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Statistical treatment of exciton decay near the interface between a molecular crystal and a metal
Chemical Physics 25 (1977) 325-331 Q North.-Holland Publishing Company
STATISTICAL TREATMENT OF EXCITON DECAY NEAR THE INTERFACE BETWEEN A MOLECULAR CRYSTAL AND A METAL W. SPANNRING and H_ BASSLER FachbereichPhysikalische Chemie der UniversitiitMarbuex D 3550 hZarbutg/Luhn,Germany Received 6 May 1977
A randomwalk model is developedto describe transport and decay of Frenkel exitons in a molecular crystal near an absorbing contact taking into account the long-range character of energy transfer (ET) via dipole coupling as well as the exponential dependence of the charge transfer (CT) - rate on the separation between excited state and acceptor layer. The stationary rate equations for the exciton density are solved numerically for 40 lattice-planes adjacent to the interface using different values for both exciton jump frequency and ET-and CT-parameters. For an authracen~metal contact it is found that the number of ET-events goes through a maximum for 2 4 i < 4, i denoting the number of the lattice plane, whereas CT-acts leading to photocarrier production proceed from the second molecular layer. The relative efficiency of ET- and CT-processes is compared.
1. Outline of the problem The behaviour of Frenkel excitons at the interface between a molecular crystal and a metal has been the subject of several recent investigations [l-7]. It has been well established that a metal contact acts as a perfect sink for excitons since it opens two non-radiative channels for deactivation of a molecular excited state: energy transfer (ET) and charge transfer (CT). Energy transfer occurs vialong-range dipole coupling of the excited states to surface plasmon modes of a metal [8] _Charge transfer is a short range effect resulting from overlap between the electronic wavefunction of the metal and the excited molecule [4,9]. Both processes have been cpnfirmed experimentally. A measure for the total loss of excitons at a metal contact is the fluorescence decrement which depends on the excitation wavelength since the number of excitons diffusing towards the interface increases as the penetration depth of the exciting light decreases [l]. Photoconductivity studies represent a selective probe for the number of excitons which dissociate in course of a CT-process to leave a free carrier inside the crystal and its counterpart iu the contact. Quantitative information on both processes has
been obtained with samples where the metal was separated from the crystal by a fatty acid mono- or multilayer assembly_ The spacer reduces the reaction velocity so that exciton decay is no longer diffusionlimited and the rate constant can be derived. This technique also supplied information on the dependence of the elementary steps on donor-acceptor distance d. In accordance with theory ET was found to follow a de3-law [2,7] whereas CT displayed an exponential decay reflecting the exponential decrease of wavefunction overlap between donor and acceptor with increasing separation [3,4] _ The presence of an exciton sink at the surface of a molecular crystal has a significant influence on the spatial distribution n(x) of excitons excited by strongly absorbed light near the interface. Whereas for negligible surface decay n(x) has a maximum at x = 0, n(x = 0) becomes zero in the case of diffusion-limited surface decay. Under stationary conditions rz(x) can be calculated by solving the differential equation II(d2n/dx2)
- kr,u + Ioo exp(-a?u) = 0,
Cl?
where D is the diffusion coefficient of the excitons in a direction perpendicular to the interface, /co = ril is their rateconstant for intrinsic decay, r. their
-326
W.Spa&g, H. Biiss!er/Exciton decny near the interface molecular crystal-metal
natural lifetime, @-the absorption coefficient and I0 the incident photon flux. Applying the boundary conditions lirn n(x) = 0 arid D(dn/dx)X=O = k,,,n(O)A, (2) x+k,, being the total rate constant for all non-radiative decay modes of an exciton at the interface and A -the spa&g betieen subsequent molecular planes parallel to the interface, n(x) is found to be [IO] n(x) = W&o
- a2D>r &dD
+ ktotidA
exp(-&)
D f &&A
1
, (3) 2. Procedure
where !d = (0~~)“~ is the exciton diffusion length. For strong exciton quenching n(x) rises lineariiy with x, n(x) = [do/(ko
- a2D)] (I;’ - ct)x,
and has a maximum at x,
= (o! - ~;‘)-‘ln(&d).
(4)
For 1, = I/cu,x, Z=Zd is obtained. The implicit assumption underlying this solution is that exciton quenching is restricted the very surface, i.e. k,,(x)
= ktot, for x = 0;
=O,
number of energy- and charge transfer acts proceeding from a particular molecular layer it will be shown that the influence of an exciton “dead zone” [I l] is much smaller than surmised if the analytical distance dependence of the relevant rate constant-is used in the calculations. The result that in most cases df practica! interest only a small fraction of exciton decay events involve the top layer of the molecular crystal is important for answering the question in how far structural or chemical surface defects, which often are present at the interface, influence exciton decay.
forx>O,
a condition which is in striking contrast to the experimental result The rateconstant kET for energy 1 transfer from anthracene singlet excitons to an evaporated gold layer has been found to be km(x) = ko(do/x)3 with k, = 1.6 X IO* s-l and do = 2 18 f 15 8, [7] which means that the exciton quenching zone has a thickness of the order of 100 .& comparable to both exciton diffusion length and penetration depth of the exciting light. The purpose of this paper is to present a solution of the problem of exciton diffusion near a strongly quenching surface taking into accounethat the rate constants for ET znd CT depend on the separation between excited state and metal contact. It was stimulated by the recent work of Agranovich and Malshukov [l l] pointing out that the existence of an exciton barrier at the interface might alter the exciton decay rate, in particular the charge transfer component, by orders of magnitude. By separately caictiating tha
Distance-dependent exciton quenching within a surface zone of finite thickness could be treated in terms of macroscopic diffusion theory by introducing a distance-dependent loss term [JcGT(x)+ k&x)] n(x) in eq. (1). However, eq. (1) can then no longer be solved in closed form. Therefore the present treatment will be based on the statistical random-walk model. The molecular crystal is treated as an array of parallel molecular planes separated by a distance A which for anthracene &-planes is 9.18 A. The interface between crystal and metal is assumed to be at a distance A/2 from the center of gravity of the first molecular plane. For each layer the stationary rate equation for the exciton density can be written as a sum of generation and loss terms, the former being direct generation by light and population by exciton transfer from both neighboring layers, the latter being intrinsic decay, non-radiative decay by both energy and charge transfer, and exciton hopping to both neighboring layers. For the ith layer 10~ exp [-(i - l)aA] + !&-+I f ni_l) - 2 Wni - [ko + k&)
+ ka(i)J
ni = 0,
(9
ni is the exciton concentration
in the ith layer and W is the exciton hopping rate to one of the lattice planes adjacent to the ith plane. It is related to the macroscopic exciton diffusion coefficient D and the diffusion length 1, by W =DfA2 = 1&A22.
(61
In accordance with experimental results the &T(i) and km(i) can be written as
W. Spanming, H. BZssierfExcitondecay near the interface molecular crystal-metal
I
DISTANCE I IWX
x; [%I ma
ma
LAYER NUMBER i
LAYER NUMBER i
Fig. 1. Computed stationary exciton density per unit area in a crystal layer near an absorbing interface as a function of the layer number i and exciton jump rate W. The distance xi of the ith layer from the interface is Xi = &2i - 1) X 9.18 A. Data are calculated for Q = 3 X lo5 cm-‘rdo = 218 A, d T 2.2 A and an incident photon flux fo = 1 photon cm-’ sS. =
k,,(i)
= k, [2doj(2i - l)A] 3 ,
(7)
= kc=,0 exp [-(2i - l)A/2dcr],
(8)
and k&i)
327
where do is the distance of an exciton from the interface at which intrinsic decay and energy transfer to the quenching layer are equally effective and dcr is
the distance at which the efficiency for exciton dissociation has dropped to e-l_ Eq_ (5) has also been used by Agranovich and Melshukov [l I] except that they introduced different W values for exciton exchange between the first and the second molecular plane (i = I ,2). They were thus able to consider explicitly the influence of a surface barrier for exciton transport to the top molecular layer. To simplify the computational treatment W is assumed to be constant for all i’s in the present analysis. The results will show that this is justified in all cases of practical significance. If 1 d i d m, eq_ (5) constitutes a set of m algebraic equations for the n,-‘s which can be solved by diagonalizing the matrix for the coefficients applying
Fig. 2. Number of energy transfer events I?ET,i per unit time and area proceeding from the ith lattice plane to an absorber at the interface as a function of layer number and exciton jump frequency IV. (a = 3 X IO5 cm-‘, do = 218 A, dCT = 2.2 A, lo = 1 photon cm-* s-t).
Gauss’ algorithm. Numerical solution is done on a TR440 computer choosing m = 40 and neglecting the influence of all layers with i> m on the exciton distribution in the m surface layers. This has no influence on the relative exciton concentration and the relative number of exciton decay events per unit time, since the kET and kcr are negligible for distances exceeding 40A = 370 .&,and has only little influence on absolute values. For the model substance crystalline anthracene a = 3 X 10’ cm-’ [12] at the Su + S, absorption peak and ld = 316 w [5]. For this choice of data eq. (4) predicts a flat maximum of n(x) at a distance around 350 A from the interface whose position depends only weakly on the magnitude of the exciton quenching rate constants. Therefore the outflow of excitons from the nzth to the m+lst layer will be balanced by the exciton influx From the m+lst layer. The influence of the m+lst and the following layers on the exciton distribution in the surface zone can therefo:m be neglected.
328
W. SPannHn& H. RiisslerfExciton decay near the interface molecular crystal-metal
3. Results and diinssion In order to get information on the counterplay between energy and charge transfer at a metal contact and on the influence of exciton diffusion on the spatial‘exciton distribution near the contact, the set of eqs. (5) was solved numerically for CY = 3 X lo5 ml and all combinations of the following set of :zameters defined by eqs. (6)-(g): W = IO*, 10’ lOlo, 10” and 1012s-1, de = 10,25,50,100,1~0 and218i%,dc,= 1.0, 1.5,2.0,2.5,3.0 and 4.0 A. For km ,,, the exciton dissociation rate at zero distance’ the value 3 X 10” s-r IS . adopted which has been extrapolated for dissociation of anthracene singlet [3] and chloranil triplet excitons [4] at an aluminum contact using photocurrent data obtained with a crystal-fatty acid-metal-assembly. In determining k,-- ,, it had been assumed that only the fraction exp(lEc,d/kT) of all exciton dissociation events contribute to charge carrier production. Scour, which is identical with the activation energy of the measured photocurrent, is the coulombic binding energy between excess carrier inside the crystal and its image charge in the metal. 3.1. Spatiar exciton distribution and energy transfer In fig. 1 the stationary exciton concentration ni is plotted for the individual lattice planes as a function of exciton hopping frequency W. The energy and charge-transfer parameters are chosen to be do = 218 & which is characteristic for a gold layer [7], and d (JT = 2.2 A, [3], respectively. Htis noteworthy that the d, is predominantly determined by the optical properties of the energy acceptorlayerexpressed in terms of real and imaginary part of the refractive index [8] _ The only emitter properties entering the theoretical expression for do areemission wavelength, luminescence quantum yield and orientation of the transition dips e moment relative to the interface, but not its osc’d ator strength. This is because the quantity (d0/x)3 denotes the fraction of molecular decay events proceeding by ET as compared to intrinsic decay. The transition oscillator strength only enters kET = ko(da/x)3 through the inverse etitter lifetime r$ = kO. The calculated exciton distribution n i has an inflection point and deviates from linearity predicted by eq. (1) for small i under the assumption that exci-
QO5
LAYER NUMBER
i
Fig. 3. Number of energy transfer events fiET,i per unit time and area proceeding from the ith lattice plane as a function of i and the energy transferparameter do. (a = 3 X 105 cm-‘, W= 10” s+,dcT =2.2 A, 10 = 1 photon cm-’ 5-l). ton quenching only occurs at the very interface. This is because deeper lying lattice planes are not only depleted by exciton diffusion towards the ‘quenching) interface but also by direct long-range energy transfer. The effect becomes more pronounced the lower the exciton jump rate is. This ilhrstrates that exciton diffusion and quenching are competititve processes: The shorter the residence time of an exciton in a particular lattice plane, the lower the relative probabihty for deactivation of a temporarily excited molecule by ET. For W = 1Or2 s-l, i.e. very rapid exciton motion the gradient of the exciton distribution approaches a constant value predicted by eq. (1). . Plots of the number of energy transfer events NET,I = n$c,,(d0/xi)3 proceeding from the ith lattice plane per unit time as a function of exciton jump rate Wand energy transfer parameter d,, are shown in figs. 2 and 3, respectively. They demonstrate that ET from deeper lying lattice planes becomes more effective as W decreases and do increases. An essential result is, that except fbr W = 10” s-l fi ET i goes through a maximum located at 2 < i < 4. T& is a consequence of the decrease of the exciton concentration with decreasing distance from the mterface which overcompensates the increase in the transfer rate. For do=218Aand W=lOL’s-l,characterizinganthracene singlet excitons near a gold contact applied to an
W Spanming DISTANCE 10 I
20 30 I ‘I_ I
& B%skr/&citon
decay near the interface molecular crystal-metal
xi Ii, 10 I
DISTANCE 50 ,
60 I
7( r
1 1 1 1 3 4 5 6 LAYER NUMBER
1 7
L 8
10 I
20 I
30 1
329 x, f&
Lo I
50 I
60 ,
I
I
I
70 ,
lo-
lo-
”
m
‘:
E -2 L-
-2
lo-
l!Y
IO-
J 1
1 2
i
Fig. 4. Number of charge transfer events A’cT,i per unit time and area proceeding from the ith lattice plane as a function of i and charge transfer parameter dcT (u = 3 X IO5 cm-l, do=218A,W=lO”s-‘,lo=lphotoncm~~~~).
&crystal face, the maximum occurs for the third molecular layer which is 23 Aapart from the metal surface. The fraction of ET everus proceeding from the top molecular layer amounts to only 6.3 to 4.6% for CT-parameters increasing from d, = 1.O to dcT = 3.0 8, The apparent conclusion is that this layer, in which the electronic levels may be affected by the adjoining metal, which can carry an interfacial charge [ 111, or which may be oxidized, is unimportant as far as energy transfer is concerned. This result is in corroboration of a previous semiquantitative estimate [2]. 3.2. iZarge transfer The situation is different for charge transfer leading to dissociation of the exciton into an excess charge
10”
; ’ I
..$,=10 d I!
IId
12315678 LAYER NUMBER
i
Fig. 5. Relative number (&,&m,i of charge transfer events per unit area and time leading to generation of free charge carriers as a function of layer number i and charge transfer p~ameterCl~~(ar=3X10Scm~1,W=1011s~1,do=218~, Io = 1 photon cm” s-r, T = 295 K, E = 3.70).
inside the crystal and its image charge in the metal. The number of CT-events, &m,i = “ik~,o exp(-xi/ &.), proceeding from the ith layer per unit time is shown in fig. 4 as a function of layer number. It has a maximum a\ i = 2 only for dcr > 2.5 A and decays exponentially with increasing layer number. The functional dependence is similar for all values of the energy transfer parameter do. The experimental value for oxidative dissociation of anthracene singlet excitons at a fatty acid-aluminum contact is da = 22 -C0.3 A_ dcr is proportional to the square root of the energy separation between the energy level of the excited electron of the molecule and the conduction band in the region of wavefunction
330
W- SPanming. H. BiissIerlExciton decay near the interface molecular crystal-metal
a sharp maximum for i = 2 and drops off rapidly. Consequently only the second molecular layer gives a significant contribution to photocarrier production. This is supported by the experimentally determined activation energy of 0.08 eV for extrinsic photocarrier production at an anthracene-gold contact [S] yielding an initial distance of 13.7 A which is exactly the distance of +e second molecular layer from the interface. Presence of a “dead zone” [ 1 l] could influence the photoelectric gain only if dcT were
-I _
1-z-
I-S-
0
2ocl 181
100 d,
3.3. Competition
behveen
energy
and chmge transfer
Fii. 7. Probabilityg that an exciton is quenched by charge transfer at a contact as a function of the energy transfer parameter do for various dCT values (in A). Arrows indicate do values for gold and aluminum contacts.
Separate summation over all ET and CT events allows determination of the probability
overlap. Since the position of conduction levels is similar for a fatty acid layer [13] and for an anthracene crystal [I41 the dcT value for dissociation of singlet excitons at a pure anthracene aluminum contact is expected to be around 2 to 2.5 a. Fig. 5 predicts that in this case about half of all CT events proceed from the top molecular layer. Existence of an energy barrier preventing exciton transfer to this layer would therefore zeduce the CT yield by about a factor two. However, the reduction in the photoelectric gain would be much smaller since the primrrrily formed charge-carrier pair is bound together by coulombic forces, the binding energy being EC = e’/ 4Exi. The fraction of carrier pairs escaping geminate recombination is cp= PO(F) exp(-E,/ko, where g(F) reflects the field-dependence of the escape probability. A plot of
that an exciton near the interface decays by charge transfer. Fig. 6 presents a plot ofg versus energy tramfer distance do_ Two conclusions can be drawn: (i) With increasing energy transfer efficiency the photoelectric yield per absorbed photon decreases. This is in agreement with experimental results by Gaehrs and Willig [6] who found that the hole production efficiency of anthracene singlet excitons is much higher at an aluminum contact for which do = 180 A has been determined * than at a gold contact (do = 218 II). The present calculations predictg(Al)/g(Au) = 2 provided that dcr is independent of the contact metal. (ii) As a result of their very low quantum efficiency for radiative decay leading to d,-, values of the order of 10 A, triplet excitons should only be quenched by charge transfer. Hence g must be close to unity .for triplet excitons. Since both singlet and triplet exciton dissociation events leading to free carrier production proceed from the second molecular layer, the escape probability rp for an initially generated carrier wiu be independent from the nature of the excitonic precursor. Consequently the ratio of triplet to singlet exciton
(~/~c&T,i= r
3
02
1
shown in fig. 5 for a dielectric constant E = 3.70 [15] and T = 295 K, indicates, that the contribution of the top lattice layer to formation of free excess carrier is negligiile. For do = 2 and 2.5 A, (~~~g~~,i displays
*This is the do v?iue derived from the data of ref. [3] for a localized emitting anthracene molecule in front of an Alabsorber.
W.Spannring~ H. Biissler/Excitondecw nearthe interfacemolecularcrystahnetal
- induced photocurrents following singlet excitation must bejT/j8 = ngTjg8, where T]is the intersystem crossing yield, which for anthracene is about 0.02 [16]. Taking Chance and Pro&s [S] result j&8 = 3.5 f 0.3 for the anthracene-gold s stem,gT/g8 = 170 and, withgT = 1, g8 = 6 X lo- Y follows. According to fig. 6 this corresponds to a CT-parameter rfcT = 1.6 8. Comparison with the anthracene-AI systems shows, that dCT must depend on the nature of the contacting metal. ‘Ihis explains why the quoted ratio for carrier production via singlet excitons gs(AI)/g8(Au) is much larger than two as expected on the basis of constant dcr (see above). It is likely that +T is particularly affected by the presence of an oxide layer, e.g. Al208 which formation depends on sample preparation. This might be the reason for differences in the ratio of the photoelectric triplet and singlet gain found by different authors [5,6].
4. Conclusion The quantitative numerical treatment of energy and charge transfer from mobile Frenkel excitons to an absorbing contact on top of a molecular crystal indicates that the long-range character of energy transfer causes a significant deviation of the spatial exciton concentration from the distribution expected if exciton quenching would only occur at the interface. In particular, it is shown, that the top layer of the molecular crystal is inactive even in the absence of any surface barrier for exciton transport. Modification of an exciton-induced photocurrent by an interfacial electric double-layer as proposed in ref. [l 11 would require that a considerable fraction of molecules in the second molecular layer carries charge. This, however, is unlikely since a field of 2 X lo7 V cm-’ , which is beyond the breakdown field, would be established if only 10% of the molecules were ionized. Moreover, it is shown that the distance parameter dcT, which characterizes the efficiency of interfacial CT-reactions and which is difficult to determine other-
331
wise, can be inferred from a measurement of the ratio of extrinsic photocurrents generated by singlet and triplet excitor& respectively. The experimental background for the present numerical results are ET and ET-data for anthracenemetal contacts. Because of the wide range of parameters used in the calculations application to other systems, for instance moiecular absorbers, should be possible.
Acknowledgement A stimulating discussion with Dr. F. Willig is gratefully acknowledged. This work was supported by the Deutsche Forschungsgemeinschaft.
References
[ 11 H. Kdmann, G. Vauhel and H. BZcsler, Phys. Stat. SOL B44 (1971) 813. [2] G. Vacbel, H. BLsler and D. Miibius, Chem. Phys. Letters 10 (lP71) 334.
[3] H. Kiliesreiterand H. BZssIer, Chem. Phys Letters 11 (1971) 411. [4] H. Killesreiter and H. B&her, Phys. Stat. Sol. B51 (1972) 657. [5] R.R. Chance and A. Prock, Phys. Stat. Sol. 8.57 (1973) 597. [6] H.J. Gaehrs and F. Wiiig, Phys. Stat Sol. A27 (1975) 355. [7] H. Kurczewska and H. B%sler, J. Luminescence, to be published. [8] R.R. Chance, A. Prock and R. Sitbey, J. Chem. Phys 62 (1975) 2245. 191 J. Sir@ and H. B5sder, Phyr Stat. Sol. 862 (1974) 147. [lo] B.I. Mulder, Phil. Res. Rept. Suppl. (1968) 4. [ll] V.M. Aganovich and A.G. Malshukov, Chem. Phys. Letters 43 (1976) 221.. [12] A. Bree and L.E. Lyons, J. Chem. Sot. (19.56) 2658. [13] B. Mann, H. Kuhn and L. v. Szentpaly. Chem. Phys. Letters 8 (1971) 82. [14] N. Karl, Advan. Solid State Phys. 14 (1974) 261. [15] G. Dix, H.W. Helbergand B. Wartenberg, Phys. Stat. Sol. A5 (1971) 633. [16] R. Aviakan and RE. Merrifield, Mol. Cryst. 5 (1968) 37.
STATISTICAL TREATMENT OF EXCITON DECAY NEAR THE INTERFACE BETWEEN A MOLECULAR CRYSTAL AND A METAL W. SPANNRING and H_ BASSLER FachbereichPhysikalische Chemie der UniversitiitMarbuex D 3550 hZarbutg/Luhn,Germany Received 6 May 1977
A randomwalk model is developedto describe transport and decay of Frenkel exitons in a molecular crystal near an absorbing contact taking into account the long-range character of energy transfer (ET) via dipole coupling as well as the exponential dependence of the charge transfer (CT) - rate on the separation between excited state and acceptor layer. The stationary rate equations for the exciton density are solved numerically for 40 lattice-planes adjacent to the interface using different values for both exciton jump frequency and ET-and CT-parameters. For an authracen~metal contact it is found that the number of ET-events goes through a maximum for 2 4 i < 4, i denoting the number of the lattice plane, whereas CT-acts leading to photocarrier production proceed from the second molecular layer. The relative efficiency of ET- and CT-processes is compared.
1. Outline of the problem The behaviour of Frenkel excitons at the interface between a molecular crystal and a metal has been the subject of several recent investigations [l-7]. It has been well established that a metal contact acts as a perfect sink for excitons since it opens two non-radiative channels for deactivation of a molecular excited state: energy transfer (ET) and charge transfer (CT). Energy transfer occurs vialong-range dipole coupling of the excited states to surface plasmon modes of a metal [8] _Charge transfer is a short range effect resulting from overlap between the electronic wavefunction of the metal and the excited molecule [4,9]. Both processes have been cpnfirmed experimentally. A measure for the total loss of excitons at a metal contact is the fluorescence decrement which depends on the excitation wavelength since the number of excitons diffusing towards the interface increases as the penetration depth of the exciting light decreases [l]. Photoconductivity studies represent a selective probe for the number of excitons which dissociate in course of a CT-process to leave a free carrier inside the crystal and its counterpart iu the contact. Quantitative information on both processes has
been obtained with samples where the metal was separated from the crystal by a fatty acid mono- or multilayer assembly_ The spacer reduces the reaction velocity so that exciton decay is no longer diffusionlimited and the rate constant can be derived. This technique also supplied information on the dependence of the elementary steps on donor-acceptor distance d. In accordance with theory ET was found to follow a de3-law [2,7] whereas CT displayed an exponential decay reflecting the exponential decrease of wavefunction overlap between donor and acceptor with increasing separation [3,4] _ The presence of an exciton sink at the surface of a molecular crystal has a significant influence on the spatial distribution n(x) of excitons excited by strongly absorbed light near the interface. Whereas for negligible surface decay n(x) has a maximum at x = 0, n(x = 0) becomes zero in the case of diffusion-limited surface decay. Under stationary conditions rz(x) can be calculated by solving the differential equation II(d2n/dx2)
- kr,u + Ioo exp(-a?u) = 0,
Cl?
where D is the diffusion coefficient of the excitons in a direction perpendicular to the interface, /co = ril is their rateconstant for intrinsic decay, r. their
-326
W.Spa&g, H. Biiss!er/Exciton decny near the interface molecular crystal-metal
natural lifetime, @-the absorption coefficient and I0 the incident photon flux. Applying the boundary conditions lirn n(x) = 0 arid D(dn/dx)X=O = k,,,n(O)A, (2) x+k,, being the total rate constant for all non-radiative decay modes of an exciton at the interface and A -the spa&g betieen subsequent molecular planes parallel to the interface, n(x) is found to be [IO] n(x) = W&o
- a2D>r &dD
+ ktotidA
exp(-&)
D f &&A
1
, (3) 2. Procedure
where !d = (0~~)“~ is the exciton diffusion length. For strong exciton quenching n(x) rises lineariiy with x, n(x) = [do/(ko
- a2D)] (I;’ - ct)x,
and has a maximum at x,
= (o! - ~;‘)-‘ln(&d).
(4)
For 1, = I/cu,x, Z=Zd is obtained. The implicit assumption underlying this solution is that exciton quenching is restricted the very surface, i.e. k,,(x)
= ktot, for x = 0;
=O,
number of energy- and charge transfer acts proceeding from a particular molecular layer it will be shown that the influence of an exciton “dead zone” [I l] is much smaller than surmised if the analytical distance dependence of the relevant rate constant-is used in the calculations. The result that in most cases df practica! interest only a small fraction of exciton decay events involve the top layer of the molecular crystal is important for answering the question in how far structural or chemical surface defects, which often are present at the interface, influence exciton decay.
forx>O,
a condition which is in striking contrast to the experimental result The rateconstant kET for energy 1 transfer from anthracene singlet excitons to an evaporated gold layer has been found to be km(x) = ko(do/x)3 with k, = 1.6 X IO* s-l and do = 2 18 f 15 8, [7] which means that the exciton quenching zone has a thickness of the order of 100 .& comparable to both exciton diffusion length and penetration depth of the exciting light. The purpose of this paper is to present a solution of the problem of exciton diffusion near a strongly quenching surface taking into accounethat the rate constants for ET znd CT depend on the separation between excited state and metal contact. It was stimulated by the recent work of Agranovich and Malshukov [l l] pointing out that the existence of an exciton barrier at the interface might alter the exciton decay rate, in particular the charge transfer component, by orders of magnitude. By separately caictiating tha
Distance-dependent exciton quenching within a surface zone of finite thickness could be treated in terms of macroscopic diffusion theory by introducing a distance-dependent loss term [JcGT(x)+ k&x)] n(x) in eq. (1). However, eq. (1) can then no longer be solved in closed form. Therefore the present treatment will be based on the statistical random-walk model. The molecular crystal is treated as an array of parallel molecular planes separated by a distance A which for anthracene &-planes is 9.18 A. The interface between crystal and metal is assumed to be at a distance A/2 from the center of gravity of the first molecular plane. For each layer the stationary rate equation for the exciton density can be written as a sum of generation and loss terms, the former being direct generation by light and population by exciton transfer from both neighboring layers, the latter being intrinsic decay, non-radiative decay by both energy and charge transfer, and exciton hopping to both neighboring layers. For the ith layer 10~ exp [-(i - l)aA] + !&-+I f ni_l) - 2 Wni - [ko + k&)
+ ka(i)J
ni = 0,
(9
ni is the exciton concentration
in the ith layer and W is the exciton hopping rate to one of the lattice planes adjacent to the ith plane. It is related to the macroscopic exciton diffusion coefficient D and the diffusion length 1, by W =DfA2 = 1&A22.
(61
In accordance with experimental results the &T(i) and km(i) can be written as
W. Spanming, H. BZssierfExcitondecay near the interface molecular crystal-metal
I
DISTANCE I IWX
x; [%I ma
ma
LAYER NUMBER i
LAYER NUMBER i
Fig. 1. Computed stationary exciton density per unit area in a crystal layer near an absorbing interface as a function of the layer number i and exciton jump rate W. The distance xi of the ith layer from the interface is Xi = &2i - 1) X 9.18 A. Data are calculated for Q = 3 X lo5 cm-‘rdo = 218 A, d T 2.2 A and an incident photon flux fo = 1 photon cm-’ sS. =
k,,(i)
= k, [2doj(2i - l)A] 3 ,
(7)
= kc=,0 exp [-(2i - l)A/2dcr],
(8)
and k&i)
327
where do is the distance of an exciton from the interface at which intrinsic decay and energy transfer to the quenching layer are equally effective and dcr is
the distance at which the efficiency for exciton dissociation has dropped to e-l_ Eq_ (5) has also been used by Agranovich and Melshukov [l I] except that they introduced different W values for exciton exchange between the first and the second molecular plane (i = I ,2). They were thus able to consider explicitly the influence of a surface barrier for exciton transport to the top molecular layer. To simplify the computational treatment W is assumed to be constant for all i’s in the present analysis. The results will show that this is justified in all cases of practical significance. If 1 d i d m, eq_ (5) constitutes a set of m algebraic equations for the n,-‘s which can be solved by diagonalizing the matrix for the coefficients applying
Fig. 2. Number of energy transfer events I?ET,i per unit time and area proceeding from the ith lattice plane to an absorber at the interface as a function of layer number and exciton jump frequency IV. (a = 3 X IO5 cm-‘, do = 218 A, dCT = 2.2 A, lo = 1 photon cm-* s-t).
Gauss’ algorithm. Numerical solution is done on a TR440 computer choosing m = 40 and neglecting the influence of all layers with i> m on the exciton distribution in the m surface layers. This has no influence on the relative exciton concentration and the relative number of exciton decay events per unit time, since the kET and kcr are negligible for distances exceeding 40A = 370 .&,and has only little influence on absolute values. For the model substance crystalline anthracene a = 3 X 10’ cm-’ [12] at the Su + S, absorption peak and ld = 316 w [5]. For this choice of data eq. (4) predicts a flat maximum of n(x) at a distance around 350 A from the interface whose position depends only weakly on the magnitude of the exciton quenching rate constants. Therefore the outflow of excitons from the nzth to the m+lst layer will be balanced by the exciton influx From the m+lst layer. The influence of the m+lst and the following layers on the exciton distribution in the surface zone can therefo:m be neglected.
328
W. SPannHn& H. RiisslerfExciton decay near the interface molecular crystal-metal
3. Results and diinssion In order to get information on the counterplay between energy and charge transfer at a metal contact and on the influence of exciton diffusion on the spatial‘exciton distribution near the contact, the set of eqs. (5) was solved numerically for CY = 3 X lo5 ml and all combinations of the following set of :zameters defined by eqs. (6)-(g): W = IO*, 10’ lOlo, 10” and 1012s-1, de = 10,25,50,100,1~0 and218i%,dc,= 1.0, 1.5,2.0,2.5,3.0 and 4.0 A. For km ,,, the exciton dissociation rate at zero distance’ the value 3 X 10” s-r IS . adopted which has been extrapolated for dissociation of anthracene singlet [3] and chloranil triplet excitons [4] at an aluminum contact using photocurrent data obtained with a crystal-fatty acid-metal-assembly. In determining k,-- ,, it had been assumed that only the fraction exp(lEc,d/kT) of all exciton dissociation events contribute to charge carrier production. Scour, which is identical with the activation energy of the measured photocurrent, is the coulombic binding energy between excess carrier inside the crystal and its image charge in the metal. 3.1. Spatiar exciton distribution and energy transfer In fig. 1 the stationary exciton concentration ni is plotted for the individual lattice planes as a function of exciton hopping frequency W. The energy and charge-transfer parameters are chosen to be do = 218 & which is characteristic for a gold layer [7], and d (JT = 2.2 A, [3], respectively. Htis noteworthy that the d, is predominantly determined by the optical properties of the energy acceptorlayerexpressed in terms of real and imaginary part of the refractive index [8] _ The only emitter properties entering the theoretical expression for do areemission wavelength, luminescence quantum yield and orientation of the transition dips e moment relative to the interface, but not its osc’d ator strength. This is because the quantity (d0/x)3 denotes the fraction of molecular decay events proceeding by ET as compared to intrinsic decay. The transition oscillator strength only enters kET = ko(da/x)3 through the inverse etitter lifetime r$ = kO. The calculated exciton distribution n i has an inflection point and deviates from linearity predicted by eq. (1) for small i under the assumption that exci-
QO5
LAYER NUMBER
i
Fig. 3. Number of energy transfer events fiET,i per unit time and area proceeding from the ith lattice plane as a function of i and the energy transferparameter do. (a = 3 X 105 cm-‘, W= 10” s+,dcT =2.2 A, 10 = 1 photon cm-’ 5-l). ton quenching only occurs at the very interface. This is because deeper lying lattice planes are not only depleted by exciton diffusion towards the ‘quenching) interface but also by direct long-range energy transfer. The effect becomes more pronounced the lower the exciton jump rate is. This ilhrstrates that exciton diffusion and quenching are competititve processes: The shorter the residence time of an exciton in a particular lattice plane, the lower the relative probabihty for deactivation of a temporarily excited molecule by ET. For W = 1Or2 s-l, i.e. very rapid exciton motion the gradient of the exciton distribution approaches a constant value predicted by eq. (1). . Plots of the number of energy transfer events NET,I = n$c,,(d0/xi)3 proceeding from the ith lattice plane per unit time as a function of exciton jump rate Wand energy transfer parameter d,, are shown in figs. 2 and 3, respectively. They demonstrate that ET from deeper lying lattice planes becomes more effective as W decreases and do increases. An essential result is, that except fbr W = 10” s-l fi ET i goes through a maximum located at 2 < i < 4. T& is a consequence of the decrease of the exciton concentration with decreasing distance from the mterface which overcompensates the increase in the transfer rate. For do=218Aand W=lOL’s-l,characterizinganthracene singlet excitons near a gold contact applied to an
W Spanming DISTANCE 10 I
20 30 I ‘I_ I
& B%skr/&citon
decay near the interface molecular crystal-metal
xi Ii, 10 I
DISTANCE 50 ,
60 I
7( r
1 1 1 1 3 4 5 6 LAYER NUMBER
1 7
L 8
10 I
20 I
30 1
329 x, f&
Lo I
50 I
60 ,
I
I
I
70 ,
lo-
lo-
”
m
‘:
E -2 L-
-2
lo-
l!Y
IO-
J 1
1 2
i
Fig. 4. Number of charge transfer events A’cT,i per unit time and area proceeding from the ith lattice plane as a function of i and charge transfer parameter dcT (u = 3 X IO5 cm-l, do=218A,W=lO”s-‘,lo=lphotoncm~~~~).
&crystal face, the maximum occurs for the third molecular layer which is 23 Aapart from the metal surface. The fraction of ET everus proceeding from the top molecular layer amounts to only 6.3 to 4.6% for CT-parameters increasing from d, = 1.O to dcT = 3.0 8, The apparent conclusion is that this layer, in which the electronic levels may be affected by the adjoining metal, which can carry an interfacial charge [ 111, or which may be oxidized, is unimportant as far as energy transfer is concerned. This result is in corroboration of a previous semiquantitative estimate [2]. 3.2. iZarge transfer The situation is different for charge transfer leading to dissociation of the exciton into an excess charge
10”
; ’ I
..$,=10 d I!
IId
12315678 LAYER NUMBER
i
Fig. 5. Relative number (&,&m,i of charge transfer events per unit area and time leading to generation of free charge carriers as a function of layer number i and charge transfer p~ameterCl~~(ar=3X10Scm~1,W=1011s~1,do=218~, Io = 1 photon cm” s-r, T = 295 K, E = 3.70).
inside the crystal and its image charge in the metal. The number of CT-events, &m,i = “ik~,o exp(-xi/ &.), proceeding from the ith layer per unit time is shown in fig. 4 as a function of layer number. It has a maximum a\ i = 2 only for dcr > 2.5 A and decays exponentially with increasing layer number. The functional dependence is similar for all values of the energy transfer parameter do. The experimental value for oxidative dissociation of anthracene singlet excitons at a fatty acid-aluminum contact is da = 22 -C0.3 A_ dcr is proportional to the square root of the energy separation between the energy level of the excited electron of the molecule and the conduction band in the region of wavefunction
330
W- SPanming. H. BiissIerlExciton decay near the interface molecular crystal-metal
a sharp maximum for i = 2 and drops off rapidly. Consequently only the second molecular layer gives a significant contribution to photocarrier production. This is supported by the experimentally determined activation energy of 0.08 eV for extrinsic photocarrier production at an anthracene-gold contact [S] yielding an initial distance of 13.7 A which is exactly the distance of +e second molecular layer from the interface. Presence of a “dead zone” [ 1 l] could influence the photoelectric gain only if dcT were
-I _
1-z-
I-S-
0
2ocl 181
100 d,
3.3. Competition
behveen
energy
and chmge transfer
Fii. 7. Probabilityg that an exciton is quenched by charge transfer at a contact as a function of the energy transfer parameter do for various dCT values (in A). Arrows indicate do values for gold and aluminum contacts.
Separate summation over all ET and CT events allows determination of the probability
overlap. Since the position of conduction levels is similar for a fatty acid layer [13] and for an anthracene crystal [I41 the dcT value for dissociation of singlet excitons at a pure anthracene aluminum contact is expected to be around 2 to 2.5 a. Fig. 5 predicts that in this case about half of all CT events proceed from the top molecular layer. Existence of an energy barrier preventing exciton transfer to this layer would therefore zeduce the CT yield by about a factor two. However, the reduction in the photoelectric gain would be much smaller since the primrrrily formed charge-carrier pair is bound together by coulombic forces, the binding energy being EC = e’/ 4Exi. The fraction of carrier pairs escaping geminate recombination is cp= PO(F) exp(-E,/ko, where g(F) reflects the field-dependence of the escape probability. A plot of
that an exciton near the interface decays by charge transfer. Fig. 6 presents a plot ofg versus energy tramfer distance do_ Two conclusions can be drawn: (i) With increasing energy transfer efficiency the photoelectric yield per absorbed photon decreases. This is in agreement with experimental results by Gaehrs and Willig [6] who found that the hole production efficiency of anthracene singlet excitons is much higher at an aluminum contact for which do = 180 A has been determined * than at a gold contact (do = 218 II). The present calculations predictg(Al)/g(Au) = 2 provided that dcr is independent of the contact metal. (ii) As a result of their very low quantum efficiency for radiative decay leading to d,-, values of the order of 10 A, triplet excitons should only be quenched by charge transfer. Hence g must be close to unity .for triplet excitons. Since both singlet and triplet exciton dissociation events leading to free carrier production proceed from the second molecular layer, the escape probability rp for an initially generated carrier wiu be independent from the nature of the excitonic precursor. Consequently the ratio of triplet to singlet exciton
(~/~c&T,i= r
3
02
1
shown in fig. 5 for a dielectric constant E = 3.70 [15] and T = 295 K, indicates, that the contribution of the top lattice layer to formation of free excess carrier is negligiile. For do = 2 and 2.5 A, (~~~g~~,i displays
*This is the do v?iue derived from the data of ref. [3] for a localized emitting anthracene molecule in front of an Alabsorber.
W.Spannring~ H. Biissler/Excitondecw nearthe interfacemolecularcrystahnetal
- induced photocurrents following singlet excitation must bejT/j8 = ngTjg8, where T]is the intersystem crossing yield, which for anthracene is about 0.02 [16]. Taking Chance and Pro&s [S] result j&8 = 3.5 f 0.3 for the anthracene-gold s stem,gT/g8 = 170 and, withgT = 1, g8 = 6 X lo- Y follows. According to fig. 6 this corresponds to a CT-parameter rfcT = 1.6 8. Comparison with the anthracene-AI systems shows, that dCT must depend on the nature of the contacting metal. ‘Ihis explains why the quoted ratio for carrier production via singlet excitons gs(AI)/g8(Au) is much larger than two as expected on the basis of constant dcr (see above). It is likely that +T is particularly affected by the presence of an oxide layer, e.g. Al208 which formation depends on sample preparation. This might be the reason for differences in the ratio of the photoelectric triplet and singlet gain found by different authors [5,6].
4. Conclusion The quantitative numerical treatment of energy and charge transfer from mobile Frenkel excitons to an absorbing contact on top of a molecular crystal indicates that the long-range character of energy transfer causes a significant deviation of the spatial exciton concentration from the distribution expected if exciton quenching would only occur at the interface. In particular, it is shown, that the top layer of the molecular crystal is inactive even in the absence of any surface barrier for exciton transport. Modification of an exciton-induced photocurrent by an interfacial electric double-layer as proposed in ref. [l 11 would require that a considerable fraction of molecules in the second molecular layer carries charge. This, however, is unlikely since a field of 2 X lo7 V cm-’ , which is beyond the breakdown field, would be established if only 10% of the molecules were ionized. Moreover, it is shown that the distance parameter dcT, which characterizes the efficiency of interfacial CT-reactions and which is difficult to determine other-
331
wise, can be inferred from a measurement of the ratio of extrinsic photocurrents generated by singlet and triplet excitor& respectively. The experimental background for the present numerical results are ET and ET-data for anthracenemetal contacts. Because of the wide range of parameters used in the calculations application to other systems, for instance moiecular absorbers, should be possible.
Acknowledgement A stimulating discussion with Dr. F. Willig is gratefully acknowledged. This work was supported by the Deutsche Forschungsgemeinschaft.
References
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[3] H. Kiliesreiterand H. BZssIer, Chem. Phys Letters 11 (1971) 411. [4] H. Killesreiter and H. B&her, Phys. Stat. Sol. B51 (1972) 657. [5] R.R. Chance and A. Prock, Phys. Stat. Sol. 8.57 (1973) 597. [6] H.J. Gaehrs and F. Wiiig, Phys. Stat Sol. A27 (1975) 355. [7] H. Kurczewska and H. B%sler, J. Luminescence, to be published. [8] R.R. Chance, A. Prock and R. Sitbey, J. Chem. Phys 62 (1975) 2245. 191 J. Sir@ and H. B5sder, Phyr Stat. Sol. 862 (1974) 147. [lo] B.I. Mulder, Phil. Res. Rept. Suppl. (1968) 4. [ll] V.M. Aganovich and A.G. Malshukov, Chem. Phys. Letters 43 (1976) 221.. [12] A. Bree and L.E. Lyons, J. Chem. Sot. (19.56) 2658. [13] B. Mann, H. Kuhn and L. v. Szentpaly. Chem. Phys. Letters 8 (1971) 82. [14] N. Karl, Advan. Solid State Phys. 14 (1974) 261. [15] G. Dix, H.W. Helbergand B. Wartenberg, Phys. Stat. Sol. A5 (1971) 633. [16] R. Aviakan and RE. Merrifield, Mol. Cryst. 5 (1968) 37.