less than the atomic dimensions without a modification of the spectrum itself, and since in addition the use of the macropotential supposes the latter to be nearly constant over atomic dimensions the model becomes inapplicable in the region of the divergency and one can hardly draw any conclusion for real space charge layers Žat finite temperatures. as has been done in w11,12x. Thus, the extreme values for Es , CY and d sc show clearly that this limit of the soliton Žbipolaron. lattice becomes meaningful only if the transition into non-degeneration for low densities and finite temperature is included. In addition in Fig. 3 also results are shown Žfor doping less than 10 19 cmy3 . obtained with the

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

201

analytical model dependence ŽEq. ŽA2.. with parameters describing a rather steep increase of the chemical potential beginning at a lower density. In this case the electric field and the capacity are significantly lower and correspondingly the extension d sc of the accumulation layer is larger. Formally the divergency of the capacitance results from the divergency of d prd m which has been discussed before. But, as will be shown separately w18x, already one-dimensionality leads to such a divergency which therefore is not a consequence of the soliton formation and the self-localization. 2.3. Bipolaron lattice and polarons In almost all other conjugated polymers than transpolyacetylene one has already an external gap Žsee part I. and the charged states are Ps and BPs with the formation energies ´ P and ´ B r2 Žper charge.. We restrict ourselves here to the case of energetically favored BPs Ž ´ B r2 - ´ P .. To describe such systems we used w20,21x a discrete version of the Brazovskii–Kirova-model w5,22,23x with inclusion of a symmetry breaking interaction Žan extension of this discrete model has been used in Ref. w24x to describe the competition between Ps and BPs.. By changing the parameters of this system Žalternating transfer integral and electron–phonon interaction. one can model the gap energy and the formation energies of polarons and bipolarons of other systems. For a given charge state the chain geometry has been optimized by minimizing the total energy for chains bearing up to 600 C-atoms. This has been performed for neutral chains and for chains with increasing even numbers of holes forming hole BPs and, for a given number of BPs, for a chain with one additional hole forming a polaron. The resulting electronic spectrum is shown in Fig. 4a for parameters for which the peculiarities are visualized well. The number of charges determines the required doping which is given in Fig. 4a per site. For small doping the P and BP states lie in the gap in between the band edges of the valence and conduction bands. For higher doping the BP states develop into the BP bands with a decrease of the lower band edge Žrelative to the midgap energy. and an increase of the upper band edge. The P states and the band edges of the valence and conduction bands increase similarly as the upper BP band edge but the distances between them decrease slightly with increasingly higher doping. The corresponding continuum model gives essentially the same results w12x. For the description of space charge layers one needs the chemical potential, i.e., the formation energies containing the lattice contribution which do not follow directly from the spectrum shown in Fig. 4a. But the calculations give just the total energy density w of a chain with a given number of BPs and from the total energy difference directly the formation energy ´ P of one P for a given number of BPs. Both are depicted in Fig. 4b Žsymbols, w and the density are given here per site.. Here we used

Fig. 4. Dependence of the energy spectrum of the discrete Brazovskii– Kirova-model on doping Ža.: Conduction and valence bands, P levels, and BP levels broadening into bands Ženergy in units of the transfer integral t .. Žb. shows the energy density w Žsymbols., its approximation by Eq. Ž9. Ždash-dotted., the BP formation energy ´ B r2 according to Eq. Ž10. Žfull., its approximation by Eqs. Ž1. and Ž2. with cs1.135 and ms 3.2 Ždashed., and the P formation energy Žsymbols. with its approximation by Eqs. Ž1. and Ž2. with cs1.22 and ms 3.2.

parameters of the model resulting in values for the formation energies typically to be expected for PPV as assumed in w8x and used by us in part I. The dependence of ´ P on the density shows clearly the same qualitative behavior as the soliton formation energy ŽFig. 1a. with a low density limit ´ P0 s 1.267 eV. Indeed, this dependence can be interpolated almost perfectly with Eqs. Ž1. and Ž2. with m s 3.2 Žsee just below. and c s 1.22. The BP formation energy follows from the derivative of the energy density with respect to the density. But, the energy density, Fig. 4b, calculated within the discrete model for a finite system Žperiodic ring. can be determined for a discrete number of added charges only. Thus, to obtain reliable changes in its slope we are interested in is almost impossible to do by direct numerically differentiating the calculated energy density. On the other hand, since ´ P can be described with the functional dependence of the soliton model, we interpolated the calculated values of the energy density by a similar expression. The corresponding expression is given again parametrically by Eq. Ž1. Žwith c s 1. and

ž

w s ´s0 n s 1 y

1 r2

qb

2 EŽ r 2 . r2 KŽ r2.

/

.

Ž 9.

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

202

Whereas in the soliton model one has b s 1 an optimum description of the calculated discrete values is obtained Žwith D s 1.95 eV, m s 3.2. with a slightly deviating value, in Fig. 4b with b s 0.9. Then instead of Eq. Ž2. the formation energy follows from ´s s EwrEp s ŽEwrEr .r ŽE prEr . as

ž

´s s ´s0 b

EŽ r 2 . r

y Ž1yb .

1yr2 K 2 Ž r2 . r

EŽ r 2 .

/

.

Ž 10 .

With this dependence the BP formation energy ´ B r2 is calculated as shown in Fig. 4b Žaccounting for the double charge appropriately., the low density limit is in this case 1.118 eV. As a further simplification, the same dependence is obtained with a negligible error by using the original and more simple soliton expressions Eqs. Ž1. and Ž2. with c s 1.135 and m s 3.2 as demonstrated in Fig. 4b also. Thus, the dependence of the P and BP formation energies on the density is qualitatively the same as that one for the soliton lattice, being especially almost constant up to a high concentration Žcorresponding to a divergency of the inverse slope. and than a relatively sharp transition into an increase with a small curvature does occur. The physical origin of this behavior rests in the fact that BPs and Ps are as solitons rather well-localized objects with only exponentially deceasing tails. Then, the repulsive interaction between them becomes important only at high densities. These general dependences can be described almost perfectly with the dependence of the soliton lattice model Eqs. Ž1. and Ž2. but with a parameter c deviating now slightly from unity Žinstead of these parametrical dependence one can use the more convenient explicit approximation given in the Appendix.. Of course, these results describe the ground state properties of the systems.

3. General description for arbitrary densities and temperatures 3.1. NoÕel approximation for polaron and bipolaron densities Due to the peculiarities of conjugated polymers with polarons and bipolarons as charged states, as outlined in part I, a statistical treatment beyond the non-degenerate limit is missing until now. Recently it has been shown by us w2x that a rather good approximation for the dependence of the polaron and bipolaron densities on the chemical potential and on the temperature is obtained by using the formal expressions of the non-degenerate limit w8x as described in part I but replacing the constant low-density limits of the formation energies of polarons and bipolarons by their values for arbitrary finite densities. This approximation has been shown to give exactly both the low-density non-degenerate limit and the high density limit for low temperatures. Furthermore, it has been proved that with the

analogous treatment of the quasi-one-dimensional conventional semiconductor the corresponding Fermi-integral describing the densities of electrons and holes is approximated with a rather small error. Thus we use for the electron Ž n. and hole Ž p . densities of polarons ŽP. and bipolarons ŽB. the expressions given in part I Žhere, to distinguish also formally between the chemical and the electrochemical potential, or Fermi energy, the latter is denoted by z . n P s n s Ny1 g P exp

zyV . y´P

½Ž ½ Ž ½Ž ½ Ž

½ ½ ½

½Ž

q g P exp y

q g B exp

kT

2 zyV . y´B

p B s n s Ny1 g B exp y

N s 1 q g P exp

kT

5

5

2 zyV . q´B kT

zyV . y´P kT

kT

kT

5 5

2Ž z y V . q ´ B kT

5

5

Ž zyV . q´P

2Ž z y V . y ´ B

q g B exp y

5

zyV . q´P

p P s n s Ny1 g P exp y

n B s n s Ny1 g B exp

kT

5

Ž 11 .

but now the formation energies depend on the densities themselves

´ P s ´ P Ž n B , p B ,n P , p P . ,

´ B Ž n B , p B ,n P , p P .

Ž 12 .

and therefore Eqs. Ž11. and Ž12. is a coupled, implicit system of equations for the densities. As before n s s n m rm follows from the monomer density n m and the extension m of one monomer unit. We have already included in Eq. Ž11. the macropotential V s Vi y e w with the constant intrinsic level Vi and the electrostatic potential w caused by deviations from the local neutrality and by an external bias. As in part I a possible e–h asymmetry has been ignored for the sake of simplicity. Principally the formation energies ŽEq. Ž12.. depend on all four densities. Practically however, one has either the case of low densities where e and h densities can become of the same order of magnitude, then the formation energies approach their low density limits ´ P0 , ´ B0 . In the other case of high densities either e or h dominate. The formation energies depend only on the respective densities and they follow w2x

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

from the partial derivatives of the energy density with respect to the densities

´ P s ´ P Ž pB , pP . s ´ B Ž pB , pP . s

E pP

Ewp Ž p B , p P . Ewn Ž n B ,n P . En P

Ewn Ž n B ,n P . En P

from theoretical ground state considerations as outlined in Section 2.3 and in the Appendix, but slightly modified as

´ B Ž n B ,n P .

,

´ B0

Ž 13 .

E pB

´ P s ´ P Ž n B ,n P . s ´ B Ž n B ,n P . s

Ewp Ž p B , p P .

,

Ž 14 .

where either Eq. Ž13. or Eq. Ž14. is to be used for p-material or n-material, respectively, and wp Ž p B , p P . and wnŽ n B , n P . are the h- or e-contributions to the energy density Žwithin the low-density limits they read as ´ P0 p P q ´ B0 p B or ´ P0 n P q ´ B0 n B .. In the following only p-material will be considered. If now, e.g., the bipolarons are energetically favored, i.e., ´ B r2 - ´ P holds for all densities, then in the ground state one has on one polymer chain only bipolarons with the transition to the bipolaron lattice at high densities or the same plus one polaron. This is no longer true at finite temperatures. Therefore, wp Ž p B , p P . and hence ŽEq. Ž13.. do not represent ground state quantities and they cannot obtained theoretically, e.g., by minimizing the total energy of one chain as done in Section 2.3. Thus the main problem to apply the approximation Ž11. to Ž14. is to find appropriate approximations for the density dependence of the polaron and bipolaron formation energies. 3.2. Parametrization of bipolaron and polaron formation energies In Ref. w2x we developed approximations for the dependence of the BP and P formation energies on the densities and used them together with the approximation Ž11., Ž12. to explain a number of electrochemical observations in oxidizing conjugated polymers w25,26x. Such approximations are possible if one of the species is energetically favored Žby, say, more than 2 kT .. As an example we consider as in Ref. w2x the case of energetically favored BPs, i.e., ´ B r2 - ´ P . The approximation should fullfil several conditions. In the limit of low temperatures the chemical potential is just y2 m s ´ B Ž n B ,n P ( 0. and one has either none P Ž n P s 0. or one P on a chain with a P formation energy ´ P Ž n B , n P ( 0. Žin the latter case one has n P s Žmean chain volume.y1 < n s .. In addition, when for finite temperature one has a finite Žnon-ground state. polaron density due to Eq. Ž13. or Eq. Ž14. one has to require E´ B rEn P s E´ PrEn B . As shown in Ref. w2x this can be achieved in the following manner. For the dominating bipolarons we use the functional dependence following

203

s 1 q b exp Ž yan sr Ž n B q n P . .

s1qb

ž

´ Bs r´ B0 y 1 b

n sr Ž n B qn P .

/

.

Ž 15 .

where ´ B0 is the low-density limit value, and in contrast to the simple dependence given in the Appendix, now ´ B depends on n B q n P , ´ Bs is the value of the formation energy for n B q n P s n s . ´ B0 , b, and either a or the ratio ´ Bs r´ B0 are the parameters which have to be obtained either from theory or empirically by fitting to electrochemical and optical data. Further, following Ref. w2x we write the P formation energy as 1 ´ P Ž n B ,n P . ' ´ B Ž n B ,n P . q D ´ P Ž n B ,n P . . 2

Ž 16 .

The main dependence on the densities is already contained in the first contribution, the BP formation energy, as demonstrated for the ground state in Section 2.3. For the dependence of the remaining contribution D ´ P one has to consider, in principle, that at finite temperatures, in spite of their larger formation energy the density of the Ps is larger than the BP density for low densities w1x, reaches a maximum value with increasing chemical potential, and decreases then again. As shown in detail in Ref. w2x this leads to a small deviation of D ´ P from its low density limit which is important to describe correctly the density dependence of the polaron density as observed in ESR measurements. But for the space charge layers considered here, it is always the total charge density that determines the screening and the resulting band bending. Thus, as the most simple approximation the low-density limit D ´ P Ž n B ,n P . f D ´ P0 ,

D ´ P0 ' ´ P0 y ´ B0 r2

Ž 17 .

can be used in Eq. Ž16.. In practice we use the more elaborate expression derived in Ref. w2x D ´ P Ž n B ,n P . s D ´ P0 q

y ´ 20 d

1 2

´ B0 b

ty1

ž

ž /

t s Ž ´ Bs r2 y D ´ P0 . r´ 20 ,

d

´ Bs r´ B0 y 1 b

ns

/

n B qn P

ns n B qn P

,

´ 20 s ´ B0 y ´ P0

Ž 18 .

With b f 8 Žsee Appendix. the narrow range of the additional parameter d f 11 . . . 14 w2x describes a small increase of D ´ P Ž n B ,n P . for high densities ŽŽ n B q n P . R n s . but for even higher densities again D ´ P0 is reached since then in any case the BPs dominate as in the ground state Žsee Section 2.3.. Fig. 5a shows the BP formation energy per charge ´ B r2 Ž15. and the P formation energy ´ P Ž16.

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G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

constant low-density values for the formation energies, the low-temperature degenerate case ŽBK. Žgiven by our approximation for zero temperature also., and our new approximation. In the non-degenerate case one has correctly the different exponential increase of both densities for low Žnegative. values of the chemical potential, but the transition of the BP density into the maximum saturation value occurs in a region of only several kT and the P density shows a symmetric maximum. On the other hand, the degenerate limit results in a zero P density, the BP density vanishes with a singular behavior at the BP formation energy per particle. In the new approximation one has the correct non-degenerate behavior, but for higher total densities the BP density becomes almost the same as in the degenerate limit and the P density peak becomes rather asymmetrically broad. In addition, in Fig. 6b the influence of varying the parameter d is shown Žbetter seen in the linear scale used here.. This parameter influences Žas long as the P formation energy exceeds ´ B r2 by more than several kT . in essence only the details of the tail in the P density. For the chosen parameters this dependence according to electrochemical observations is described well with d s 12.

Fig. 5. The BP formation energy Eq. Ž15., the P formation energy ŽEqs. Ž16. and Ž18.. Ža., and D ´ P ŽEq. Ž18.. Žb. as functions of the total density of BPs and Ps for different parameters d and for D ´ P s const. Other parameters are given in the text, T s 300 K.

with Ž18. with low density limits 1.15 eV and 1.2 eV, respectively. The remaining parameters here and in the following examples are according to Ref. w2x, see Appendix, b s 8 and ´ Bs r´ B0 s 1.326 Žor c s 1 in Eqs. Ž1. and Ž2. equivalent to Eq. Ž15... Both formation energies show the general dependence discussed already for the soliton lattice. The dependence of ´ P on the parameter d in Eq. Ž18. is better seen in Fig. 5b showing D ´ P Ž n B ,n P .. Its increase for higher densities and the subsequent decrease to approximately the low density value reflects both the results of numerical ground state calculations ŽSection 2.3. and the fact that Žfor room temperature. one has more polarons as in the ground state leading to a further increase of their formation energy Žwhereas at highest densities again the BP dominate.. This dependence is the best one to describe simultaneous cyclic voltammograms and ESR measurements in the oxidizing process of conjugated polymers as shown in Ref. w2x. Consequences of varying the parameter d are seen below. Fig. 6 shows the resulting hole BP and P densities by using Eqs. Ž15., Ž16. and Ž18. in the general approximation Eq. Ž11.. Both densities are shown in dependence on the chemical potential Žat room temperature and for zero macropotential. from the lowdensity limit up to high densities. Fig. 6a gives the comparison between the following different approximations: the non-degenerate case ŽDSS. w8x, i.e., Eq. Ž11. with the

Fig. 6. Ža. BP and P densities as functions of the electrochemical potential for the soliton lattice model ŽBK., and for T s 300 K with the full non-degenerate statistics ŽDSS. w8x and with the present model. In Žb. for the present model the influence of the parameter d on both densities is shown Žlinear scale..

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

205

3.3. Surface electric field and differential capacity With the approximations given in the preceding subsections one obtains the total charge density

r s e Ž p P y n P q 2 p B y 2 n B y NAy .

Ž 19 .

as a function of the macropotential V. For the sake of simplicity only acceptor doping is considered here and as in part I we suppose the existence of a constant density of ionized acceptors Žtraps, counterions.. The bulk value V b is determined by this density. Considering semiconductorlike doping Ž- 10 19 cmy3 . one gets Ž z y V b . from bulk neutrality r s 0 by using in Ž19. the non-degenerate limit Žthe Fermi energy lies in the gap sufficiently far from the P and BP formation energies.. Then according to part I Ž z y V b . follows from Žnotation slightly modified according to the more general formulation used here. sinh

Vb y z kT

y ´ 20 r kT

qe

sinh 2

n i P s 2 n m exp Ž y´ P0rkT . ,

Vb y z

s

kT

´ 20 s ´ B0 y ´ P0

NAy 2 niP

,

Ž 20 .

As in part I the essential informations on the space charge layers follow already from the first integral of the Poisson equation resulting in the dependence of the Žsurface. electric field Žor total charge per unit area. on the potential Žband bending. at the surface. The bulk neutrality boundary conditions are the same as before in part I V Ž`. s V b , V X Ž`. s 0 Žin contrast to Section 2 now it is more convenient to choose the zero of the potential not at V b but at the Fermi energy z .. The numerical integration requires some care since the Žfor acceptor doping. hole P and BP densities are given only implicitly by the coupled equations as explained above. The differential capacity follows as before from the second Eq. Ž8.. In the following results are discussed. The dependence of the electric field on the band bending in shown only in the accumulation region, in the depletion region the nondegenerate results given in part I are valid and in the inversion region one has the same behavior as in the accumulation region. Fig. 7a–Fig. 9a show the electric field and Fig. 7b–Fig. 9b the resulting differential capacity. For the latter one has to have in mind that a value CY R 10 mFrcm2 corresponds to an extension of the accumulation layer d sc Q 0.35 nm Žfor e s 4. which is of the order of the interchain spacing. This gives a limit for applying the description with a macropotential itself. In all cases we use for the low density limit of the P formation energy as in Ref. w1x, ´ P0 s 1.2 eV, n s s 5 = 10 20 cmy3, e s 4, and the parameter b s 8 as explained above. In Fig. 7 with ´ B0 r2 s 0.9 eV and NAys 10 17 cmy3 the different approximations are compared with one another: the non-degenerate approximation Ž11. with the constant low-density limits for the formation energies, either with the denominator N s 1 in Eq. Ž11. ŽDSS with N s 1. or with the full denominator ŽDSS., the degenerate approxi-

Y Fig. 7. Surface electric field E Ža. and differential capacity C Žb. as functions of the band bending V y z for different models: BK, DSS, and the present model; DSS with N s1 Žsee Eq. Ž11.. is the approximation used in part I, T s 300 K.

mation ŽBK. Žas in Section 2 and at the same time this is the low temperature limit of our description. and finally the model described here. DSS with N s 1 is the approximation used in part I and the exponential dependence of the electric field shown in Fig. 7 is the same as in Fig. 7 of part I in the accumulation region Žbut there much larger intervals are shown on both axes.. The denominator in Eq. Ž11. arise within the non-degenerate approximation from the fact that only a maximum of Ps and BPs can be distributed over the chains. Its inclusion leads to a much smaller increase of the field for higher band bending and, surprisingly to a maximum in the capacity. The degenerate limit has been discussed already in Section 2, it is characterized by a field going to zero at V y z s ´ B0 r2 and a diverging capacity, a smaller band bending cannot be described. Our new approximation coincides with the non-degenerate limit for smaller band bending and approaches the degenerate limit smoothly for stronger accumulation. Also this best approximation shows the maximum of the capacity which is not known in three dimensional systems. In contrast to the densities themselves shown in Fig. 6 Žand which are directly accessible in electrochemical doping experiments. the difference between our new full description and the non-degenerate approximation with inclusion of the denominator is less important for the surface electric field and for the differen-

206

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

tial capacity as seen directly from Fig. 7. In Fig. 7 we used the parameter d s 12. A variation of d in the range as in Fig. 6 cannot be seen in Fig. 7. Only for a much smaller difference between the P and BP formation energies Žbut ´ B0 r2 must be still several kT smaller than ´ P , see above. the field and the capacity are modified slightly by some percent. Thus, the results are not sensitive to the exact value of this parameter. In Fig. 7 the BPs are energetically strongly favored Žthe smaller ´ B0 r2 is the more are the BPs favored.. Now in Fig. 8 we compare at first different higher BP formation energies, all with the same P formation energy. One sees that the field and the maximum capacity decrease when the BPs are less favored, especially when ´ P y ´ B0 r2 becomes comparable with kT. This is immediately understood having in mind that one has then relatively more Ps with only one charge which are less effective in the screening than the BPs. Furthermore, in Fig. 8 also a variation of the doping in a range typical for semiconducting applications is demonstrated. As in the case of normal inorganic semiconductors the range where degeneration becomes important is not influenced by the doping since in the strong accumulation case one has already more charges than dopants. It has been seen in Fig. 7 that the new approximation, considering the BP lattice formation, gives qualitatively

Y Fig. 9. Surface electric field E Ža. and differential capacity C Žb. as functions of the band bending V y z for the full non-degenerate approximation ŽDSS., for the present model with ds12, and with Eqs. Ž15. and Ž17. with values of a and b describing an increase of the formation energies already at lower densities Žsee also Fig. 10 and Fig. 3., T s 300 K.

the same result as the full non-degenerate approximation Žincluding the denominator in the densities. and only small deviations between them. As discussed already in Section 2.2 and in the Appendix one should expect lower fields and capacities if the increase of the BP formation energy begins already at lower densities. Fig. 9 shows the corresponding results. Since in this example the BPs are strongly favored the Ps are unimportant and one can use D ´ P s D ´ P0 . With Eq. Ž15. and the parameters b s 8 and a s 0.5 used already in Fig. 3 both the field and the capacity become smaller and even the maximum of the capacity is now only at about 10 mFrcm2 , a value up to which especially the use of a macropotential can be regarded as reasonable.

4. Discussion and conclusions

Y Fig. 8. Surface electric field E Ža. and differential capacity C Žb. as functions of the band bending V y z for a given P formation energy and different BP formation energies for 10 17 cmy3 doping. For the lowest BP formation energy the influence of different doping Ž10 16 , 10 17 , 10 18 cmy3 from left to right. is shown also, T s 300 K.

In this paper, as an extension of part I, for the first time a general description has been developed for space charge layers in conjugated polymers with polarons and bipolarons as the charged states. The theory is valid for finite Žroom. temperature and for arbitrary densities including the transition to extremely high densities Ždegeneration in

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

the sense of statistics. with the formation of the bipolaron lattice. The zero temperature degenerate limit is treated separately for the SSH Žsoliton. model including doping. This limit yields Ži. for accumulation and for the highly doped case for depletion extremely high surface electric fields and differential capacities connected with rather small band bending, the corresponding extension of the space charge layer is of the order of the interchain spacing indicating that such a treatment reaches the limits set by using a description with a macropotential. Žii. As a peculiarity, the capacity diverges in the limit of vanishing density. This divergency results from one-dimensionality and the T s 0 limit, only the type of the divergency is determined by the nature of the self-localizing solitonic states. Žiii. Due to this divergency it is not clear whether the rather high field and capacity and the low extension of the space charge layer will occur in the finite Žroom. temperature case also; since in addition in this limit a position of the Fermi level in the gap Žfor finite T and doping as in a usual semiconductor. cannot be described, the general treatment with the transition from low to high densities is needed. Numerical ground state energy calculations have been carried out for the Brazovskii–Kirova-model to describe systems with polarons and bipolarons including the formation of the bipolaron lattice at high densities. From these results, Ži. interpolating the calculated energy density by a generalization of the SSH formulae one can calculate the dependence of the BP formation energy on the density. Žii. Both this BP formation energy and the directly obtained formation energy for one additional P can be described with an negligible error by the formulae of the soliton model and Žiii. for the latter a simpler explicit approximation can be used. For the general case, since the statistics for such systems is missing due to the properties of solitonic Žpolaronicrbipolaronic. states, we use the approximate dependence of the P and BP densities on T and on the chemical potential developed by us in Ref. w2x. This approximation makes use of the functional dependence of the non-degenerate limit but the two formation energies have to be replaced by their dependence on both densities. The quality of this approximation has been proved in Ref. w2x and is connected with one-dimensionality. The desired dependences ´ P Ž n B , n P ., ´ B Ž n B , n P . deviate from their ground state dependence, reasonable approximations are used. Calculations of the surface electric field and of the differential capacity lead to following results: Ži. Deviations from the non-degenerate case presented in part I do occur only for the strong accumulation Žinversion. case when due to the band bending the electrochemical potential approaches ´ B r2 Žabout one to two kT .. Žii. The singularity in the differential capacity disappears but there remains a maximum which is unknown in usual three-dimensional semiconductors. Žiii. For typical parameters Žin-

207

crease of the formation energies at about 0.4 n s . the maximum is larger than 10 mFrcm2 Žextension of the space charge layer below the atomic spacing. at fields above the break through limit Žsome 10 6 Vrcm., Živ. details of the model Žour parameter d . and the doping level are unimportant in this region, the maximum decreases slightly when the Žsmaller. BP formation energy approaches the P formation energy. Žv. On the other hand, if the increase of the formation energies begins already at lower densities, the maximum of the capacity and the magnitude of the surface electric field remain in a reasonable region. In such cases the formation of the BP lattice should be possible in space charge layers. Actually, until now the needed parameters are not known sufficiently well. Especially the rather short range interaction of the BPs, resulting in the almost constant formation energy up to rather high densities, results from models which do not include other interactions, especially the Coulomb interaction. Thus the question whether in a space charge layer a BP lattice can occur or not remains open and the theory presented here should be helpful to analyze future relevant experimental results. But in most cases it will be sufficient to use the non-degenerate formulation given in part I.

Acknowledgements We thank S.A. Brazovskii, N.N. Kirova, A. Saxena, W. Streitwolf, W. Brutting, and M. Schwoerer for discussions. ¨ Financial support from the Deutsche Forschungsgemeinschaft and from the Bayerische Forschungsstiftung ŽFOROPTO. is gratefully acknowledged.

Appendix A. Explicit approximation for the soliton lattice model The expressions Ž1., Ž2. represent only an implicit Žparametrical. connection between the density and the chemical potential and they are somewhat inconvenient due to the lack of an explicit dependence of the chemical potential on the density. Using their expansion in lowest order w17x Ž E ™ 1 q 1r2ŽlnŽ4rr X . y 1r2. r X 2 , K ™ lnŽ4rr X ., r X s '1 y r 2 ™ 0. the explicit dependence is then in the low density limit Ža similar expansion has been used in Ref. w27x.

m ´s0

™1q

ž

16 n s n

/

q 4 exp Ž y4n srn .

Ž A1.

As shown in Fig. 10 the exact dependence Žalmost constant below n Q n s , sharp transition into an increase with small curvature. is described correctly up to high densities nrn s Q 0.7 and shows a small deviation for even higher densities. This expansion is a little bit inconvenient, since it

G. Paasch et al.r Synthetic Metals 104 (1999) 197–209

208

But the density at which the transition takes place and the slope for higher densities can vary. Such variations can be easily modelled with Eq. ŽA2.. as demonstrated in Fig. 10. Therefore, the corresponding expressions for the dependence of the electric field on the band bending and for the capacity are given here also. With the same boundary conditions for the macropotential as in Section 2.2 a bulk acceptor doping gives the electrochemical potential yßr´s0 s 1 q b expŽyan srNAy. , the charge density depends then on the macropotential as

r s en s

ž ln

ya

ß y Ž V q ´s0 . m0 b

y

NAy ns

/

0

Ž A3.

and the electric field is given by

E s sgn Ž yV .

´s0re LD

° 'ab ~E ¢

1

ž ž yln

ß y Ž V q ´s0 . ´s0 b

//

¶ • ß

1r2

ž ž //

yE1 yln Fig. 10. Dependence of the chemical potential on the density for the soliton lattice model: comparison of the exact expressions Eqs. Ž1. and Ž2. with the low density expansion Eq. ŽA1. and with the approximation Eq. ŽA2.. In Žb. different values of c in Eqs. Ž1. and Ž2. and of b and a in Eq. ŽA2. are used.

cannot be solved for nrn s and it is not integrable. But the main features are described rather good already with the approximation

m ´s0

s 1 q bexp Ž yan srn . s 1 q b

ž

msr´s0 y 1 b

V

1

y bm 0

ln

ß y ´s0

ž / ´s0 b

Ž A4. where E1 is the integral exponential function and the Debey length L D is given in Eq. Ž7.. The capacity follows from the second Eq. Ž8. with Eqs. ŽA3. and ŽA4..

References

n srn

/

ß y ´s0 ´s0 b

.

Ž A2. with b s 8 and a s 3.2 as demonstrated in Fig. 10a. In the second equation in ŽA2. m s s mŽ n s . is used as a second parameter instead of a. Considering b and a as parameters, Eq. ŽA2. can be used to approximate Ž1., Ž2. with c / 1. The accuracy of the approximation ŽA2. has been discussed already in w2x and is demonstrated in Fig. 10b for c s 0.9, 1.0, 1.1. With Eq. ŽA2. one has not only an explicit approximation for Eqs. Ž1. and Ž2. but one can also interpolate numerical results for systems with polarons and bipolarons Žsee Section 2.3.. Moreover, theoretical models as for the soliton lattice or the generalization with symmetry breaking contain a lot of unrealistic simplifications Žtight binding, neglect of Coulomb interaction. which presumably will not change the general features of the dependence ´sŽ n.: almost constant up to a high density and then a sharp transition into an increase with a small curvature.

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