Quasi-one-dimensional devices utilizing two conducting polymers

Synthetic Metals, 15 (1986) 23 - 47

23

QUASI-ONE-DIMENSIONAL DEVICES UTILIZING TWO CONDUCTING POLYMERS A. SAXENA and J. D. GUNTON Physics Department, Temple University, Philadelphia, PA 19122 (U.S.A.) (Received October 2, 1985;in revised form January 14, 1986; accepted January 16, 1986)

Abstract Expressions for the chemical potential (Fermi level) are obtained as a function of the Peierls gap and the doping level for those quasi-onedimensional (Q1D) semiconductors and metals whose excitations not only include electrons and holes but in addition one or more of the following: solitons, polarons and bipolarons. The possibility of fabricating Q1D devices such as Q1D p - n junctions, Q1D Schottky barriers and Q1D heterojunctions is considered. Some specific examples of Q1D devices, including a Q1D S c h o t t k y barrier involving a junction between polaronic trans-(CH)~ Ca metal) and solitonic trans-(CH)~ (a semiconductor), are discussed. The photoresponse, as well as J - V and C - V characteristics of the Q1D devices, are discussed in terms of the chemical potential of the material on each side of the junction.

1. Introduction In this paper we derive expressions for the chemical potential (Fermi level) as a function of the Peierls gap and the doping level for the quasi-onedimensional (QID) semiconductors and metals. Our motivation for this is to provide necessary information for optimizing the performance of a variety of Q I D solid state devices such as Q1D p - n homojunctions, Q1D Schottky barriers and Q I D heterojunctions. It is important to note that such devices can be ' t u n e d ' by varying the doping level and the choice of the material. The temperature and doping dependence of the Peierls gap in Q1D materials is k n o w n to determine their conducting properties [1, 8, 40] (e.g., metallic, semiconducting or superconducting). For example, at low doping levels trans-(CH)x is known to be a solitonic semiconductor, whereas at high dopant concentrations it becomes a polaronic metal [2]. Similarly, quarter-filled-band, large ' U ' Q1D conductors [3, 4], such as (NMP)~(Phen)I_x(TCNQ), are semiconductors up to room temperature. According to theoretical calculations [4, 39] the Peierls gap is expected to vanish at 0379-6779/86/$3.50

© Elsevier Sequoia/Printed in The Netherlands

24 higher temperatures. A wide variety of Q1D conducting polymers is now available. The dominant non-linear excitations in these materials are solitons [5], polarons [6, 12] and bipolarons [7, 11, 12]. Each distinct conducting polymer has a different value of the Peierls gap and a different set of excitations. This well-known fact suggests the possibility of fabricating a whole gamut of Q1D solid state devices, which includes (a) Q1D p - n junctions (b) Q1D Schottky barriers and (c) Q1D heterojunctions. These devices are expected to exhibit somewhat different electrical, photovoltaic and transport properties from those of the usual devices fabricated from inorganic semiconductors and metals. These differences arise from the presence of localized electronic states in the Peierls gap and the existence of highly mobile charged defects (solitons, polarons and bipolarons) that may contribute to the transport properties [8, 40]. To our knowledge, one a t t e m p t to fabricate a Q1D p - n junction [9] and also a Q1D heterojunction [10] has been reported so far. Recently a Q I D Schottky barrier has also been fabricated [38]. However, we do not know of any other Q I D devices. The statistics of various non-linear excitations have been recently worked out by Conwell and Howard [3, 4, 11, 12]. In Section 2 we make use of these statistics and the relations among the chemical potentials of solitons, polarons, bipolarons, electrons and holes to obtain explicit expressions for the Fermi energy (chemical potential) in terms of the Peierls gap and the dopant concentration in the following cases: (1) The half-filled-band Peierls-distorted Q1D conductors with U < t and doubly degenerate ground states. (Here U denotes the Coulomb repulsion for a second electron on a site and t is the inter-site transfer integral.) The prototype of such a material is trans-(CH)x [8]. The dominant excitations are solitons. (2) Highly doped trans-(CH)x and lightly doped cis-(CH)x [6, 13]. The excitations are polarons. (3) Highly doped cis-(CH)~. The excitations are bipolarons. (4) Half-filled-band Q1D conductors with U < t and a non-degenerate ground state. Examples of polymers in this category are polypyrrole (PPy) [14], polyparaphenylene (PPP) [15] and polythiophene (PT) [16]. Both the polarons and the bipolarons can coexist in these materials. (5) Q1D quarterfilled-band Peierls-distorted conductors with U > t. Examples of this type include those charge transfer salts in which the interchain potential is not important. The dominant excitations in this case are fractional charge solitons [19] with +le/2] charge. (6) Q1D quarter-filled-band, large 'U' conducting polymers with significant interchain potential. Some of the examples in this category are quinolinium-di-tetracyanoquinodimethanide [Qn(TCNQ)2 ] [17] and (N-methylphenazinium)~(phenazine),_~tetracyanoquinodimethanide [(NMP)~(phen)I_x(TCNQ) ] [18] with x = 0.5. The stable excitations in this case are bipolarons with + ]el charge [4]. We do not know of any materials where solitons and bipolarons are observed to coexist as dominant excitations. Solitons and polarons can coexist in trans-(CH)~; however, it is predominantly a solitonic semiconductor at low doping levels and a polaronic metal at high dopings. Therefore we do not consider the coexistence of solitons and polarons here. Since fractional

25 charge polarons have not been observed in any of the organic conductors, we will not discuss this case either. We derive the expressions for Fermi levels in Section 2 for the cases mentioned above. In Section 3 we then discuss the performance of the Q1D devices in terms of the Fermi levels on the two sides of the junction. In particular, we discuss the performance of a Q1D S c h o t t k y barrier device fabricated by using polaronic trans-(CH)x (a metal) on one side of the junction and solitonic trans-(CH)x (a semiconductor) on the other. We also discuss examples of Q1D p - n junctions and Q I D heterojunctions. The d o m i n a n t contribution to the current transport in these devices is expected to come from solitons, polarons and bipolarons. This feature makes them generically different from the three-dimensional devices. The values of photovoltage, barrier height, etc., obtained in the examples mentioned above show, in principle, that these are reasonable solid state devices. The expressions for the Fermi level derived in the next Section take into account all of the charge carriers mentioned above. Therefore those physical quantities of the junctions that explicitly depend on the Fermi level should provide quantitatively correct results. As previously done by many workers in this field [11, 21, 22, 26, 27, 38] we first invoke in Section 3 the equations used to explain the J - V characteristics of junctions fabricated from ordinary semiconductors such as silicon and GaAs to understand qualitatively the behavior of Q1D junctions. These equations are derived with the assumption that the charge transport is due to electrons and holes everywhere in the proximity of the junction. However, in Q I D devices the charge is also transported via solitons, polarons and bipolarons. We have neglected this contribution, although it could be important in certain cases. We hope to study the effects of such carriers in a future publication.

2. Calculation of the Fermi level For the calculation of the Fermi level we need to know the electron concentration Pn in the conduction band and the hole concentration Ph in the valence band. (The subscript h is used to denote holes rather than the more c o m m o n subscript p, which we will use later to denote polarons.) These concentrations are given by

Pn = Nn exp{(Pn--Zc)/kT} = Nn e x p ( ( p n - A)/kT}

(1)

Ph = Nh exp{(Ev -- pn)/kT} = Nh exp(--(A + p,)/kT}

(2)

with 2A being the Peierls gap. The effective densities of states per unit length for electrons and holes are denoted by N . and Nh, respectively. Ec and E v denote the b o t t o m of the conduction band and the top of the valence band, respectively, while Pn denotes the Fermi energy and k is the Boltzmann constant. If one considers the electrons and holes as a one-dimensional Maxwell-Boltzmann gas, one has

26

Nn = 2(21rme*kT/h 2) 1/2

(3)

and

Nh = 2( 27rm~k T/h 2) 1/2

(4)

where me* and mh* are the effective masses of the electrons and the holes, respectively. In the following Na and Nd denote the acceptor and donor concentrations, respectively. Furthermore, N and L denote the number of m o n o m e r units on the polymer chain and the length of the chain, respectively. In what follows we invoke the charge balance equations and the statistics for the various cases mentioned earlier to obtain the Fermi level. In each case we obtain separate expressions for p- and n-type doping. The details of the calculations are given in the Appendix. In all the cases discussed below it is important to note that the Fermi level has a non-trivial dependence on the doping (N d or Na) and the Peierls gap (A). (See eqns. (6), (S), (10), (14), (16), (lS), (20), (21), (23), (24), (26), (28), (30) and (32)). As a consequence, as noted in the Introduction, these Q1D devices can be 'tuned' by varying the doping level and the Peierls gap for optimal device performance. In general A varies with doping. In polypyrrole and polyparaphenylene the gap is seen to increase with doping [14]. On the other hand, the greatly enhanced conductivity of (NMP)x(Phen)I_x(TCNQ) with x = 0 . 5 4 is attributed to the decrease of the Peierls gap with doping [39]. It is difficult to determine the effect of doping on A in polyacetylene due to the inhomogeneous nature of the doping.

2.1. Solitons The presence of solitons introduces states at the mid-gap [8, 11] (Fig. 1). In real systems there is a band of soliton states centered about the

t

T

~v

(a)

(b)

(a)

T

I (b)

Fig. 1. Schematic band diagram for a Q1D polymer with solitons. (a) For n-type doping negatively charged solitons (S-) exist and the mid-gap states are completely filled. (b) For p-type doping positively charged solitons (S +) exist and the mid-gap states are empty. The difference E c - - E v is equal to the Peierls gap A. An arrow indicates an electron with its appropriate spin. Fig. 2. Band diagram for a QID polymer with polarons. (a) For n-type doping negatively charged polarons (P-) exist. The upper polaron levels are half filled, whereas the lower polaron levels are completely filled. (b) For p-type doping positively charged polarons (P÷) exist. The upper polaron levels are empty but the lower polaron levels are half filled.

27 mid-gap. In the present analysis, however, we assume for simplicity that all the solitons introduce states exactly at the mid-gap.

(a) n-Type doping The soliton states are completely filled (Fig. l(a)). neutrality condition [11] is given by P, + 2Psf(0) = Ph + Nd + Ps

The charge (5)

where Ps denotes the soliton concentration on the chain, f(0) is the Fermi distribution evaluated at the mid-gap. Substituting the explicit expressions (see the Appendix) for the quantities in eqn. (5), we obtain the Fermi level in an n-doped solitonic Q I D semiconductor

eL;

us, n = k T I n

e

2Nn e_~Z~ + N e_~%

×

+

.

.

.

.

.

.

2

tN e- ~ a - N e_~ZS_Nd 1 \ '~ 2L

--I

tl

(6)

where fi = 1/kT and E s = 2A/~r is the creation energy for a soliton. We assume that A does not vary with d o p a n t concentration.

(b) p-Type doping In this case the soliton states are e m p t y (Fig. l(b)). Hence the charge balance is given by

ph + ps= Pn + Na"

(7)

Positively-charged solitons exist in this case. The Fermi level in the p-doped solitonic semiconductor is then given by

bts'p = --kT ln t

NaN

t

(8)

Nhe-~ a + - - e-DEs 2L

2.2. Polarons Polarons introduce two states in the Peierls gap: one below Ec and the other above E v (Fig. 2). As in the above treatment we assume that all the polarons are in these two levels and do not form bands.

(a) n-Type doping In this case the polaron state close to Ec is half filled and the other level close to E v is completely filled (Fig. 2(a)). The charge neutrality condition

[12] is

28

(9)

Ph + 2pp + Nd = Pn + 3 p p

where pp denotes the polaron concentration. Substituting the expressions for the quantities in eqn. (9) we obtain the Fermi level of an n-doped polaronic Q1D material:

11P ' n =

k T In - -

(N -- N d/p) e -~Ep + LNn e-~z~ 2lpLNn e-a(A +Ep)

+ [{(N --Ndlp) e-~Ep + LNne-~/'} 2 + 4NdlpLNne-~(z~+Ep)],/2 ] 21pLNn e -~(z~+ Ep)

(lO)

where Ep = X/~(2A/~) is the creation energy of a polaron, lp denotes the polaron length in terms of m o n o m e r units. Since the upper level in the gap is half filled, the energy per polaron decreases (within the SSH model [5]) as the density of polarons increases with doping according to the following relation: Ep = V~(2AHr ) -- (ll~r)Wp(Nd)

(11)

The polaron bandwidth Wp (within the continuum TLM model [20]) is given by

(12)

Wp(Nd) = 4 v / 2 A exp(R/x/2~0)

where R and Go = a(2t/A) are the mean separation between polarons and the electronic coherence length, respectively. The distance between the two monomers on the chain is denoted b y a. Thus the Fermi level in a polaronic c o n d u c t o r has a non-trivial dependence on b o t h the Peierls gap and the d o p a n t concentration. (b) p-Type doping The upper level in the gap is e m p t y and the lower level is half filled (Fig. 2(b)). The charge balance [12] is given by

(13)

Ph + 2pp = Pn + Pp + Ya

Positive polarons are predominant in this case. The Fermi level of a p-doped polaronic c o n d u c t o r is then given by

(N - - N a/p) e--~EP + I~ "p = --k T In - +

L N a e -~A

21pLNhe -fl(A+Ep)

[ { ( N - Na/p) e -~Ep + LNh e-~Z~}2 + 4-NalpLNh e -~(a÷Ep)] 1/2 ] 21pLNh e -~C/' +Ep)

]

(14)

29

2.3. Bipolarons As in the case of polarons, bipolarons also introduce two levels in the gap (Fig. 3). These levels are closer to the mid-band than in the polaronic case.

(a) n-Type doping Both the bipolaron levels are filled (Fig. 3(a)). The charge neutrality condition [12] is given by Ph + 2pB + N d = Pn + 4pB

(15)

where PB is the concentration of bipolarons. Substituting the expressions for the quantities in eqn. (15) gives the Fermi level for an n-doped Q1D bipolaronic c o n d u c t o r as a solution of the following cubic equation:

lLNnl-----2-B N e-~(~+~B) 2 l e~" + l ( +

n e_~

e~ - 2

Nd/B)2 e-~EB f e2e~

=0

(16)

where EB is the creation energy of a bipolaron. EB is smaller than 2Ep by an a m o u n t equal to the binding energy of polarons in the bipolaron. IB denotes the length of a bipolaron in terms of m o n o m e r units.

(b) p-Type doping Both the bipolaron levels in this case are e m p t y (Fig. 3(b)). The charge balance [ 12] equation is Ph + 2pB = Pn + N,

(17)

Doubly-charged positive bipolarons are the dominant excitations. The Fermi level of a p-doped bipolaronic conductor is then given by the solution of the following cubic equation:

E, T

I

"Y/////////////////~ Ev ~/////////////////~, (a)

(b)

Fig. 3. Band diagram for a Q I D polymer with bipolarons. (a) For n-type doping negatively charged bipolarons ( B - - ) exist. Both the upper and the lower bipolaron levels are completely filled. (b) For p-type doping positively charged bipolarons (B ÷+) exist. Both the bipolaron levels are empty. Note that the bipolaron levels are closer to the mid-gap compared to the polaron levels shown in Fig. 2.

30

tLI~I------~Be-~(Z~+EB)fe-3~n+t(N +

I LNh e - ~ I e - ~ a

Na

2

2

NTB) e-~EB f e- 2~a

- 0

(18)

2.4. Coexisting polarons and bipolarons In this case both the polarons and the bipolaron states are present in the gap (Fig. 4). The thermal equilibrium ratio of the polarons and bipolarons depends on their formation energies [12] according to e x p { ( E B 2Ep)/2kT}. Furthermore, at low doping Pp/PB is the product of this exponential factor and V~B (see Appendix and ref. 12). Therefore pp goes through a m a x i m u m as a function of doping. The expressions for pp and PB at low doping are different from those at high doping depending on whether PBlB~ NIL or PBlB~-NIL. As a result the statistics are somewhat different in the two cases and we discuss them separately.

(a) n-Type doping The charge neutrality condition [12] (Fig. 4(a)) is given by Ph + 2pp + 2pB + Nd = Pn + 3pp + 4PB

(19)

LOW doping regime. The number PBlB~ NIL. Polarons are the dominant

of bipolarons is very small, that is excitations, but their number is also much smaller than the n u m b e r of m o n o m e r units on the polymer chain, that is pplp ~ NIL. Substituting the quantities in eqn. (19) we obtain the Fermi level n

'

=

kT

In - - ~

e ~(EB-Ep) + _ _

e-~tA-EB)

N

Ill

+ __

e~(EB-Ep) .[. _ _

4

e-~(A-EB)

N

+

"dL I"21 e-~EB

(20)

N

"//'//////////////////~ ~< "////////////////////~

It

I

T T

11

T

[v "///////////////////~ "///////////////////~, (a)

(b)

Fig. 4. Band diagram for a Q1D polymer with coexisting polarons and bipolarons. (a) For n-type doping negatively charged polarons ( P - ) a n d bipolarons ( B - - ) exist (see Figs. 2(a) and 3(a)). (b) For p-type doping positively charged polarons (P+) and bipolarons (B ++) exist (see Figs. 2(b) and 3(b)).

31

High doping regime. In this case pplp ~ PBl~ and PBIB = N/L. Bipolarons are the d o m i n a n t excitations. The Fermi level is obtained from the solution of the following cubic equation: LNn e_~Z~ e 3 ~ + 2 --Nd N ~

e2t3pn +

_

e~(EB--Ep) +

Ip

e-~(A-EB)

e~Un

N

- N d e ~Es = 0.

(21)

(b) p-Type doping Only the polaron states close to E v are half filled. All other states in the gap are e m p t y (Fig. 4(b)). Positive polarons and bipolarons exist in this case. The charge balance [12] is given by Ph + 2pp + 2p~ = Pn + Pp + Na

(22)

Low doping regime. The conditions stated in the case of the n-type doping also hold in the present case. A small number of polarons are present. The Fermi level is given by n

In - - ~

= --kT

e/3(EB - E p ) .[.

e--~(A--EB)

N +

--

e ~(EB-Ep) + - -

4

e-~(A-EB)

+ - -

N

(23)

eflEB

N

High doping regime. Bipolarons dominate the polarons. The Fermi level is thus given by the solution of the following cubic equation: e- ~

e

3~ttn +

N

--N a

e- 2 ~

~ - - N a e ~EB = 0

+

-- e~(EB-Ep ) +

lp

e -B(A-EB)

e-HUn

N (24)

2.5. Solitons in the quarter-filled-band materials In the following n- and n + refer to the number of fractionally charged negative and positive solitons, respectively. (a) n-Type doping The charge neutrality eqn [3] for such a material can be expressed as Ph + No + (n- + n +)/2N = Pn + n - / N

(25)

Substituting explicit expressions for the quantities in the above equation (see the Appendix), the Fermi energy is obtained as a solution of the following equation:

32

1

4

1 e_2#v=

(26) where the creation energy for a fractionally charged (+le/21) soliton is E, = A/~. (b) p-Type doping

The charge balance is expressed as (27)

p , + (n- + n+)/2N = p , + Na + n+/N

The Fermi level is the solution of the following equation

ph--pn= (Na-- ~-~s)--(2 e-#Es)e~(~n12) +

' e-~Es

e-~(Pr,/2) +

e-2/3P-n

81, (28)

2.6. Bipolarons in the quarter-filled-band materials

The interchain potential in quarter-filled-band (U >> t) semiconductors gives rise to stable bipolarons consisting of pairs of (--e/2,--e/2) or (+e/2, +e/2) fractional charges. In what follows nB- and nB÷ denote negative and positive bipolarons respectively and nB= nB-+ nB ÷. (a) n-Type doping

The charge balance equation [4] is given by Ph

+gd

+

(29)

nB/N = Pn + 2riB-IN

and the Fermi energy is obtained from a solution of the following equation --

Ph - - Pn +

Nd =

e -~EB

1--

l

--e

-2~

4

eb'~ + e-~EB{1 + 4 e 2 ~ }

(30)

(b) p-Type doping

The charge neutrality condition is Ph + nB/N = Pn + Na + 2nB/N

(31)

The Fermi energy is a solution of the following equation: 1

2 e - ~ E B { e - 2 ~ - 4} Ph

--

Pn

--

iV.

=

l e - ~ + e-~EB{4 + e - 2 ~ }

(32)

33 3. Quasi-one-dimensional devices As we noted in the Introduction, the possibility of fabricating various solid state electronic devices using conducting polymers has drawn considerable attention. These polymers are known to be either semiconducting or metallic depending on the dopant concentration and the temperature [ 1 , 8 , 4 0 ] . In fact, p - n junctions [ 2 1 - 2 5 ] and Schottky-type barrier junctions [ 2 6 - 31] have been fabricated which have an interface with a conducting polymer on one side of the junction and a three-dimensional inorganic semiconductor or metal on the other side. We discuss here the possibility of fabricating a variety of solid state electronic devices involving different combinations of the Q1D conducting polymers (mentioned in Sections 2.1 -2.6) on the two sides of the junction. There is one control parameter, namely the doping level, which can be tuned to achieve optimal device performance. (The choice of the polymer, to a certain extent, also affects the device performance because every distinct species of polymer has a different value of A. As we have noted before in Section 2, A in many of the conducting polymers is known to vary with doping.) In particular, we consider Q I D p - n homojunctions, Q I D Schottky barriers and Q1D heterojunctions, since these form the basis for understanding most of the more elaborate Q I D solid state electronic devices such as solar cells, field effect transistors (FETs) and thyristors. A p - n junction experimentally fabricated from p- and n-doped trans(CH)x was f o u n d to be unstable [9] due to migration of dopant ions to the interface. This instability may not exist in Q1D p - n junctions fabricated from polymers with a ring structure, such as polypyrrole. Also, a heterojunction device between two conjugated organic polymers [10], namely (CH)x and poly(N-methyl pyrrole), (PNR), was reported to exhibit rectifying characteristics. Recently a Schottky barrier-type device has been fabricated between the Q I D conducting polymers [38]. In the following Section we discuss the J - V and C - V behavior, as well as the photoresponse, of Q I D devices in terms of the explicit dependence of their Fermi levels on the dopant concentration. We note, however, that the experimentally fabricated devices may not exactly follow the behavior predicted here, which could arise from poor transport due to solitons, etc. As indicated below, this results from transport via a hopping mechanism that may not be as efficient as transport due to electrons and holes. The presence of an insulating layer at the junction and interfacial surface states, which we do not discuss in this paper, may also affect junction properties. Quantities such as photovoltage, which explicitly depend on the difference between the Fermi levels on the two sides of the junction, are n o t affected by a particular transport mechanism nor the type of levels the electrons actually populate [11]. On the other hand, photoconductivity and current-voltage characteristics are very much dependent on the nature of the d o m i n a n t excitation and its transport mechanism. For instance, in Q I D devices consisting of trans-(CH)x the main contribution to

34

the conductivity comes from solitonic transport. The charge transport takes place mainly by the Kivelson mechanism [33], i.e., phonon-assisted electron hopping among solitons. Free soliton motion, however, may also contribute. Similarly interchain bipolaron hopping in PPy, PPP and PT has been suggested as a mechanism for spinless charge transport [ 1 ]. In this Section we do not a t t e m p t to develop a 'one-dimensional' theory for Q I D device characteristics which incorporates the effects of the non-linear excitations mentioned above. Instead, we invoke the 'threedimensional' theory as a qualitative guide to explain certain features of Q1D devices. This three-dimensional theory takes into account only electrons and holes and holds for all the ordinary semiconductors. On the other hand, a one-dimensional theory should take into account the contribution to the current from solitons, polarons and bipolarons as well as electrons and holes. Such a one-dimensional theory is appropriate for the Q1D conducting polymers. Clearly the three-dimensional theory cannot be adopted for a quantitative understanding of the Q I D devices, so there is a need for a complete one-dimensional theory.

3.1. Q1D p - n ]unction Non-uniform doping across a p - n junction induces an electrostatic potential ~b. Within the semiclassical treatment [32] we assume that the change in electrostatic energy of an electron, eA¢, over a distance of the order of the lattice constant be small compared with the band gap Eg. We further assume that the field in the depletion layer is n o t strong enough to induce tunneling of electrons from valence band to conduction band levels. The potential drop across the junction is given by eA¢ = pe(oo) -- pe(--oo)

(33)

where the electrochemical potential is given by

#(x) = p + e¢(x)

(34)

and # is the thermal equilibrium value of the chemical potential. Also, one has for the carrier concentrations pn(°°) = Nn(T) exp(--[Ec -- e¢(oo) -- p]/kT~

(35)

and ph(°°) = Nh(T) e x p ( - - [ p - - E v + eC(--oo)]/kT)

(36)

For a p - n junction with solitonic semiconductors on both sides, pn(oo) and ph(oo) are determined by the charge balance equation Pn(°°) = Nd -- (2f(0) -- 1)p s-

(37)

and p.(oo) = Na

-

ps,,

(a8)

35

where Ps- and ps÷ d e n o t e the concentrations of 'negative' solitons and 'positive' solitons, respectively. Making use of the above four equations, we can express the potential dr op across the j u n c t i o n in terms of the d o p a n t and soliton co n cen trat i ons as (Na -- (2f(0)

eAdp = edp(~) -- ec~(--m) = Eg + k T l n S- n ---- P n ' --

-- 1)p~-}(Na

S +, p Pn

-- p~+)

(39)

where ps--,, is the Fermi level o f a soiitonic c o n d u c t o r on the n-side of the j u n c t i o n and PnS+' p is the same q u a n t i t y on the p-side of the junction. Corresponding equations for polaronic and ot her types of Q1D semiconductors can similarly be obtained. The i m p o r t a n t feature is that the device characteristics d ep en d on the quant i t y A¢. The depletion width is proport i onal to (A¢) 1/2. When a voltage V is applied across a three-dimensional p - n junction the total electrical current is given by j = e ( J gen + J g n e n ) { e x p ( e V / k T )

- - 1}

(40)

The generation current is proportional to e x p { - - e A ¢ / k T ) . A similar dependence is e x p e c t e d f or a Q1D device. In principle, by varying the d o p a n t concentration one can vary eA(b so as to obtain the optimal device performance. Now we consider an example o f a Q1D p - n h o m o j u n c t i o n {similar to the one depicted in Fig. 5) which is fabricated from p-doped trans-(CH)x and n-doped trans-(CH)x. F o r low doping, positive and negative solitons are the d o m i n a n t excitations on the p- and n-sides of the junction, respectively. The Peierls gap (A) for trans-(CH)~ at r o o m t e m p e r a t u r e [33] is about 1.5 eV. The typical n u m b e r of m o n o m e r s on a chain, N, is cited in the literature to be a p p r o x i m a t e l y 3000. The distance a0 between two carbon atoms, which is the separation between two m o n o m e r units on the chain, is about 1.22 A for trans-(CH)x. Thus the typical chain length L = N a o = 3.66 X 10 -v m. The linear density of carbon atoms on the chain equals 1/ao ~- 8.2 X 109/m. As shown by Kivelson and Heeger [2], trans-(CH)~ is a solitonic sem i conduct or below a critical doping Yc -~ 0.053 (i.e., a density o f 4.34 X 10S/m). We

Ecp

ft

~

.................. ~o

E ............... F p Evp

1'

r . . . . ~ - . . . . . . . . . . . . ev, ~. . . . . . EF

? tl"

"El' E,,.

Fig. 5. F o r m a t i o n o f a Q I D p - n h o m o j u n c t i o n b e t w e e n p- a n d n - d o p e d p o l y p y r r o l e (PPy) (a) b e f o r e c o n t a c t . E F P a n d EFn c o r r e s p o n d t o t h e F e r m i level, d e n o t e d in t h e t e x t b y Pn, o n t h e p- a n d t h e n-side, respectively. (b) T h e Q 1 D p - n h o m o j u n c t i o n a f t e r c o n t a c t in t h e r m a l e q u i l i b r i u m . Vb d e n o t e s t h e built-in voltage across t h e j u n c t i o n .

36 choose y = 0.01 on both sides of the junction, that is Na = Nd = Y/ao = 8.20 × 10V/m. The effective masses of holes and electrons are taken to be equal to the free electron mass for simplicity. Using these parameters in eqns. (6) and (8), we obtain ps, n = 0.705 eV and ps,p = --0.547 eV, respectively. These values are obtained with the mid-gap energy chosen to be zero. The potential drop, ez~¢ across the p - n junction is then found from eqn. (33) to be equal to 1.25 eV. Knowing this, it is then possible to obtain the J - V and C - V characteristics (see for example ref. 34). In the calculation of #n above we chose the effective electron (or hole) mass, me*, equal to the free electron mass, m e. We also studied the effect of the variation of me* on PnS, n • For values of me* equal to 0.5me, 5me and 50me ' /~S,n in eqn. (6) turned out to be 0.714, 0.684 and 0.654 eV, respectively. This means that ps,~ gets closer to the mid-gap as me* increases. To our knowledge the value of me* is not yet accurately determined experimentally. We think that it is probably greater than me in trans-(CH)x. We also studied the effect of the variation of Nd on gs, ~. As we increased Nd to 3%, #s, n shifted towards the conduction band edge. In this specific case for values of me* equal to 0.5me, 5me and 50me,/~s,n turned out to have the values 0.742, 0.713 and 0.683 eV, respectively. This means that as we increase the doping, trans-(CH)x tends to be more metallic than semiconducting. This is consistent with the prediction of Kivelson and Heeger [2] of a s e m i c o n d u c t o r - m e t a l transition at a doping level of 5.3%. As an example of a p - n homojunction fabricated from a conducting polymer (Fig. 5) that has a ring structure, we consider a junction between p-(PPy) and n-(PPy). At room temperature PPy is a semiconductor with a Peierls gap of 4.0 eV [14]. At low doping polarons and bipolarons coexist with creation energies 1.46 eV and 2.47 eV, respectively. The distance between the monomers on the PPy chain, a0, is about 3.7 A. The linear density of monomers on the chain is 1/ao -- 2.67 × 109/m. For 1% doping on each side of the junction we have Na = Nd = 2.67 × 10V/m. Substituting these quantities in eqns. (20) and (23), we obtain p~'~ = 1.67 eV and .u~'p = --1.67 eV, respectively. Note that the values of p~' ~ and #e,p are symmetrically placed about the mid-gap, unlike the solitonic case discussed above. This is due in part to the s y m m e t r y between the charge balance eqns. (9) and (13) for the p- and n-doped polaronic case and in part to our choice of equal values for the effective masses of the electrons and the holes. The potential drop across the junction, eA¢, equals 2.34 eV. This homojunction is depicted in Fig. 5(b). 3.2. Q I D S c h o t t k y barrier According to the fully developed three-dimensional t h e o r y of a m e t a l semiconductor contact due to Crowell and Sze (which combines both the diffusion and thermionic theories), the J - V characteristics [34] are given by

J =

qNnvR 1+

--

VD

e - ~ b ( e ~ q v - 1)

(41)

37

where VR is the recombination velocity at the top of the barrier and Cb is the barrier height. VD is an effective diffusion velocity associated with the transport of electrons from the edge of the depletion region to the top of the barrier, and is defined by

VD

= I f flq exp(--fl(~bb--E¢)} dx l;m p

(42)

where/~ is the electron mobility and Xm is the position of the maximum of the potential barrier. In this theory the only charge carriers are electrons and holes. It does not take into account any other species of charge carriers such as solitons. By taking into account the effects of optical p h o n o n scattering in the region between the top of the barrier and the metal, as well as the quantum mechanical reflection of electrons, this equation for J can be written as J = A**T 2 exp(--qOb/kT)(exp(qV/kT

) - - 1}

(43)

where A** = fpfqA*/(1

+ fpfQVR/VD)

Here A* is the Richardson coefficient and fp is the probability of electron emission over the potential maximum of the semiconductor into the metal w i t h o u t electron o p t i c a l - p h o n o n backscattering, fQ is the ratio of the total current flow, including tunneling and quantum mechanical reflection, to the current flow neglecting these effects. Because of the image force rounding of the barrier, the current can be written in terms of the 'ideality factor' n as J = (A**T 2 exp(--qcPb/kT)}{exp(qV/nkT

) - - 1}

(44)

The barrier height is given by Ob = ~m - - ~sc

(45)

where 0m and Csc are the work functions of the metal and semiconductor, respectively. As noted above, one does not expect this three-dimensional theory to be completely applicable to the one-dimensional case. The reason is that in certain regions of the Q I D devices such as the depletion region, electrons and holes are no longer the dominant current carriers. Instead, the current is likely to be carried by solitons, polarons and bipolarons. As indicated earlier, the transport mechanisms of these excitations are quite different from that of electrons and holes. This would be reflected in the J - V a n d C - V characteristics. Therefore the Schottky barriers fabricated from three-dimensional metals and polyacetylene, etc. do not follow the theory of Crowell and Sze [34]. We are probably considerably overstating the electron-hole contribution to the transport and neglecting solitons, etc. Thus in this sense, we only have a qualitative theory.

38 For a Q I D Schottky barrier, say involving a polaronic metal and a solitonic semiconductor, ¢m and ¢s¢ can be directly related to the Fermi levels for which we have obtained explicit expressions in terms of the doping level and the Peierls gap of the Q I D material. Since both the prefactor and n in eqn. (44) have a dependence on the doping, in principle the optimal device performance can again be obtained by varying the doping. As Conwell [11] has pointed out, the photovoltage from a Schottky barrier is equal to the difference between the chemical potentials (Fermi levels) of the metal and the semiconductor. Similarly the open circuit voltage is yo c =/2e(SC ) - - # e ( m )

(46)

Hence Cb, Vo¢ and the photovoltage are directly related to the difference in the Fermi levels. This provides one with a great deal of flexibility in terms of tuning the device characteristics by varying both the doping level of the metal, as well as that of the semiconductor in the device. In Fig. 6 a S c h o t t k y - t y p e junction between n-doped trans-(CH)x (a polaronic metal at high doping level) and trans-(CH)x (a solitonic semic o n d u c t o r at low doping level) is shown. For the n-doped trans-(CH)~ at room temperature we obtained in the previous Section the value /~S,n= 0.705 eV. We choose on the metallic side of the junction y = 0.15 (which is well above Yc and corresponds to Nd -~ 1.23 × 109/m). The average spacing between the polarons at y = 0.15 is 1/y -~ 7 m o n o m e r units. We choose in our calculations the extent of a polaron, /p, to be 5. Using these parameters in eqn. (10) we obtain pP, n = 0.563. This corresponds to a barrier height of approximately 0.14 eV (or a photovoltage equal to 0.14 V). The values of /~s, n and pP' n are insensitive to a small variation in Ip. On the other hand the junction deteriorates with decreasing temperature, that is Cb decreases (pe, n increases and ps, ~ decreases). Both pS, n and pnP'n decrease with lowering of the doping level.

E~ m EF

T ..............

.................

E~F

b .......

Tl

~. . . . . . . . . . . . . . . . . . . . . . .

v EF

C (a)

(b)

Fig. 6. Formation of a Q1D Schottky barrier between n-doped polaronic trans-(CH)x (a metal) and n-doped solitonic trans-(CH)x (a semiconductor). The superscripts ' m ' and 'so' refer to the metal and the semiconductor, respectively. (a) Before contact and (b) after contact in thermal equilibrium. ~bb and Vb denote the barrier height and the built-in potential, respectively.

39

3.3. QID hetero]unctions A heterojunction is a junction formed between two semiconductors having different energy band gaps. The current-voltage relationship for a three-dimensional heterojunction [34] is given by J = J0(1 -- V/Vb~){exp(qV/kT) -- 1}

(47)

where

Jo = ( qA *TVbi/k ) exp(--q Vbi/k T)

(48)

The quantity of interest for a heterojunction is the built-in potential Vbi = Vbl + Vb2- The depletion width and junction capacitance also depend on V~. Vbl and Vb2 depend on the Fermi level. The desired values of Vbl and Vb2 can be obtained by choosing appropriate values of the doping levels on both sides of the junction. For Q1D c o n d u c t o r s , unlike three-dimensional semiconductors, there are two possibilities of fabricating a heterojunction: (i) choose two different conductors or (ii) choose the same conductor with different doping levels on either side of the junction. This is possible because in some of the conducting polymers the Peierls gap is known to vary with the doping level. In such a case varying the doping level changes both the Fermi level and the Peierls gap. In Fig. 7 a heterojunction fabricated from p-doped trans-(CH)x and pdoped polypyrrole (PPy) is depicted. The chemical potentials for p-doped trans-(CH)x and p-doped PPy were obtained in Section 3.1 as #s, p = --0.547 eV and pP'P =--1.166 eV. The built-in potential Vbi = #S,p_#P,p is about 0.51 eV. A similar Q I D rectifying device recently fabricated [10] from pdoped trans-(CH)x and p-doped poly(N-methylpyrrole), PNR, resulted in a very low value of the photovoltage of about 10 mV. The Peierls gap for PNR is about the same as for PPy, that is approximately 3.0 eV. One possible reason for this low value of the observed photovoltage as compared with our estimated value could be the poor transport due to solitons and polarons. However, one cannot preclude other factors such as the non-abrupt nature of the junction due to the polymeric nature of the material. The presence of interracial surface states also tends to deteriorate the junction.

VI~

E¢

Eel

EF~ . . . . . . . . . . . .

EvI

- . . . . . . . . . . . .- .- .~ . . . . . . . . . . . .

................. EFT_ --

(a)

?

EF

/Z~iEi

Ik

Ev2 (b)

Fig. 7. Formation of a Q1D heterojunction between p-doped trans-(CH)x and p-doped polypyrrole (PPy). (a) Before contact and (b) after contact in thermal equilibrium. Total built-in voltage is given by Vb = Vbl + Vb2.

40 4. Conclusions In summary, the new results discussed here include the possibility of fabricating a variety of quasi-one-dimensional solid state electronic devices whose performance can be optimized by changing a control parameter, namely the dopant concentration. We have provided necessary information for the fabrication of Q1D p - n homojunctions, QID Schottky barriers and Q1D heterojunctions. This has been achieved by deriving expressions for the chemical potential (Fermi level) as a function of doping and the Peierls gap for the QID semiconductors and metals whose excitations not only include as usual electrons and holes, but in addition, one or more of the following: solitons, polarons and bipolarons. The devices mentioned above are important because as is well known, they form the basis of most of the more elaborate Q1D solid state electronic devices such as solar cells, field effect transistors, etc. [ 34]. We have discussed two different examples of QID p - n junctions, one fabricated from trans-(CH)x and the other from PPy. In fact, an attempt [9] to fabricate a trans-(CH)x p - n junction has been reported in the literature. This attempt was unsuccessful, possibly due to the poor transport by solitons and the non-abrupt nature of the junction. The presence of interfacial surface states and the migration of dopant ions across the junction could be another possibility. Recently a Q1D Schottky device consisting of metallic PPy and p-doped PT has been fabricated [38]. We have considered one such device fabricated from polaronic trans-(CH)x (a metal) and solitonic trans-(CH)~ (a semiconductor). So far only one example of a Q1D heterojunction [10] has been reported in the literature, namely a device fabricated from trans-(CH)~ and poly(N-methylpyrrole), PNR. We have discussed a heterojunction fabricated from trans-(CH)x and PPy. Due to the inefficient transport by solitons, etc. and the experimental difficulties involved in producing an abrupt junction with polymers, the values of barrier height, photovoltage, etc., predicted here may be difficult to achieve experimentally. Though we have discussed only a few prototypical examples, many more different devices involving quarter-filled-band and other types of Q1D polymers can possibly be fabricated. There is clearly a need for further studies involving fabrication of these devices. In the discussion of Q1D devices we briefly addressed the question of how solitons as well as other non-linear excitations affect the conductivity and the carrier transport. In principle, spinless transport of these charged species affects both the nature of the junction and the device characteristics. The presence of a dopant-ion array may hinder the transport of charge carriers. On the other hand, interchain hopping of solitons [33] and bipolarons [ 1 ] has been proposed as a possible mechanism for spinless charge transport. In some of the conducting polymers such as (NMP).s4(Phen).46TCNQ [35], the offchain hopping (three-dimensional) of solitons is negligible compared to the on-chain (one-dimensional) hopping. Moreover, intramolecular phonon-assisted electron hopping between solitons contributes

41

substantially to the transport and on-chain hopping mobilities. This observation indicates that the Q1D devices fabricated from such conducting polymers would have significantly improved performance. In particular, the conversion efficiency (or quantum yield) of a Q1D solar cell is expected to be considerably enhanced. We emphasize that the existing three-dimensional theories are inapplicable to the case of Q1D devices. They only provide a qualitative guide to the behavior of these devices. There is a need for a complete onedimensional theory that takes into account the transport due to the nonlinear excitations in the conducting polymers. Apart from the Q1D organic polymers, there are other kinds of Q I D metals known in the literature. This indicates the possibility of fabricating still different kinds of Q I D devices, in particular Schottky barriers. Some examples of these are (i) alkali-metal [36] chains on a Si(100) surface (with overlayer plasmons present) and (ii) linear chain metals [37] such as NbSe3 and TaS 3 (with a charge density type of mechanism present).

Acknowledgements We would like to thank Dr. E. M. Conwell for useful correspondence. This work was supported by Office of Naval Research Grant no. N00014-83K-0382.

References 1 J. L. Br6das, B. Themans, J. M. Andre, R. R. Chance and R. Silbey, Synth. Met., 9 (1984) 265. 2 S. Kivelson and A. J. Heeger, Phys. Rev. Lett., 55 (1985) 308. 3 I. A. Howard and E. M. Conwell, Phys. Rev. B, 31 (1985) 2140. 4 E. M. Conwell and I. A. Howard, Phys. Rev. B, 31 (1985) 7835. 5 W.-P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. Lett., 42 (1979) 1698;Phys. Rev. B, 22 (1980) 2209;M. J. Rice, Phys. Lett., 71A (1979) 152. 6 S. A. Brazovskii and N. N. Kirova, Pis'ma Zh. Eksp. Theo. Fiz., 33 (1981) 6 [JETP Lett., 33 (1981) 4]; D. K. Campbell and A. R. Bishop, Phys. Rev. B, 24 (1981) 4859; A. R. Bishop, D. K. Campbell and K. Fesser, Mol. Cryst. Liq. Cryst., 77 (1981) 319. 7 A. R. Bishop and D. K. Campbell, in A. R. Bishop, D. K. Campbell and B. Nicolaenko (eds.), Nonlinear Problems: Present and Future, North-Holland, Amsterdam, 1982, p. 195. 8 S. Etemad, A. J. Heeger and A. G. MacDiarmid, Ann. Rev. Phys. Chem., 33 (1982) 443. 9 C. K. Chiang, S. C. Gau, C. R. Fincher, Jr., Y. W. Park, A. G. MacDiarmid and A. J. Heeger, Appl. Phys. Lett., 33 (1978) 18. 10 H. Koezuka, K. Hyodo and A. G. MacDiarmid, J. Appl. Phys., 58 (1985) 1279. 11 E. M. Conwell, Synth. Met., 9 (1984) 195. 12 E. M. Conwell, Synth. Met., 11 (1985) 21. 13 Y. Onodera, Phys. Rev. B, 30 (1984) 775. 14 J. L. Br6das, J. C. Scott, K. Yakushi and G. B. Street, Phys. Rev. B, 30 (1984) 1023; J. L. Br6das, B. Themans and J. M. Andre, Phys. Rev. B, 27 (1983) 7827;J. C. Scott, P. Pfluger, M. T. Krounbi and G. B. Street, Phys. Rev. B, 28 (1983) 2140.

42 15 M. Peo, S. Roth, K. Dransfeld, B. Tieke, J. Hocker, H. Gross, A. Grupp and H. Sixl, Solid State Commun., 35 (1980) 119; J. L. Br~das, R. R. Chance and R. S. Silbey, Mol. Cryst. Liq. Cryst., 77 (1981) 319 and Phys. Rev. B, 26 (1982) 5843. 16 J. L. Br4das, B. Themans, J. G. Fripiat, J. M. Andr~ and R. R. Chance, Phys. Rev. B, 29 (1984) 6761 ; T. C. Chung, J. H. Kaufman, A. J. Heeger and F. Wudl, Phys. Rev. B, 30 (1984) 702. 17 R. P. McCall, D. B. Tanner, J. S. Miller, I. A. Howard and E. M. Conwell, Mol. Cryst. Liq. Cryst., in press. 18 A. J. Epstein and J. S. Miller, Solid State Commun., 27 (1978) 325; A. J. Epstein, J. W. Kaufer, H. Rommelmann, I. A. Howard, E. M. Conwell, J. S. Miller, J. P. Pouget and R. Comes, Phys. Rev. Lett., 49 (1982) 1037. 19 M. J. Rice and E. J. Mele,Phys. Rev. B, 25 (1982) 1339. 20 H. Takayama, Y. R. Lin-Liu and K. Maki, Phys. Rev. B, 21 (1980) 2388. 21 M. Ozaki, D. Peebles, B. R. Weinberger, A. J. Heeger and A. G. MacDiarmid, J. Appl. Phys., 51 ( 1 9 8 0 ) 4 2 5 2 . 22 T. Tani, P. M. Grant, W. D. Gill, G. B. Street and T. C. Clarke, Solid State Commun., 33 (1980) 499. 23 J. Tsukamoto, H. Ohigachi, K. Matsumura and A. Takahashi, Synth. Met., 4 (1982) 177. 24 B. R. Weinberger, M. Akhtar and S. C. Gau, Synth. Met., 4 (1982) 187. 25 D. L. Peebles, J. S. Murday, D. C. Weber and J. Milliken, J. Phys. (Paris) Colloq., 44 (1983) C3-591. 26 P. M. Grant, T. Tani, W. D. Gilt, M. Krounbi and T. C. Clarke, J. Appl. Phys., 52 (1981) 869. 27 H. Koezuka and S. Etoh, J. Appl. Phys., 54 (1983) 2511. 28 T. Ogama and H. Koezuka, J. Appl. Phys., 56 (1984) 1036. 29 J. Kanicki, Mol. Cryst. Liq. Cryst., 105 (1984). 203. 30 J. R. Waldrop, M. J. Cohen, A. J. Heeger and A. G. MacDiarmid, Appl. Phys. Lett., 38 (1981) 53. 31 M. Ozaki, D. L. Peebles, B. R. Weinberger, C. K. Chiang, S. C. Gau, A. J. Heeger and A. G. MacDiarmid, Appl. Phys. Lett., 35 (1979) 83. 32 N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, Philadelphia, 1976. 33 S. Kivelson, Phys. Rev. B, 25 (1982) 3798. 34 S. M. Sze, Physics of Semiconductor Devices, Wiley-Interscience, New York, 1981. 35 I. A. Howard and E. M. Conwell, Mol. Cryst. Liq. Cryst., 120 (1985) 63. 36 T. Aruga, H. Tochihara and Y. Murata, Phys. Rev. Lett., 53 (1984) 372. 37 P. Momceau (ed.), Properties of Inorganic Quasi-One-Dimensional Materials, Reidel, Boston, 1985. 38 K. Kaneto, S. Takeda and K. Yoshino, Jpn. J. Appl. Phys., 24 (1985) L553. 39 I. A. Howard and E. M. Conwell, Phys. Rev. B, 27 (1983) 6205; J. Phys. (Paris) Colloq., 44 (1983) C3-1487. 40 J. Simon and J.-J. Andr4, in J. M. Lehn and Ch. W. Rees (eds.), Molecular Semiconductors: Photoelectrical Properties and Solar Cells, Springer-Verlag, Berlin, 1985.

Appendix

(1) S o l i t o n s For n-type f i e d as

doping

Pn % ( 2 f ( 0 ) - - 1)p s- = g d

Pn>~ Ph a n d Ps = P s - - E q u a t i o n

(5) can be simpli-

(A1)

43 where Pn is given by eqn. (1). The Fermi function and the negative soliton concentration are given by

f(e)= {1 + exp[fl(e --p)l} -I

(A2)

and [11]

Ps

2L exp fl Ps

(A3)

7r

respectively. Also, Ps- =/~n where Pn denotes the Fermi energy. Substitution of eqns. (A2) and (A3) in (A1) provides the following quadratric equation:

IN n e ~' +

N -¢(2&/~r)--Ndle6~ Nd

-N - e -~(2~/~') I e ~ n + l N n e -~z~- - - e 2L 2L

=0

(ha)

The positive solution (exp(2fl/~n) > 0) of the above equation is then given by eqn. (6). Similarly, for p-type doping Pp>>Pn and Ps-~Ps*- Equation (7) reduces to Ps+ = Na -- Ph

(A5)

where Ph is given by eqn. (2). The concentration of the positive solitons [11] is given by

Equations (A5) and (A6) together with the relation #s+----Pn lead to eqn. (8).

(2) Polarons In the case of n-type doping Pn >>Ph and pp -~ pp-. Equation (9) can be written as Pp- = Nd -- Pn

(A7)

that is, negative polarons are lost as electrons go into the conduction band. To obtain the polaron concentration, we use [12]

pp=Zp

+ kTlnl

-PP

(N/L ) -- pplp

1

(A8)

Substitution of eqns. (1) and (A7) and the condition p , - = Pn in (A8) results in the following quadratic equation:

(IpLNn e-~(EP +A)} e ~

+ {(N --Yd/p) e-~EP +

LNn e -~n}

e~-Nd

=0

(A9)

44 The positive solution of the above equation gives eqn. (10). For p-type doping Ph >>Pn and pp -~ pp+. These relations simplify eqn. (13) to Pp. = Na -- Ph

(A10)

that is, positive polarons are lost as holes go into the valence band. Equations (2), (A8), (A10) and the condition lap. = - - p . then lead to ~he following quadratic equation:

(IpLNh e -~(Ep+ A)) e - ~

+ {(N--Na/p) e -~Ep + LATh e - ~

e - ~ n - - Na = 0 (All)

The solution of ( A l l ) gives eqn. (14).

(3) Bipolarons For n-type doping Pn >> Ph and PB -~ PB--. Equation (15) simplifies to 2PB-- = Nd -- Pn

(A12)

To obtain the bipolaron concentration, we use [12]

lab = E B + k T l n t

P_B t (N/L)

- - PBIB

I

(A13)

Substitution of eqns. (1) and (A12) and the condition laB-- = 2lan in (A13) leads to eqn. (16). For p-type doping p~ >>Pn and PB -~ PB÷+. Equation (17) can be expressed as 2Pm+ = Na -- Ph Using eqns. (2), (A14) and the condition laB. . . . eqn. (18).

(A14) 2lan in (A13) we obtain

(4) Coexisting polarons and bipolarons In the case of n-type doping, eqn. (19) simplifies to 2PB-- + Pp- + P n - - N d = 0

(A15)

In the low doping regime pp and PB are obtained from [12]

lap = Ep + k T In (N/L) -- pplp -- PBlB

(A16)

and laB = E~ + k T In (N/L) -- pplp -- PBIB

(A17)

respectively. Furthermore, lap-= lan and lab- - = 2lan. Equations (A16) and (A17) are equivalent to

45 pp = ( N / L ) exp(--flEp) exp(flpn)

(A18)

and PB = ( N / L ) exp(--flEB) exp(2~/~n)

(A19)

respectively. Substitution of eqns. (1), (A18) and (A19) in (A15) results in the following quadratic equation:

N e - L

--L

Nd0

A20>

The solution of the above equation is eqn. (20). In the high doping regime PBIB = N / L and pplp < pBIB . Equations (A16) and (A17) then provide

N ( e-~ e~(~B-Ep) 1 ]

(A21)

and

PB=

N( ~

1+

1 e -2~un

e~EB)

(A22)

Equations (1), (A15), (A21) and (A22) lead to (21). For p-type doping eqn. (22) simplifies to Ph + Pp+ + 2PB+* --Na = 0

(A23)

In the low doping regime (A16) and (A17) and the conditions PB++ = --2pn and pp+ = --lan provide pp+ = ( N / L ) exp(--~pn) exp(--~Ep)

(A24)

and pB++ = ( N / L ) exp(--2~Pn) exp(--~EB)

(A25)

Substitution of eqns. (2), (A24) and (A25) in (A23) leads to the following quadratic equation: 2N e_~EB e - ~

tL

f

+

lL

e-~EP+

--N~

0

(A26)

The solution of this equation gives eqn. (23). In the high doping regime the polaron and bipolaron concentrations are obtained from eqns. (A16) and (A17) as PP = and

/~L l 1e~Un e~(EB-EP) + eZe~e ~EB f

(A27)

46

1 ~EB f PB = ~N I 1 + e2~Pne

(A28)

respectively. Equations (2), (A23), (A27) and (A28) then lead to (24). 5. S o l i t o n s in t h e q u a r t e r - f i l l e d - b a n d m a t e r i a l s

For n-type doping eqn. (25) can be rewritten as Ph -- Pn = (n-/2N)

--

(A29)

(n +/2N) - - N d

Using the condition p - = p ~ / 2 [3], one has the following relation between n- and n + n + / 2 N = ( n - / 4 N ) exp(--/~pn)

(A30)

and

lsll

n-/N = --

1+

f e-e(z~/,,-~/2)

-1- e - ~ 2

(A31)

+

where ls denotes the length of a soliton in terms of monomer units. Substitution of (A30) and (A31) in (A29) results in eqn. (26) where P

1/

n

-~"

d(qa)

- -

7r ~/2

1 1 + -- exp(--flpn) exp{fl(4t : cos:qa + A:)l/2}

1

1 [/2 J

(A32)

and

Ph

2

~ 0

d(qa) 1

1 + -- exp(--/3pn ) exp(--[J(4t2cos:qa

(A33) + A2) 1/2}

2 Statistics for fractionally charged particles are used in eqns. (A32) and (A33). In the case of p-type doping eqn. (28) can be simplified as Ph - - Pn = ( n + / 2 N ) - - ( n - / 2 N ) + N a

(A34)

Use of (A30) and (A31) in (A34) provides eqn. (28) where Pn and Ph are given by (A32) and (A33), respectively. ( 6 ) B i p o l a r o n s in t h e q u a r t e r - f i l l e d - b a n d m a t e r i a l s

For n-type doping eqn. (29) can be rewritten as Ph -- Pn + Nd

=

( n B - / N ) - - (nB+/N)

(A35)

Using equations akin to (A13) as well as the relations/1B = Pn and #~ = --gn [4], we obtain n B- = 4ns + exp(2fl#n)

(A36)

47 Starting f r o m the following expression

n)ex

/2 B Y/B- -

2

+

we o b t a i n for the negative bipolarons

nB ~ =

4N - - exp(--flEn) ls exp(~pn) + exp(--flEB)(1 + 4 exp(2~Pn)}

(A38)

where lB is the length of a bipolaron in terms of m o n o m e r units. Equations (A35), (A36) and (A38) lead to eqn. (30), where

1 ,.'/2 Ph = -- / ~ 0

d(qa) 1 1 + -- exp(--~pn) e x p { - - ( 4 t : cos2qa + A2) 1/2) 2

(A39)

and 1 / d(qa) p~ = -lr 1 ./2 1 + -- exp(--/3pn ) e x p ( ( 4 t 2 cos2qa + A2) 1/2}

(A40)

F o r p-type doping eqn. (31) can be written as Ph - - Pn - - N a = ( n B + / N ) - - ( n B - / N )

(A41)

Starting from (A37) with nB- replaced by nB+, we obtain for the positive bipolarons N -

nn+=

-

L

exp(--~EB) exp(--2~Pn)

exp(--~pn) + exp(--~EB){4 + exp(--2~pn)}

(A42)

Substitution of (A36) and (A42) in (A41) results in eqn. (32). Ph and pn are given by (A39) and (A40), respectively.