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Optical and electrooptical absorption in conducting polymers
Thin Solid Films 403 – 404 (2002) 419–424
Optical and electrooptical absorption in conducting polymers N. Kirova*, S. Brazovskii ˆ 100, Universite´ Paris-Sud, 91405 Orsay Cedex, France LPTMS, Bat.
Abstract We describe effects of the long-range Coulomb interactions on absorption and photoconductivity of polymeric semiconductors. The final-state interaction suppresses the 1D edge singularity of band-to-band transitions, reducing it to the final value. This value meets the averaged subgap absorption through the high excitonic series below the gap. After a dip, the absorption increases at the primary exciton peak. At moderately strong electric fields, higher excited states are destroyed, while the primary one is preserved. Our results explain the rounding of the e–h absorption edge observed in various polymeric semiconductors. Moreover, they quantitatively agree with new photoconductivity data on PPV-type polymers where new spectral features appear under electric field. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Conducting polymers; Optics; Electrooptics; Excitons
1. Introduction Most of the current activity in the physics and applications of conjugated polymers is centered around their ability to emit light in the optical range of approximately 2 eV. Here we are facing a curious and almost unknown situation: while these optical materials are already commercialized, there is no consensus yet on the nature of the light-emitting excitons. Thus, for nearly a decade, the estimations for exciton binding energies in phenyl-type polymers ranged within almost an order of magnitude from 0.2 to 0.9 eV. A substantial body of experimental and theoretical work has emerged, supporting each of these extremes, and a resolution of these issues remains as a fundamental challenge (see w2,5x and references therein). Determination of the exciton binding energy (Eb), and more generally the fundamental edge nature, is critically important to answering these questions, and thereby to understanding the electronic structure of semiconducting polymers. We believe that our recent interpretations of several new, key experiments allow unambiguous fixing of the * Corresponding author. Tel.: q33-169-15-73-32; fax: q33-16915-65-25. E-mail address: [email protected] (N. Kirova).
physical picture. At the microscopic level, taking into account the peculiarities of phenyl-based polymers, we identified the phenomenon of band inversion, which proves the domination of the band picture of strongly delocalized electrons and holes. In this way, a picture of a one-dimensional semiconductor becomes adequate to describe the vicinity of the fundamental edge for light absorption and emission w1,2x. More recently, we could demonstrate success for this semiconducting picture by explaining a striking observation of a new spectral peak appearing in photoconductivity under strong electric field w3,4x. This article is mostly devoted to this effect. First of all, we shall briefly recall the underlying microscopic picture of the most exploited family of phenyl-based polymers. We have suggested a theoretical picture of PPV-type conjugated polymers w1,2x, which unifies features of both a semiconductor band model (delocalized excitations), and a molecular exciton model (localized excitations), while invoking the effects of the long-range Coulomb interactions — for both bound and free e–h states on absorption and photoconductivity spectra in polymeric semiconductors. The necessary discrete energy levels appear naturally, in addition to the delocalized states, mainly from long-range Coulomb interactions. On this basis, we identified the most important optical
0040-6090/02/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 5 6 3 - 2
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
420
features in direct and in transient photo-induced absorption. On this microscopic basis, we arrive at an understanding of the absorption profile as one which is dominated by the shallow exciton unresolved from the band edge. Moreover, final-state interactions above the fundamental edge Eg (the threshold for generation of free carriers) do not change the density of states (DOS), but they strongly affect the band edge profile, namely the square root edge singularity disappears, being saturated at a finite value at the scale of the primary exciton binding energy Eb. This rounding of absorption above the gap exactly meets the mean value from the subgap absorption by the dense series of higher excited excitonic states En. The reason that the Coulomb attraction bleaches the absorption singularity is as follows. On first sight, the attractive electron and hole have a higher probability to be in one place, which determines the light-emitting probability. However, this is not the case for a light quantum particle: the uncertainty principle enforces their relative motion and the long-range force then accelerates their falling upon each other. Finally, they pass by with a high velocity vsdxydt, preventing their annihilation, the probability of which is P;1yv;6x. Our results explain the long-standing observation of the e–h absorption-edge rounding observed in various polymers and other 1D semiconductors. Now we summarize effects of external electric field F. In the experimentally significant region of intermediate fields, only the primary exciton is preserved, for which we determine the ionization probability: Gs
w 4 Eb3y2m*1y2 z Eb | expxy y 3 " e"F ~
(1)
Eq. (1) provides the field dependence of the photoconductivity due to the cold ionization of the exciton. It describes the main F-dependence of the excitonic peak in photoconductivity. All other states are not bound and show only a smoother F-dependence via the carrier mobility (Fig. 1). We have estimated the exciton binding energy from new photoconductivity measurements w3x, which quantitatively agree with the results based on our band picture. 2. Band structure and optical excitations in phenylbased polymers Today, among the family of phenyl-based polymers, the PPV one is the most studied and most likely to be processible in applications. The PPV chain consists of phenyl rings and dimers of carbon atoms providing the vinylene linkage. The energy bands may be viewed as originating from the intra-ring electronic levels at "T and "2T, as well as from vinyl dimer levels at "T1.
Fig. 1. Photocurrent excitation spectra in PPV at various external fields; the normalization factors of the curves are indicated on the right. The inset compares the excitation spectra of pristine PPV to that of the same sample after photo-oxidation by exposure in air to 400 mWycm2 of light from a Xe lamp (from w3x).
We use standard values for hopping integrals between adjacent atoms within the ring Ts2.4 eV, within the dimer T1s2.6 eV and between them ts2.1 eV. The ring states are classified in terms of their parity g,g* and u,u* with respect to the chain axis. The u,u* states have zero amplitude at the connection to the vinyl group, and hence they do not hybridize with the dimer and do not acquire a band dispersion. The g,g* states strongly hybridize with ones from vinyl, forming wide bands. We have found the analytical solution providing the dispersion equation for free electrons and transition intensities w1,2x. The robust Coulomb corrections were included by the rigid displacements of all D, L, D*, L* bands by "0.6 eV correspondingly. The resulting band structure is presented in Fig. 2. The bands L1, L1* are flat, originating from the u,u* states of the phenyl ring, while the dispersion of bands D1, D1* is remarkably high. The especially strong hybridization in PPV is an apparent consequence of the resonance T levels of the ring and T1 levels of the dimer. For a finite system, we have also found the ‘edge states’ localized near the chain ends. Their energies appear in the gaps between D1,D2 and D1*,D2*. There are several important consequences of the picture: ● Delocalization breaks the intra-ring correlation. The state responsible for highest intra-ring excitation f6 eV now contributes to the lowest inter-chain excitation f2 eV.
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
421
strongly depends on ´, the static dielectric susceptibility of the media, which can actually be several-fold higher than the optical one (f2). Hence, Eb is not universal for the polymer, but it can change for the films with respect to solutions. The long-range Coulomb interaction also originates the series of higher excitons with localization radii Rn and binding energies: ansna*b; E*b Ebns 2wnqwlnŽanya.xy1x2
Fig. 2. Band structure for PPV and peak assignments for experimental features presented in Fig. 3.
● High bandwidth reduces the lowest intramolecular transition at 5 eV to the range of visible optics f2 eV. ● The effective mass is surprisingly small, which gives rise to solid-state physics of highly mobile particles, the shallow bounds states, etc. ● Curious states with zero dispersion are predicted at high energy, which determines most of the photoinduced features observed. In terms of superconductivity, their creation corresponds to a high photoinduced doping. This band picture naturally allows for three different excitons confined by long-range Coulomb interactions. Exciton 1 (EX1) corresponds to a weakly bound state of a delocalized electron and hole in the HOMO–LUMO bands D1 and D*1 w2x. This one-dimensional Wannier– Mott exciton is well described within the effective mass approximation. Its localization radius ab and binding energy Eb, respectively, are: absab*
1 ; lnabya
EbsEb*ln2
ab a
(4) ns1,2,«
which resembles the hydrogen series. There is also a series of optically forbidden excitons. Exciton 2 (EX2) corresponds to a more tightly bound state of L1D*1 or D1L*1. Now the non-dispersive electron L1* or hole L1 plays the role of an immobile Coulomb center, similar to optically activated acceptors in semiconductors, which binds the delocalized particle of the opposite sign. With a two-fold higher effective mass m * with respect to EX1, the EX2 radius is reduced to values of the order of the unit cell, where the dispersion of the dielectric susceptibility ´ will shift it to still lower values. Supposing ´*s1.5, we find an effective radius of ;10 A and binding energy Eb2;0.8 eV. Exciton 3 (EX3) corresponds to an intra- and interring bound state of a non-dispersive electron and hole. The L,L* state is localized within one ring, being related to the u,u* states of the electron and hole near "T. This is a Frenkel exciton. In addition to intramolecular L1L*1 excitons, we expect, at even higher energies, ionized decoupled states, where L1 and L*1, belong to different rings. The assignment of the optical features from Fig. 3 is presented together with the band diagram in Fig. 2.
(2)
where a*bs
´" ; m *e 2
E*bs
m *e 4 "2´2
(3)
´ is the dielectric susceptibility H to the chain, " is Planck’s constant, m * is the electron effective mass at the zone center in k-space (for PPV, m *f0.1me w2x), e is the electron charge and a is the unit cell length. Here, the logarithm is a characteristic enhancement for the Coulomb binding in the 1D case w6x. Note that Eb
Fig. 3. Optical transitions in MEH-DSB (from w1x).
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
422
y
d 2c 1 y cqfjcs´c dj2 )j)
(5)
where we have introduced the dimensionless variables jsxya*b, ´sŽEyEg.yE*bsyk2 and f sFyF * (Eg is the bandgap). The leading order in the ln2 factor of Eq. (2) corresponds to approximating the Coulomb term by the d function: y
Fig. 4. The wave function of the primary exciton; the dashed line indicates the potential.
c )j)
Usy
e2 qeFx ´)x)
This shows a well at small x, which is separated from the unbinding region by a barrier Ubsy2eyeFy´ with a top at xsyyey´F; see Fig. 4. Relatively long-lived states can exist if their energy is below the barrier. Thus, at strong fields F)F0sF*Ža8byab.3;106y107 eV, ŽF*sEb*yab*., the primary and all higher excitons would be destroyed, so that all states will be delocalized. Hence, the photoconductivity will depend on F only via the mobility. At intermediate fields F *-F-F0, the higher excited states n are destroyed, while the primary 1Bu exciton is preserved within a finite lifetime ts Gy1, determined by the tunneling probability G through the barrier. Finally, at F-F *, a part of states with n)1 will be preserved. We shall concentrate on the experimentally significant region of intermediate fields when all higher states n/0 lie above the barrier, hence being ¨ delocalized. The Schrodinger equation for the excitons is:
|
c )j)
(6)
dj
The parameter V0 has to be determined self-consistently. For the primary exciton, the exact solution of Eqs. (5) and (6) can be written as a combination of two Airy functions:
3. Field-induced exciton ionization The long debated question on the origin of the absorption edge, and consequently on the luminescent states, seems to have been finally elucidated by new experiments on photoconductivity which resolve the excitonic peak in the external electric field w3x (Fig. 1). The most striking observation is the appearance of a narrow peak in the photocurrent spectrum Iphoto near the absorption edge at F)105 Vycm. Its emergence just below the absorption edge suggests a contribution to the carrier density from exciton dissociation by the external field. At very high fields, the bound state would be destroyed. At intermediate fields, there is a barrier to field ionization, but the carriers can dissociate by tunneling. Consider the e–h pair at a distance x. Under the applied constant electric field F and at the long-range Coulomb attraction, the potential energy is:
´yV0dŽj., V0s
S pV0
B
k2 E B k2qfj E FAiC F : j)0 D f2y3 G D f2y3 G
BiC
Tf T pV AiC k 1y3
cŽj.sU V
f
0 1y3
B
E B k2qfj E FBiC F D f2y3 G D f2y3 G 2
: j-0
For yk 2 y f -j-0, i.e. at the escape direction but still within the potential well, the solution can be approximated as: cŽj.s
w Žk2qfj. 3y2yk3 z V0 2 x | exp 1y2 2 1y4 y3 ~ 2k Žk qfj. f
(7)
The self-consistency equation for V0 wEq. (6)x, and the normalization integral which determines k are expressed as: 1f
1 w 1 53 f2 z xln | ; y k y ka 96 k6 ~
V0f21y2k3y2
(8)
The main first term on the right-hand side of Eq. (8) returns us to the expression in Eq. (2). The second term is the field correction, which provides the small (quadratic only) Stark effect: dEbfy
ŽeFx0.2 Eb*ln7Žabya.
Note the absence of the linear Stark-effect characteristic for 3D hydrogen-like excitons because there is no level degeneracy. The escape probability G is determined by the wave function flow ±c±2y at the point j1syk 2 y f where the bound state exits from below the barrier. We find Gs
w 4 Eb3y2m*1y2 z Eb | expxy y 3 " e"F ~
(9)
which provides the field dependence of the photoconductivity due to the cold ionization of the exciton. For Fs105 eVycm and Ebs0.1 eV, we estimate Gf2.8=1012 sy1.
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4. Unbound states: quasi-classical approximation
continuous spectrum, we obtain:
Contrary to photoconductivity, the absorption a is not essentially affected by F. It is worth discussing this in view of confusions between the excitonic peak and the band-to-band edge singularity. In spite of a relatively small contribution to the total intensity (see below), higher excitonic levels determine the band edge profile w2x. Smearing these transitions to the continuous spectrum, we obtain a flat shape afconstant at EsEg instead of the usually expected singularity (EyEg)y1y2. For the long-range Coulomb interaction, the quasiclassical (WKB) approximation correctly reproduces a sequence ;1yn 2 of excitonic states, while missing the primary exciton wEq. (2)x. For higher excited states, this approximation becomes adequate, even quantitatively. These states can hardly be resolved as separate levels, but they provide a major continuous contribution to the band edge shape. In the WKB approximation, the local motion is controlled by the action S related to the classical momentum P:
aexsC8)cn)2dŽEyEgyEbn.´C)cn)2
Ps
dS s dx
y
B
E e2 qEyEgF D ´)x) G
m*C
(10)
For the bound states EyEgsyEbn, the quasi-classical quantization is given by the integration between the turning points "xns"e 2 y ´Ebn: xn
|
2p"ns2
yxn
y EE
Pdxsp"
* b
(11)
bn
and hence EbnsE*by4n2. The normalized excitonic wave functions are: An B S SE Csin ,cos F, "G yP D " * 3 * y 2 ´ m Ebn 2 ym Eb A2ns s p e2 p a*bn3
cns
(12)
The absorption coefficient for the x-polarized optical transition to the nth allowed excitonic state is: 2 a1y2 aŽexn.sCwcnŽ0.x2s C 3 *3y2 p n ab
(13)
where Csconstant ; M2bb where Mbb is the dipole matrix element between the basic wave functions, which is the same as for the band-to-band absorption w2x. The ratio of the absorption aex arising from all higher excitonic states to the primary exciton absorption is: ` wc Ž0.x2 aex 8ns1 n ab s s * 2 wc1ByuŽ0.x a1Bu ab
y aa ;0.1 b * b
(14)
In spite of the relatively small contribution to the total intensity, the higher excitonic levels determine the band edge profile. Smearing these transitions to the
dn dEbn
1 a1y2 s C *3y2 * sconstant p ab Eb By continuity, this non-zero value should also be preserved for E)Eg: this is different from the model of short-range interactions, where the bleaching suppresses the intensity at Eg to zero. The question arises: what happens with the traditionally expected a;(EyEg)y1y2 singularity above the edge? While the unbound states are the plane waves at large distances, at shorter ones they show a dip due to increasing velocity. While formal expressions for C coincide with those described above, now they are considered at E)0 when they are valid at all x. Since the wave function is delocalized, its normalization comes from the large x, where the interaction is negligible. Hence, the normalized wave function is: CsCEs
B EyEg C yL D EyEgqe2y
2
Ž
E1y4 S F cos " ´)x) G
.
the magnitude of which is suppressed at small x as ±C±2;6x. This comes from the enhanced gradients (faster motion) near xs0. Eq. (10) shows that while on average NPMfym*ŽEyEg., at the origin, PŽ0.; ym*E*a*ya4NPM. The excitation probability wex(E) is constant only at energy values much higher than the excitonic scale: wexŽE.;wcEŽ0.x2s2
y EyEEyE qe yŽ´a. g 2
g
However, at lower energy values, it is suppressed as wexŽE.;yEyEg. This suppression exactly compensates for the diverging factor of the DOS dnydE;1yyEyEg. Finally, for the band-to-band absorption, we obtain the formula we mentioned in Section 2: dn 2 m* s C dE p" EyEgqe2y´a 1 a1y2 ™ C *3y2 * p ab Eb sconstant
abbsC)cE)2
y
(15)
Note that the 1yyEyEg singularity is rounded at the energy scale e 2 y ´a, which is several-fold ;a*bya larger than the excitonic scale. The absorption saturates at the band edge, thus meeting the averaged absorption from the high excitonic series below the edge. (This is different from the model of short-range interactions, where the bleaching suppresses the intensity at Eg to zero). The crossover happens at an energy E*; E*ba*bya;Eb. Here we have exploited the numerical feature: the enhancement factors a*bya for E * and ln2(R y
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
424
a) for the primary exciton are numerically similar (;4). In summary, the optical absorption increases from high energy values, saturates above the band edge at the scale of the binding energy and continues as a kind of a plateau through the edge down to the excitonic level yEb1fyEb*, and after a dip, it sharply increases at y Eb. Since the law (EgyE)y1y2 at E)Eg became limited to high enough values, it can be shadowed by a systematic increase of the band-to-band matrix elements across the total D–D* band, as found in w2x. The edge singularity is then dominated by the exciton alone. Our transparent derivations agree with elaborated analysis carried out in w6x, which was not extended to the finite F. As above, for Fs0, we obtain: asC)CEŽ0.)2
dn 1 m* fC dE p" yEyEgyFaqe2y´a
1 a1y2 B F a2 E f C *3y2 * C1y * *2 F p ab Eb D F a G
(16)
which reduces to the law in Eq. (15) at Fs0. The field dependence comes only through the correction ;yFa. This is small, as given by the field energy only within the core region a, thus providing a small Franz–Keldysh effect. 5. Conclusions The small exciton binding energy in PPV indicates that the el–el interaction is of long-range nature, compatible with a ‘band model’ approach to the electronic
excitations. The theoretical analysis indicates that whenever the sharp optical features are observed (stimulated emission, high field photoconductivity), they originate from the well-ordered, highly conjugated fraction of the polymer. In this region, the physics are dominated by the band picture and the consequent long-range bound states, which lead to narrow shallow excitons. The excitons also dominate at the onset of the direct absorption, while being smeared by contributions from lessordered regions, most probably by variation of ´. A practical consequence is that these contemporary materials still have high potential for improving their quality and applications. Together with earlier analysis w2x, we can classify the typical conjugated polymers as particular 1D semiconductors. In this spirit, their properties must be universal. Quantitatively, they can be described in terms of a few parameters, such as the gap position and the effective mass, which can be taken from experiments, or if desired, exercised in ab initio calculations. Note the possibility of some polymers being indirect gap semiconductors, as explained in w5x in relation to ‘Ag excitons’ described in quantum chemistry. References w1x N. Kirova, S. Brazovskii, A. Bishop, D. McBranch, V. Klimov, Synth. Met. 101 (1999) 188. w2x N. Kirova, S. Brazovskii, A.R. Bishop, Synth. Met. 100 (1) (1999) 29. w3x D. Moses, J. Wang, A.J. Heeger, N. Kirova, S. Brazovskii, Synth. Met. 119 (2001) 503. w4x N. Kirova, S. Brazovskii, Synth. Met. 119 (2001) 651. w5x S. Brazovskii, N. Kirova. In: Proceedings of the Nobel Symposium at Penn University, 2001, Synth. Met., in press. w6x R.J. Elliot, R. Loudon, J. Phys. Chem. Solids 8 (1959) 382.
Optical and electrooptical absorption in conducting polymers N. Kirova*, S. Brazovskii ˆ 100, Universite´ Paris-Sud, 91405 Orsay Cedex, France LPTMS, Bat.
Abstract We describe effects of the long-range Coulomb interactions on absorption and photoconductivity of polymeric semiconductors. The final-state interaction suppresses the 1D edge singularity of band-to-band transitions, reducing it to the final value. This value meets the averaged subgap absorption through the high excitonic series below the gap. After a dip, the absorption increases at the primary exciton peak. At moderately strong electric fields, higher excited states are destroyed, while the primary one is preserved. Our results explain the rounding of the e–h absorption edge observed in various polymeric semiconductors. Moreover, they quantitatively agree with new photoconductivity data on PPV-type polymers where new spectral features appear under electric field. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Conducting polymers; Optics; Electrooptics; Excitons
1. Introduction Most of the current activity in the physics and applications of conjugated polymers is centered around their ability to emit light in the optical range of approximately 2 eV. Here we are facing a curious and almost unknown situation: while these optical materials are already commercialized, there is no consensus yet on the nature of the light-emitting excitons. Thus, for nearly a decade, the estimations for exciton binding energies in phenyl-type polymers ranged within almost an order of magnitude from 0.2 to 0.9 eV. A substantial body of experimental and theoretical work has emerged, supporting each of these extremes, and a resolution of these issues remains as a fundamental challenge (see w2,5x and references therein). Determination of the exciton binding energy (Eb), and more generally the fundamental edge nature, is critically important to answering these questions, and thereby to understanding the electronic structure of semiconducting polymers. We believe that our recent interpretations of several new, key experiments allow unambiguous fixing of the * Corresponding author. Tel.: q33-169-15-73-32; fax: q33-16915-65-25. E-mail address: [email protected] (N. Kirova).
physical picture. At the microscopic level, taking into account the peculiarities of phenyl-based polymers, we identified the phenomenon of band inversion, which proves the domination of the band picture of strongly delocalized electrons and holes. In this way, a picture of a one-dimensional semiconductor becomes adequate to describe the vicinity of the fundamental edge for light absorption and emission w1,2x. More recently, we could demonstrate success for this semiconducting picture by explaining a striking observation of a new spectral peak appearing in photoconductivity under strong electric field w3,4x. This article is mostly devoted to this effect. First of all, we shall briefly recall the underlying microscopic picture of the most exploited family of phenyl-based polymers. We have suggested a theoretical picture of PPV-type conjugated polymers w1,2x, which unifies features of both a semiconductor band model (delocalized excitations), and a molecular exciton model (localized excitations), while invoking the effects of the long-range Coulomb interactions — for both bound and free e–h states on absorption and photoconductivity spectra in polymeric semiconductors. The necessary discrete energy levels appear naturally, in addition to the delocalized states, mainly from long-range Coulomb interactions. On this basis, we identified the most important optical
0040-6090/02/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 5 6 3 - 2
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
420
features in direct and in transient photo-induced absorption. On this microscopic basis, we arrive at an understanding of the absorption profile as one which is dominated by the shallow exciton unresolved from the band edge. Moreover, final-state interactions above the fundamental edge Eg (the threshold for generation of free carriers) do not change the density of states (DOS), but they strongly affect the band edge profile, namely the square root edge singularity disappears, being saturated at a finite value at the scale of the primary exciton binding energy Eb. This rounding of absorption above the gap exactly meets the mean value from the subgap absorption by the dense series of higher excited excitonic states En. The reason that the Coulomb attraction bleaches the absorption singularity is as follows. On first sight, the attractive electron and hole have a higher probability to be in one place, which determines the light-emitting probability. However, this is not the case for a light quantum particle: the uncertainty principle enforces their relative motion and the long-range force then accelerates their falling upon each other. Finally, they pass by with a high velocity vsdxydt, preventing their annihilation, the probability of which is P;1yv;6x. Our results explain the long-standing observation of the e–h absorption-edge rounding observed in various polymers and other 1D semiconductors. Now we summarize effects of external electric field F. In the experimentally significant region of intermediate fields, only the primary exciton is preserved, for which we determine the ionization probability: Gs
w 4 Eb3y2m*1y2 z Eb | expxy y 3 " e"F ~
(1)
Eq. (1) provides the field dependence of the photoconductivity due to the cold ionization of the exciton. It describes the main F-dependence of the excitonic peak in photoconductivity. All other states are not bound and show only a smoother F-dependence via the carrier mobility (Fig. 1). We have estimated the exciton binding energy from new photoconductivity measurements w3x, which quantitatively agree with the results based on our band picture. 2. Band structure and optical excitations in phenylbased polymers Today, among the family of phenyl-based polymers, the PPV one is the most studied and most likely to be processible in applications. The PPV chain consists of phenyl rings and dimers of carbon atoms providing the vinylene linkage. The energy bands may be viewed as originating from the intra-ring electronic levels at "T and "2T, as well as from vinyl dimer levels at "T1.
Fig. 1. Photocurrent excitation spectra in PPV at various external fields; the normalization factors of the curves are indicated on the right. The inset compares the excitation spectra of pristine PPV to that of the same sample after photo-oxidation by exposure in air to 400 mWycm2 of light from a Xe lamp (from w3x).
We use standard values for hopping integrals between adjacent atoms within the ring Ts2.4 eV, within the dimer T1s2.6 eV and between them ts2.1 eV. The ring states are classified in terms of their parity g,g* and u,u* with respect to the chain axis. The u,u* states have zero amplitude at the connection to the vinyl group, and hence they do not hybridize with the dimer and do not acquire a band dispersion. The g,g* states strongly hybridize with ones from vinyl, forming wide bands. We have found the analytical solution providing the dispersion equation for free electrons and transition intensities w1,2x. The robust Coulomb corrections were included by the rigid displacements of all D, L, D*, L* bands by "0.6 eV correspondingly. The resulting band structure is presented in Fig. 2. The bands L1, L1* are flat, originating from the u,u* states of the phenyl ring, while the dispersion of bands D1, D1* is remarkably high. The especially strong hybridization in PPV is an apparent consequence of the resonance T levels of the ring and T1 levels of the dimer. For a finite system, we have also found the ‘edge states’ localized near the chain ends. Their energies appear in the gaps between D1,D2 and D1*,D2*. There are several important consequences of the picture: ● Delocalization breaks the intra-ring correlation. The state responsible for highest intra-ring excitation f6 eV now contributes to the lowest inter-chain excitation f2 eV.
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
421
strongly depends on ´, the static dielectric susceptibility of the media, which can actually be several-fold higher than the optical one (f2). Hence, Eb is not universal for the polymer, but it can change for the films with respect to solutions. The long-range Coulomb interaction also originates the series of higher excitons with localization radii Rn and binding energies: ansna*b; E*b Ebns 2wnqwlnŽanya.xy1x2
Fig. 2. Band structure for PPV and peak assignments for experimental features presented in Fig. 3.
● High bandwidth reduces the lowest intramolecular transition at 5 eV to the range of visible optics f2 eV. ● The effective mass is surprisingly small, which gives rise to solid-state physics of highly mobile particles, the shallow bounds states, etc. ● Curious states with zero dispersion are predicted at high energy, which determines most of the photoinduced features observed. In terms of superconductivity, their creation corresponds to a high photoinduced doping. This band picture naturally allows for three different excitons confined by long-range Coulomb interactions. Exciton 1 (EX1) corresponds to a weakly bound state of a delocalized electron and hole in the HOMO–LUMO bands D1 and D*1 w2x. This one-dimensional Wannier– Mott exciton is well described within the effective mass approximation. Its localization radius ab and binding energy Eb, respectively, are: absab*
1 ; lnabya
EbsEb*ln2
ab a
(4) ns1,2,«
which resembles the hydrogen series. There is also a series of optically forbidden excitons. Exciton 2 (EX2) corresponds to a more tightly bound state of L1D*1 or D1L*1. Now the non-dispersive electron L1* or hole L1 plays the role of an immobile Coulomb center, similar to optically activated acceptors in semiconductors, which binds the delocalized particle of the opposite sign. With a two-fold higher effective mass m * with respect to EX1, the EX2 radius is reduced to values of the order of the unit cell, where the dispersion of the dielectric susceptibility ´ will shift it to still lower values. Supposing ´*s1.5, we find an effective radius of ;10 A and binding energy Eb2;0.8 eV. Exciton 3 (EX3) corresponds to an intra- and interring bound state of a non-dispersive electron and hole. The L,L* state is localized within one ring, being related to the u,u* states of the electron and hole near "T. This is a Frenkel exciton. In addition to intramolecular L1L*1 excitons, we expect, at even higher energies, ionized decoupled states, where L1 and L*1, belong to different rings. The assignment of the optical features from Fig. 3 is presented together with the band diagram in Fig. 2.
(2)
where a*bs
´" ; m *e 2
E*bs
m *e 4 "2´2
(3)
´ is the dielectric susceptibility H to the chain, " is Planck’s constant, m * is the electron effective mass at the zone center in k-space (for PPV, m *f0.1me w2x), e is the electron charge and a is the unit cell length. Here, the logarithm is a characteristic enhancement for the Coulomb binding in the 1D case w6x. Note that Eb
Fig. 3. Optical transitions in MEH-DSB (from w1x).
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
422
y
d 2c 1 y cqfjcs´c dj2 )j)
(5)
where we have introduced the dimensionless variables jsxya*b, ´sŽEyEg.yE*bsyk2 and f sFyF * (Eg is the bandgap). The leading order in the ln2 factor of Eq. (2) corresponds to approximating the Coulomb term by the d function: y
Fig. 4. The wave function of the primary exciton; the dashed line indicates the potential.
c )j)
Usy
e2 qeFx ´)x)
This shows a well at small x, which is separated from the unbinding region by a barrier Ubsy2eyeFy´ with a top at xsyyey´F; see Fig. 4. Relatively long-lived states can exist if their energy is below the barrier. Thus, at strong fields F)F0sF*Ža8byab.3;106y107 eV, ŽF*sEb*yab*., the primary and all higher excitons would be destroyed, so that all states will be delocalized. Hence, the photoconductivity will depend on F only via the mobility. At intermediate fields F *-F-F0, the higher excited states n are destroyed, while the primary 1Bu exciton is preserved within a finite lifetime ts Gy1, determined by the tunneling probability G through the barrier. Finally, at F-F *, a part of states with n)1 will be preserved. We shall concentrate on the experimentally significant region of intermediate fields when all higher states n/0 lie above the barrier, hence being ¨ delocalized. The Schrodinger equation for the excitons is:
|
c )j)
(6)
dj
The parameter V0 has to be determined self-consistently. For the primary exciton, the exact solution of Eqs. (5) and (6) can be written as a combination of two Airy functions:
3. Field-induced exciton ionization The long debated question on the origin of the absorption edge, and consequently on the luminescent states, seems to have been finally elucidated by new experiments on photoconductivity which resolve the excitonic peak in the external electric field w3x (Fig. 1). The most striking observation is the appearance of a narrow peak in the photocurrent spectrum Iphoto near the absorption edge at F)105 Vycm. Its emergence just below the absorption edge suggests a contribution to the carrier density from exciton dissociation by the external field. At very high fields, the bound state would be destroyed. At intermediate fields, there is a barrier to field ionization, but the carriers can dissociate by tunneling. Consider the e–h pair at a distance x. Under the applied constant electric field F and at the long-range Coulomb attraction, the potential energy is:
´yV0dŽj., V0s
S pV0
B
k2 E B k2qfj E FAiC F : j)0 D f2y3 G D f2y3 G
BiC
Tf T pV AiC k 1y3
cŽj.sU V
f
0 1y3
B
E B k2qfj E FBiC F D f2y3 G D f2y3 G 2
: j-0
For yk 2 y f -j-0, i.e. at the escape direction but still within the potential well, the solution can be approximated as: cŽj.s
w Žk2qfj. 3y2yk3 z V0 2 x | exp 1y2 2 1y4 y3 ~ 2k Žk qfj. f
(7)
The self-consistency equation for V0 wEq. (6)x, and the normalization integral which determines k are expressed as: 1f
1 w 1 53 f2 z xln | ; y k y ka 96 k6 ~
V0f21y2k3y2
(8)
The main first term on the right-hand side of Eq. (8) returns us to the expression in Eq. (2). The second term is the field correction, which provides the small (quadratic only) Stark effect: dEbfy
ŽeFx0.2 Eb*ln7Žabya.
Note the absence of the linear Stark-effect characteristic for 3D hydrogen-like excitons because there is no level degeneracy. The escape probability G is determined by the wave function flow ±c±2y at the point j1syk 2 y f where the bound state exits from below the barrier. We find Gs
w 4 Eb3y2m*1y2 z Eb | expxy y 3 " e"F ~
(9)
which provides the field dependence of the photoconductivity due to the cold ionization of the exciton. For Fs105 eVycm and Ebs0.1 eV, we estimate Gf2.8=1012 sy1.
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
423
4. Unbound states: quasi-classical approximation
continuous spectrum, we obtain:
Contrary to photoconductivity, the absorption a is not essentially affected by F. It is worth discussing this in view of confusions between the excitonic peak and the band-to-band edge singularity. In spite of a relatively small contribution to the total intensity (see below), higher excitonic levels determine the band edge profile w2x. Smearing these transitions to the continuous spectrum, we obtain a flat shape afconstant at EsEg instead of the usually expected singularity (EyEg)y1y2. For the long-range Coulomb interaction, the quasiclassical (WKB) approximation correctly reproduces a sequence ;1yn 2 of excitonic states, while missing the primary exciton wEq. (2)x. For higher excited states, this approximation becomes adequate, even quantitatively. These states can hardly be resolved as separate levels, but they provide a major continuous contribution to the band edge shape. In the WKB approximation, the local motion is controlled by the action S related to the classical momentum P:
aexsC8)cn)2dŽEyEgyEbn.´C)cn)2
Ps
dS s dx
y
B
E e2 qEyEgF D ´)x) G
m*C
(10)
For the bound states EyEgsyEbn, the quasi-classical quantization is given by the integration between the turning points "xns"e 2 y ´Ebn: xn
|
2p"ns2
yxn
y EE
Pdxsp"
* b
(11)
bn
and hence EbnsE*by4n2. The normalized excitonic wave functions are: An B S SE Csin ,cos F, "G yP D " * 3 * y 2 ´ m Ebn 2 ym Eb A2ns s p e2 p a*bn3
cns
(12)
The absorption coefficient for the x-polarized optical transition to the nth allowed excitonic state is: 2 a1y2 aŽexn.sCwcnŽ0.x2s C 3 *3y2 p n ab
(13)
where Csconstant ; M2bb where Mbb is the dipole matrix element between the basic wave functions, which is the same as for the band-to-band absorption w2x. The ratio of the absorption aex arising from all higher excitonic states to the primary exciton absorption is: ` wc Ž0.x2 aex 8ns1 n ab s s * 2 wc1ByuŽ0.x a1Bu ab
y aa ;0.1 b * b
(14)
In spite of the relatively small contribution to the total intensity, the higher excitonic levels determine the band edge profile. Smearing these transitions to the
dn dEbn
1 a1y2 s C *3y2 * sconstant p ab Eb By continuity, this non-zero value should also be preserved for E)Eg: this is different from the model of short-range interactions, where the bleaching suppresses the intensity at Eg to zero. The question arises: what happens with the traditionally expected a;(EyEg)y1y2 singularity above the edge? While the unbound states are the plane waves at large distances, at shorter ones they show a dip due to increasing velocity. While formal expressions for C coincide with those described above, now they are considered at E)0 when they are valid at all x. Since the wave function is delocalized, its normalization comes from the large x, where the interaction is negligible. Hence, the normalized wave function is: CsCEs
B EyEg C yL D EyEgqe2y
2
Ž
E1y4 S F cos " ´)x) G
.
the magnitude of which is suppressed at small x as ±C±2;6x. This comes from the enhanced gradients (faster motion) near xs0. Eq. (10) shows that while on average NPMfym*ŽEyEg., at the origin, PŽ0.; ym*E*a*ya4NPM. The excitation probability wex(E) is constant only at energy values much higher than the excitonic scale: wexŽE.;wcEŽ0.x2s2
y EyEEyE qe yŽ´a. g 2
g
However, at lower energy values, it is suppressed as wexŽE.;yEyEg. This suppression exactly compensates for the diverging factor of the DOS dnydE;1yyEyEg. Finally, for the band-to-band absorption, we obtain the formula we mentioned in Section 2: dn 2 m* s C dE p" EyEgqe2y´a 1 a1y2 ™ C *3y2 * p ab Eb sconstant
abbsC)cE)2
y
(15)
Note that the 1yyEyEg singularity is rounded at the energy scale e 2 y ´a, which is several-fold ;a*bya larger than the excitonic scale. The absorption saturates at the band edge, thus meeting the averaged absorption from the high excitonic series below the edge. (This is different from the model of short-range interactions, where the bleaching suppresses the intensity at Eg to zero). The crossover happens at an energy E*; E*ba*bya;Eb. Here we have exploited the numerical feature: the enhancement factors a*bya for E * and ln2(R y
N. Kirova, S. Brazovskii / Thin Solid Films 403 – 404 (2002) 419–424
424
a) for the primary exciton are numerically similar (;4). In summary, the optical absorption increases from high energy values, saturates above the band edge at the scale of the binding energy and continues as a kind of a plateau through the edge down to the excitonic level yEb1fyEb*, and after a dip, it sharply increases at y Eb. Since the law (EgyE)y1y2 at E)Eg became limited to high enough values, it can be shadowed by a systematic increase of the band-to-band matrix elements across the total D–D* band, as found in w2x. The edge singularity is then dominated by the exciton alone. Our transparent derivations agree with elaborated analysis carried out in w6x, which was not extended to the finite F. As above, for Fs0, we obtain: asC)CEŽ0.)2
dn 1 m* fC dE p" yEyEgyFaqe2y´a
1 a1y2 B F a2 E f C *3y2 * C1y * *2 F p ab Eb D F a G
(16)
which reduces to the law in Eq. (15) at Fs0. The field dependence comes only through the correction ;yFa. This is small, as given by the field energy only within the core region a, thus providing a small Franz–Keldysh effect. 5. Conclusions The small exciton binding energy in PPV indicates that the el–el interaction is of long-range nature, compatible with a ‘band model’ approach to the electronic
excitations. The theoretical analysis indicates that whenever the sharp optical features are observed (stimulated emission, high field photoconductivity), they originate from the well-ordered, highly conjugated fraction of the polymer. In this region, the physics are dominated by the band picture and the consequent long-range bound states, which lead to narrow shallow excitons. The excitons also dominate at the onset of the direct absorption, while being smeared by contributions from lessordered regions, most probably by variation of ´. A practical consequence is that these contemporary materials still have high potential for improving their quality and applications. Together with earlier analysis w2x, we can classify the typical conjugated polymers as particular 1D semiconductors. In this spirit, their properties must be universal. Quantitatively, they can be described in terms of a few parameters, such as the gap position and the effective mass, which can be taken from experiments, or if desired, exercised in ab initio calculations. Note the possibility of some polymers being indirect gap semiconductors, as explained in w5x in relation to ‘Ag excitons’ described in quantum chemistry. References w1x N. Kirova, S. Brazovskii, A. Bishop, D. McBranch, V. Klimov, Synth. Met. 101 (1999) 188. w2x N. Kirova, S. Brazovskii, A.R. Bishop, Synth. Met. 100 (1) (1999) 29. w3x D. Moses, J. Wang, A.J. Heeger, N. Kirova, S. Brazovskii, Synth. Met. 119 (2001) 503. w4x N. Kirova, S. Brazovskii, Synth. Met. 119 (2001) 651. w5x S. Brazovskii, N. Kirova. In: Proceedings of the Nobel Symposium at Penn University, 2001, Synth. Met., in press. w6x R.J. Elliot, R. Loudon, J. Phys. Chem. Solids 8 (1959) 382.