- Email: [email protected]
Distortion of excitonic states by lattice defects and violation of the quasi-momentum conservation law in excitonic absorption
8 May 2000
Physics Letters A 269 Ž2000. 245–251 www.elsevier.nlrlocaterpla
Distortion of excitonic states by lattice defects and violation of the quasi-momentum conservation law in excitonic absorption A.M. Ratner B. Verkin Institute for Low Temperature Physics & Engineering 61164 KharkoÕ, Ukraine Received 10 March 2000; accepted 29 March 2000 Communicated by V.M. Agranovich
Abstract Absorption spectrum, related to creating excitons with nonzero wave vector inside a wide band, is considered for an insulator crystal with lattice defects. It is shown that the main contribution to absorption is determined not by the scattering section of defects but by the exciton state distortion within the core or a close vicinity of a defect. Its contribution to absorption differs from that of an impurity center by a great multiplier dependent on the size of the region of distortion. Due to this, lattice defects with a very low concentration Ž; 10 14 cmy3 ., present even in a perfect crystal, can manifest themselves in luminescence excitation spectrum. This can be used for the diagnostics of lattice defects in insulator crystals Že.g., scintillators. exposed to irradiation. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The present consideration is concerned with absorption spectrum related to generation of excitons in a wide band originated by an allowed atomic transition Žspatial dispersion, inessential for the phenomena under consideration, is not taken into account.. Along with a strong narrow absorption peak, related to creating excitons with the zero wave vector k Žat the band bottom or top., there is a weak absorption in the spectral region corresponding to generation of excitons with k / 0 inside the band. According to a conventional viewpoint, this weak absorption consists of two contributions of different nature: First, there is absorption accompanied by the simultaneous generation of an exciton and phonon E-mail address: [email protected] ŽA.M. Ratner..
with wave vectors k / 0 and yk; such transitions meet the quasi-momentum conservation law but are weakly pronounced commensurably to a weak exciton–phonon interaction. This contribution predominates for a perfect crystal. Second, absorption with creating excitons with k / 0 is made possible by the scattering of generated plane waves resulting in their partial dephasing. This manifests itself mainly in a smearing of the strong absorption lines related to creating excitons with k s 0. Such an effect is usually described in terms of an imaginary addition to the exciton energy or wave vector, no matter what is the specific cause of scattering: exciton–phonon interaction w1x, lattice defects w2;3x or interaction between excitons w4x. The present consideration is focussed on the case of a lattice with static defects. The above-mentioned conventional way Žthrough exciton scattering., in
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 6 1 - 9
246
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
which excitonic absorption is influenced by lattice defects, usually cannot result in a noticeable effect. Indeed, for a reasonable concentration of lattice defects Ž n F 10 16 cmy3 . and their scattering section Ž s F 10y1 4 cm2 ., the corresponding free path length of excitons Ž1rn s G 10y2 cm. much exceeds that related to the exciton–phonon scattering. The purpose of this Letter is to show that, despite such an obvious argument, defects of a real crystal can essentially contribute to excitonic absorption accompanied by creating excitons with k / 0. The point is that the influence of lattice defects on excitonic absorption is not reducible to the effect of exciton scattering, that is to the asymptotic behavior of plane waves far from a lattice defect. It is just the exciton state distortion within the core or a close vicinity of a defect which produces the main contribution to absorption spectrum. Illustratively speaking, the contribution of a lattice defect to absorption differs from that of an impurity center by a great multiplier dependent on the distortion size Žthe number of sites in the defect vicinity where the excitonic state is distorted.. Due to this, lattice defects with a very low concentration of 10 14 cmy3 Žpresent even in a perfect crystal., can manifest themselves in luminescence excitation spectrum.
2. General relations To elucidate qualitative effects, caused by the distortion of excitonic states by lattice defects, we will describe excitons within the simplest model neglecting such details Žspatial dispersion, anisotropy of exciton state etc.. which, being taken into account, could not eliminate the main qualitative conclusions. Let us consider the dipole transition between the ground and excited states of a wide-gap insulator Žof the type of a rare-gas or alkali-halide crystal.. In the excited state Žwith a single exciton., the crystal is described by the Hamiltonian H s H0 q V divided into two parts: the site Hamiltonian H0 s Ý H0 n and the rest V which has the meaning of the exchange interatomic interaction Žthe dipole–dipole interaction is neglected compared to exchange.. Here H0 n is the Hamiltonian of the nth atom including the singleelectron part of the crystal potential responsible for
the crystalline shift of the atomic level w5x. The crystal state C meets the Schrodinger equation ¨
ž ÝH
0n qV
n
/ C s EC .
Ž 1.
Eq. Ž1. is solved in the atomic basis formed by the eigenfunctions, Cn , of the Hamiltonian H0 n . By definition, H0 nCn s W Ž n . Cn .
Ž 2.
The site level W Ž n. depends on n due to lattice defects and plays the role of potential energy for an exciton. For brevity, we write down the site wavefunction of the crystal, Cn , neglecting interatomic exchange Žalready taken into account by the term V in Eq. Ž1..:
Cn s cn
Ł wm ,
Ž 3.
m/n
where wn and cn stand for the ground and excited state, respectively, of the nth atom. Irrespective of the spatial extension of the excited state cn , the overlap integral between cn and cm for m / n is small Žit cannot exceed a small overlap between the ground-state single-electron wavefunctions centered at adjacent sites.. This imparts atomic character to crystal states and justifies the use of atomic basis. The ground-state wavefunction of the crystal is:
Cgr s Ł wn .
Ž 4.
n
Substituting the expansion
C s Ý C Ž n . Cn
Ž 5.
n
into Eq. Ž1. and neglecting terms quadric in the mentioned overlap integral, we obtain equation for an excitonic state in the site representation yV0
Ý C Ž n q l . q C Ž n.
W Ž n. y E s 0
Ž 6.
< l
Ž a is the minimal interatomic distance and ŽyV0 . the corresponding exchange integral; exchange interaction is allowed for only for adjacent sites.. This equation differs from a usual form by the coordinate dependence of the site level W Ž n. which is condi-
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
tioned by lattice defects breaking the translation symmetry of the crystal. To investigate the exciton wavefunction C Ž n., it is useful to assume its slow spatial variation and to rewrite Ž6. in the continual approximation 1 y 2m
DC Ž r . q W Ž r . y E C Ž r . s 0 ,
Ž 7.
with effective mass m s 1rV0 za2 Ž z is coordination number and E is exciton energy counted from the band bottom.. The above conventional relations were written down in order to specify absorption coefficient k , related to the transitions from the ground state Ž4. to all excited states Ž5.:
kf
8p
v
3" ncD v
Ý < M0 a < 2 ,
Ž 8.
a
where c is light velocity, n is the refractive index of the crystal, M0 a stands for the matrix element of dipole moment between the ground state Ž4. and the excited state Ž5. numbered by index a . The summation in Ž8. is extended over all the excited states a for which transition energy gets into the spectral interval D v . Using the ground state Ž4. and excited state Ž5., this matrix element can be rewritten with allowance for Ž3. in the form M0 a s Ý Ca Ž n . d 0 a Ž n . .
Ž 9.
n
Here d 0 a is atomic dipole moment which can depend on n in so far the translation symmetry of the crystal is broken by lattice defects. Below, Eqs. Ž8. – Ž9. will be investigated separately for a continuous excitonic spectrum and for the discrete spectrum of excitons localized near lattice defects.
3. Light absorption with creating free excitons distorted by lattice defects Lattice defects, present in a crystal in a low concentration, make additive contributions to absorption conditioned by the distortion of excitonic states in the core of defects. Let us consider a crystal with
247
a single defect; its contribution to absorption depends on its type. We will restrict ourselves to the most interesting type of lattice defects – vacancy clusters Žsmall pores.; their concentration can be controlled via irradiation or annealing of the crystal w6;7x. It is convenient to investigate the matrix element Ž9., appearing in Eq. Ž8., separately for the lowest excitonic band and higher bands. 3.1. The lowest excitonic band For the lowest excitonic band of rare-gas or alkali-halide crystals, the radius of the excitonic site state, r , does not exceed the interatomic distance a. Such site states cannot be strongly distorted by the neighborhood of a small pore; this permits one to assume d 0 a to be independent of the site position n. Thus, the exciton state distortion is connected with the coordinate dependence of the first multiplier in Ž9.: C a Ž n. s 0 for empty sites n positioned inside the pore. Outside the pore, the exciton wavefunction C a Ž n. is a linear combination of plane waves with the same < k <. This wavefunction together with its analytic extension to the inward region of the pore will be denoted as C˜a Ž n.. For k / 0, the summation of C˜a Ž n. over all sites of the crystal results in zero. Hence, the sum Ž9., taken over all sites except those inside the pore, can be presented in the form M0 a s yZa d 0 a Ny1 r2 , Za s 'N
Ý C˜a Ž n . ,
Ž 10 . Ž 11 .
ng V
where V designates the region inside the pore. The quantity Ž11. has the meaning of an effective size of distortion Žexpressed in the number of sites.. After substituting Eqs. Ž10. and Ž11. into Ž8., the summation in Eq. Ž8. should be extended over the states with a s k corresponding to the transition frequency within a given interval d v . After multiplying by the number, Ndef , of identical lattice defects per unit volume, the expression Ž8. takes up the form
kŽ v. f
4p e 2 m 0 nc
f 01 g Ž v . Ndef < Z < 2 .
Ž 12 .
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
248
Here e and m 0 are the free-electron charge and mass, g Ž v . is the excitonic density of states within the lowest band of the width B Ž g Ž v . ; 1rB ., f 01 is the corresponding atomic oscillator strength. As can be seen from Ž12., the contribution of distorted excitons to absorption mainly coincides with that of impurity centers with the concentration Ndef < Z < 2 and the same oscillator strength Žsupposed their absorption band has the width B .. For a pore with a small radius R, meeting the inequality < k < R < 1, the exciton scattering can be neglected and the exciton wavefunction beyond the pore is simply Ny1 r2 expŽ ikn.. In this case the effective distortion size Ž11. coincides with the true size of the vacancy cluster: < Z k < s Z0 for < k < R < 1 ,
Ž 13 .
where Z0 is the number of sites within the pore. For < k < R ); 1, scattering of excitons should be taken into account. This is easy to make for a large pore with R 4 a in the continual approximation. In the Schrodinger equation Ž7., the potential W Ž r . ¨ should be put equal to 0 for r ) R and to ` for r F R. The normalized solution of this equation in the region r G R, turning to a plane wave on infinity, is of the form w8x Ck Ž r . s
(N v
0
k
Ý
3.2. Higher bands of excitons Consider now excitons of higher bands with the site state radius r large compared with the interatomic distance a. Since the site excitonic state is distorted within the distance r from the pore surface, the simple expression Ž11. for Z is inapplicable. To retain the illustrative estimation Ž12., the summation in Ž11. should be extended over the sphere of the radius R q r within which the exciton potential W Ž n. differs from that in ideal lattice. Within this sphere, the dipole matrix element d 0 a , high sensitive to a variation of the excited-state wavefunction with several radial nodes w5x, significantly varies Žwith changing sign. from site to site. Taking this into account, the sum Ž9. can be roughly estimated, under assumption on random phases of the summands, as a quantity proportional to the square root from the number of sites inside the sphere of radius R q r . In such an approach, the estimation Ž12. holds if the distortion size Z is defined as Z f Ž 4pr3v0 .
`
i
differs from Z0 by a small multiplier 6Ž kR .y2 Žhowever, < Z < f 8p Ž Rra.Ž ka.y2 much exceeds unity..
1r2
Ž Rqr .
3r2
.
Ž 16 .
Ž 2 l q 1 . Fk l Ž r . Pl Ž cos u . ,
ls0
Ž 14 . where cos u s krrkr, Pl is Legendre polynomial, v0 is volume per lattice site; Fk l Ž r . denotes an orthogonal set of functions from which only the lowest term, Fk 0 Ž r . s Ž1rr .sin k Ž r y R ., is used below. Within the continual approximation, one has to change in the expression Ž11. from summation to integration over the region 0 F u F p , 0 F r F R and to use as an integrand the analytic extension of Eq. Ž14. obtained for r G R. Integration over u annihilates all summands in Ž14. with l / 0, and Eq. Ž11. takes the form < Zk < s
6Z0
Ž kR .
2
ž
1y
sin kR kR
/
,
R4a
Ž 15 .
Ž a is interatomic distance.. For kR < 1, < Z < turns to Z0 in accordance with Ž13.. In the opposite case, < Z <
4. Light absorption with generation of excitons localized near lattice defects For a deep enough potential well, W, the Schrodi¨ nger equation Ž7. has solutions with discrete energy levels lying below the band bottom. Such a discrete level a makes a contribution to absorption which can be estimated, proceeding from Ž8. and Ž9., by Eq. Ž12. with Za s Ca Ž r . d rr v0 .
H
Ž 17 .
For the lowest discrete state with a large localization radius r Žnot to be confused with the site state radius r ., Eq. Ž17. can be rewritten as Za f Ž 4pr3 . r 3rv0 .
Ž 18 .
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
The discrete spectrum for sure exists in two cases. Ži. Excitons localized near a pore in a rare-gas crystal. In rare-gas crystals ŽNe, Ar, Kr, Xe., there exist surface exciton bands the bottom of which is shifted by the value of 0.1 to 0.3 eV relative to the bottom of the corresponding bulk band. The wave vector of a surface exciton has an imaginary normal component k z and two real tangential components k x , k y . A similar excitonic state, localized near a pore, has nearly the same energy shift relative to the bulk band bottom, a radial multiplier with nearly the same penetration depth L s Ž2 < k z <.y1 and an angular multiplier – spherical function instead of the tangential multiplier, expŽ ik x x q ik y y ., of the true surface-exciton state. According to Ž17., only the lowest angular state makes a nonzero contribution to absorption characterized by the distortion size 3
Z f Ž 4pr3v0 . Ž R q L . y R 3 ,
Ž 19 .
with the penetration depth L G a. Žii. Excitons localized near a charged lattice defect. It is known w6;7x that a lattice defect in rare-gas crystals Žin particular, a small pore. can capture a light conduction-band electron which forms a broad cloud with radius R ch ) a. Higher-band excitons with a large polarizability can be localized near such charged defects. The radius of such a localized excitonic state can be estimated under condition Egy En < En y E loc ,
Ž 20 .
where En is the position of the n th band Žthe lowest band being numbered by n s 1., Eg is the dielectric gap width Ž En Eg for n `., and Eloc is the level of the localized exciton. Inequality Ž20. means that the electron and hole, forming an exciton, are weakly bound as compared to the binding of the hole with the negative charge. In the lowest approximation with respect to the electron–hole binding, the hole is localized on a hydrogen-like level near the negative charge. For estimations, let the charge be uniformly distributed within the sphere of the radius R ch . Then the
™
™
249
energy and radius of the localized state can be approximated with accuracy 10% Žin atomic units.: E loc f y
0.7
´ R ch
,
rs
0.8
´ < Eloc <
f 1.1 R ch for 6 -
m R ch ´
- 60.
Ž 21 .
Here m is the effective mass of holes Žassumed to be isotropic., e is the crystal permittivity. With allowance for Ž21., the distortion size Ž18. can be written Žin atomic units. as Zf
2.2
´ 3 < Eloc < 3 v0
f
6 R 3ch v0
.
Ž 22 .
As for rare-gas crystals, the estimation Ž21–22. is rather rough in view of a highly anisotropic character of the effective mass of holes w5x. For example, in the case of argon Ž e s 1.66, m ; 2 ., a charge with ˚ Ž1.6 times the interatomic distance. creradius 6 A ates an excitonic state with localization energy of ˚ and effective distortion size about 1 eV, radius 7 A Z f 40.
5. Comparison with experiment First it is useful to make a rough general estimation for the contribution of lattice defects to excitonic absorption in rare-gas crystals. According to Eqs. Ž16., Ž19., Ž22., the effective distortion size Z exceeds 10 even for a small vacancy cluster with radius a Žconsisting of 6 vacancies.. The contribution of such defects to absorption, proportional to Z 2 , exceeds at least by two orders of magnitude the absorption of impurity centers for the same concentration Ndef , atomic oscillator strength f and the absorption band half-width D v . For f f 1 and D v s 0.5 eV, such vacancy clusters in the concentration Ndef s 10 14 cmy3 give rise to a noticeable absorption coefficient exceeding 1 cmy1 . Structural defects of such a low concentration are present even in a perfect crystal. To illustrate the above inferences, we will analyze the luminescence excitation spectrum of an argon crystal of a good structure quality w7x presented in Fig. 1. This excitation spectrum, measured in the
250
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
Fig. 1. Two-site exciton luminescence excitation spectrum of solid argon w7x which mainly reproduces the absorption spectrum in the region of excitonic absorption. Bold line relates to a perfect crystal with a low concentration of vacancy clusters Ž Ndef ; 3= 10 14 cmy3 .. Thin line relates to the same sample after a strong irradiation sufficient to enhance Ndef by an order of magnitude. The dashed vertical lines indicate the bottom position for the lowest exciton band Ž E1 ., for the next higher band Ž E2 . and the dielectric gap Eg . Short lines labeled ‘‘S’’ show the bottom positions for surface excitons. The table attached to the figure divides the frequency axis into intervals and indicates for every of them the squared absorption size and the corresponding optical density of the sample before irradiation. The downfalls of the excitation spectrum in the region of low optical density Žbold line. disappear after irradiation.
luminescence band of two-site self-trapped excitons, mainly reproduces the spectrum of weak absorption since almost all excitations, generated by absorbed light, turn to two-site self-trapped excitons. In the figure, the excitation spectrum is shown by bold line for a perfect crystal and by thin line for the same crystal after a strong irradiation sufficient to produce a noticeable number of lattice defects Žvacancy clusters.. Abscissa axis is divided into spectral regions F1, F2, where absorbed light generates free excitons, and L1, L2, where excitons are generated in a localized state. The region L1 lies on the left-hand side from the lowest band bottom labeled as E1 Žspin-orbital splitting, insignificant for argon, is neglected.. Absorption in this region corresponds to generation of lowest-band excitons, localized near the surface of pores, and is characterized by distortion size Ž19.. In the region F1, covering the
energy extension Žabout 0.6 eV. of the lowest band, absorbed light produces free excitons with distortion size Ž15.. In the region F2, free excitons with distortion size Ž16. are created in the second band. In the region L2, where excitons of the second band are generated below its bottom E2 in localized states, the most essential contribution to absorption spectrum is made by charged defects with distortion size Ž22.. The excitation spectrum, shown in the figure, can be qualitatively explained under assumption that the crystal in the initial state Žbefore irradiation. contained rather small vacancy clusters with R s a Ž Z0 f 6. in a very low concentration Ndef s 3 = 10 14 cmy3 . The squared distortion size, estimated according to Ž19., Ž15., Ž16. and Ž22., is indicated in the table attached to the figure. In the second line of the table, optical density k d is estimated with the use of Eq. Ž12. for a crystal of the thickness d s 0.1 cm Žatomic oscillator strength is near to 0.8 and 0.2 for the lowest and second band, respectively.. The optical density, estimated in such a way in the spectral regions L1, F1, L2, F2, qualitatively explains the character of the excitation spectrum presented in the figure. The table attached to the figure is related to the perfect crystal before irradiation Žits excitation spectrum is shown by bold line.. In the regions L1 and L2, where the initial optical density exceeds unity, the excitation efficiency actually achieved saturation and, as seen from the figure, did not grow after the number of lattice defects was enhanced by irradiation Žnew photo-produced defects even somewhat diminished the excitation efficiency for the two-site exciton luminescence which can be assigned to the development of a competing relaxation channel w7x.. The picture is of opposite character in the regions F1 and F2 where the initial optical density is small Žespecially in the region F1., so that excitation efficiency is low and far from saturation. In these regions, excitation efficiency became several times greater after irradiation; this is indicative of the corresponding photoinduced increase Žby an order of magnitude. of the number of vacancy clusters. It should be emphasized that such a low concentration of lattice defects Ž; 10 14 or 10 15 cmy1 . is for sure insufficient to influence upon absorption or excitation spectrum through scattering of excitons
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
Žthe corresponding free path length is of the macroscopic scale of 1 cm.. In conclusion it can be noted that a high sensitivity of absorption spectrum to lattice defects of insulator crystals can be used for the diagnostics of photoproduced defects Žvacancy aggregates. and for the study of their dose dependence, which is important, in particular, for enhancement of radiation hardness of scintillation crystals.
Acknowledgements The author is thankful to A.N. Ogurtsov and E.V. Savchenko for fruitful discussions. This work was partly supported by the grant INTAS 96-626.
251
References w1x V.M. Agranovich, Theory of Excitons, Nauka, Moscow, 1968 Žin Russian.. w2x V.L. Bonch-Bruyevich, I.P. Zvyagin, R. Kaiper, A.G. Mironov, R. Enderlain, B. Esser, Electronic Theory of Unordered Semiconductors, Nauka, Moscow, 1981 Žin Russian.. w3x M. Wubs, J. Knoester, Chem. Phys. Lett. 284 Ž1998. 63. w4x I.M. Sokolov, A. Blumen, G. Oshanin, P. Argyrakis, in: M. Schreiber ŽEd.., Proc. 2nd Int. Conf. on Excitonic Processes in Condensed Matter, Dresden University Press, 1996. w5x A.M. Ratner, Phys. Rep. 269 Ž1996. 197. w6x A.N. Ogurtsov, E.V. Savchenko, J. Becker, M. Runne, G. Zimmerer, J. Lumin. 76 Ž1998. 478. w7x A.N. Ogurtsov, A.M. Ratner, E.V. Savchenko, V. Kisand, S. Vielhauer, J. Phys.: Condens. Matter 12 Ž2000. 2769. w8x L.D. Landau, E.M. Lifshits, Quantum Mechanics, Nauka, Moscow, 1963 Žin Russian..
Physics Letters A 269 Ž2000. 245–251 www.elsevier.nlrlocaterpla
Distortion of excitonic states by lattice defects and violation of the quasi-momentum conservation law in excitonic absorption A.M. Ratner B. Verkin Institute for Low Temperature Physics & Engineering 61164 KharkoÕ, Ukraine Received 10 March 2000; accepted 29 March 2000 Communicated by V.M. Agranovich
Abstract Absorption spectrum, related to creating excitons with nonzero wave vector inside a wide band, is considered for an insulator crystal with lattice defects. It is shown that the main contribution to absorption is determined not by the scattering section of defects but by the exciton state distortion within the core or a close vicinity of a defect. Its contribution to absorption differs from that of an impurity center by a great multiplier dependent on the size of the region of distortion. Due to this, lattice defects with a very low concentration Ž; 10 14 cmy3 ., present even in a perfect crystal, can manifest themselves in luminescence excitation spectrum. This can be used for the diagnostics of lattice defects in insulator crystals Že.g., scintillators. exposed to irradiation. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The present consideration is concerned with absorption spectrum related to generation of excitons in a wide band originated by an allowed atomic transition Žspatial dispersion, inessential for the phenomena under consideration, is not taken into account.. Along with a strong narrow absorption peak, related to creating excitons with the zero wave vector k Žat the band bottom or top., there is a weak absorption in the spectral region corresponding to generation of excitons with k / 0 inside the band. According to a conventional viewpoint, this weak absorption consists of two contributions of different nature: First, there is absorption accompanied by the simultaneous generation of an exciton and phonon E-mail address: [email protected] ŽA.M. Ratner..
with wave vectors k / 0 and yk; such transitions meet the quasi-momentum conservation law but are weakly pronounced commensurably to a weak exciton–phonon interaction. This contribution predominates for a perfect crystal. Second, absorption with creating excitons with k / 0 is made possible by the scattering of generated plane waves resulting in their partial dephasing. This manifests itself mainly in a smearing of the strong absorption lines related to creating excitons with k s 0. Such an effect is usually described in terms of an imaginary addition to the exciton energy or wave vector, no matter what is the specific cause of scattering: exciton–phonon interaction w1x, lattice defects w2;3x or interaction between excitons w4x. The present consideration is focussed on the case of a lattice with static defects. The above-mentioned conventional way Žthrough exciton scattering., in
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 6 1 - 9
246
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
which excitonic absorption is influenced by lattice defects, usually cannot result in a noticeable effect. Indeed, for a reasonable concentration of lattice defects Ž n F 10 16 cmy3 . and their scattering section Ž s F 10y1 4 cm2 ., the corresponding free path length of excitons Ž1rn s G 10y2 cm. much exceeds that related to the exciton–phonon scattering. The purpose of this Letter is to show that, despite such an obvious argument, defects of a real crystal can essentially contribute to excitonic absorption accompanied by creating excitons with k / 0. The point is that the influence of lattice defects on excitonic absorption is not reducible to the effect of exciton scattering, that is to the asymptotic behavior of plane waves far from a lattice defect. It is just the exciton state distortion within the core or a close vicinity of a defect which produces the main contribution to absorption spectrum. Illustratively speaking, the contribution of a lattice defect to absorption differs from that of an impurity center by a great multiplier dependent on the distortion size Žthe number of sites in the defect vicinity where the excitonic state is distorted.. Due to this, lattice defects with a very low concentration of 10 14 cmy3 Žpresent even in a perfect crystal., can manifest themselves in luminescence excitation spectrum.
2. General relations To elucidate qualitative effects, caused by the distortion of excitonic states by lattice defects, we will describe excitons within the simplest model neglecting such details Žspatial dispersion, anisotropy of exciton state etc.. which, being taken into account, could not eliminate the main qualitative conclusions. Let us consider the dipole transition between the ground and excited states of a wide-gap insulator Žof the type of a rare-gas or alkali-halide crystal.. In the excited state Žwith a single exciton., the crystal is described by the Hamiltonian H s H0 q V divided into two parts: the site Hamiltonian H0 s Ý H0 n and the rest V which has the meaning of the exchange interatomic interaction Žthe dipole–dipole interaction is neglected compared to exchange.. Here H0 n is the Hamiltonian of the nth atom including the singleelectron part of the crystal potential responsible for
the crystalline shift of the atomic level w5x. The crystal state C meets the Schrodinger equation ¨
ž ÝH
0n qV
n
/ C s EC .
Ž 1.
Eq. Ž1. is solved in the atomic basis formed by the eigenfunctions, Cn , of the Hamiltonian H0 n . By definition, H0 nCn s W Ž n . Cn .
Ž 2.
The site level W Ž n. depends on n due to lattice defects and plays the role of potential energy for an exciton. For brevity, we write down the site wavefunction of the crystal, Cn , neglecting interatomic exchange Žalready taken into account by the term V in Eq. Ž1..:
Cn s cn
Ł wm ,
Ž 3.
m/n
where wn and cn stand for the ground and excited state, respectively, of the nth atom. Irrespective of the spatial extension of the excited state cn , the overlap integral between cn and cm for m / n is small Žit cannot exceed a small overlap between the ground-state single-electron wavefunctions centered at adjacent sites.. This imparts atomic character to crystal states and justifies the use of atomic basis. The ground-state wavefunction of the crystal is:
Cgr s Ł wn .
Ž 4.
n
Substituting the expansion
C s Ý C Ž n . Cn
Ž 5.
n
into Eq. Ž1. and neglecting terms quadric in the mentioned overlap integral, we obtain equation for an excitonic state in the site representation yV0
Ý C Ž n q l . q C Ž n.
W Ž n. y E s 0
Ž 6.
< l
Ž a is the minimal interatomic distance and ŽyV0 . the corresponding exchange integral; exchange interaction is allowed for only for adjacent sites.. This equation differs from a usual form by the coordinate dependence of the site level W Ž n. which is condi-
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
tioned by lattice defects breaking the translation symmetry of the crystal. To investigate the exciton wavefunction C Ž n., it is useful to assume its slow spatial variation and to rewrite Ž6. in the continual approximation 1 y 2m
DC Ž r . q W Ž r . y E C Ž r . s 0 ,
Ž 7.
with effective mass m s 1rV0 za2 Ž z is coordination number and E is exciton energy counted from the band bottom.. The above conventional relations were written down in order to specify absorption coefficient k , related to the transitions from the ground state Ž4. to all excited states Ž5.:
kf
8p
v
3" ncD v
Ý < M0 a < 2 ,
Ž 8.
a
where c is light velocity, n is the refractive index of the crystal, M0 a stands for the matrix element of dipole moment between the ground state Ž4. and the excited state Ž5. numbered by index a . The summation in Ž8. is extended over all the excited states a for which transition energy gets into the spectral interval D v . Using the ground state Ž4. and excited state Ž5., this matrix element can be rewritten with allowance for Ž3. in the form M0 a s Ý Ca Ž n . d 0 a Ž n . .
Ž 9.
n
Here d 0 a is atomic dipole moment which can depend on n in so far the translation symmetry of the crystal is broken by lattice defects. Below, Eqs. Ž8. – Ž9. will be investigated separately for a continuous excitonic spectrum and for the discrete spectrum of excitons localized near lattice defects.
3. Light absorption with creating free excitons distorted by lattice defects Lattice defects, present in a crystal in a low concentration, make additive contributions to absorption conditioned by the distortion of excitonic states in the core of defects. Let us consider a crystal with
247
a single defect; its contribution to absorption depends on its type. We will restrict ourselves to the most interesting type of lattice defects – vacancy clusters Žsmall pores.; their concentration can be controlled via irradiation or annealing of the crystal w6;7x. It is convenient to investigate the matrix element Ž9., appearing in Eq. Ž8., separately for the lowest excitonic band and higher bands. 3.1. The lowest excitonic band For the lowest excitonic band of rare-gas or alkali-halide crystals, the radius of the excitonic site state, r , does not exceed the interatomic distance a. Such site states cannot be strongly distorted by the neighborhood of a small pore; this permits one to assume d 0 a to be independent of the site position n. Thus, the exciton state distortion is connected with the coordinate dependence of the first multiplier in Ž9.: C a Ž n. s 0 for empty sites n positioned inside the pore. Outside the pore, the exciton wavefunction C a Ž n. is a linear combination of plane waves with the same < k <. This wavefunction together with its analytic extension to the inward region of the pore will be denoted as C˜a Ž n.. For k / 0, the summation of C˜a Ž n. over all sites of the crystal results in zero. Hence, the sum Ž9., taken over all sites except those inside the pore, can be presented in the form M0 a s yZa d 0 a Ny1 r2 , Za s 'N
Ý C˜a Ž n . ,
Ž 10 . Ž 11 .
ng V
where V designates the region inside the pore. The quantity Ž11. has the meaning of an effective size of distortion Žexpressed in the number of sites.. After substituting Eqs. Ž10. and Ž11. into Ž8., the summation in Eq. Ž8. should be extended over the states with a s k corresponding to the transition frequency within a given interval d v . After multiplying by the number, Ndef , of identical lattice defects per unit volume, the expression Ž8. takes up the form
kŽ v. f
4p e 2 m 0 nc
f 01 g Ž v . Ndef < Z < 2 .
Ž 12 .
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
248
Here e and m 0 are the free-electron charge and mass, g Ž v . is the excitonic density of states within the lowest band of the width B Ž g Ž v . ; 1rB ., f 01 is the corresponding atomic oscillator strength. As can be seen from Ž12., the contribution of distorted excitons to absorption mainly coincides with that of impurity centers with the concentration Ndef < Z < 2 and the same oscillator strength Žsupposed their absorption band has the width B .. For a pore with a small radius R, meeting the inequality < k < R < 1, the exciton scattering can be neglected and the exciton wavefunction beyond the pore is simply Ny1 r2 expŽ ikn.. In this case the effective distortion size Ž11. coincides with the true size of the vacancy cluster: < Z k < s Z0 for < k < R < 1 ,
Ž 13 .
where Z0 is the number of sites within the pore. For < k < R ); 1, scattering of excitons should be taken into account. This is easy to make for a large pore with R 4 a in the continual approximation. In the Schrodinger equation Ž7., the potential W Ž r . ¨ should be put equal to 0 for r ) R and to ` for r F R. The normalized solution of this equation in the region r G R, turning to a plane wave on infinity, is of the form w8x Ck Ž r . s
(N v
0
k
Ý
3.2. Higher bands of excitons Consider now excitons of higher bands with the site state radius r large compared with the interatomic distance a. Since the site excitonic state is distorted within the distance r from the pore surface, the simple expression Ž11. for Z is inapplicable. To retain the illustrative estimation Ž12., the summation in Ž11. should be extended over the sphere of the radius R q r within which the exciton potential W Ž n. differs from that in ideal lattice. Within this sphere, the dipole matrix element d 0 a , high sensitive to a variation of the excited-state wavefunction with several radial nodes w5x, significantly varies Žwith changing sign. from site to site. Taking this into account, the sum Ž9. can be roughly estimated, under assumption on random phases of the summands, as a quantity proportional to the square root from the number of sites inside the sphere of radius R q r . In such an approach, the estimation Ž12. holds if the distortion size Z is defined as Z f Ž 4pr3v0 .
`
i
differs from Z0 by a small multiplier 6Ž kR .y2 Žhowever, < Z < f 8p Ž Rra.Ž ka.y2 much exceeds unity..
1r2
Ž Rqr .
3r2
.
Ž 16 .
Ž 2 l q 1 . Fk l Ž r . Pl Ž cos u . ,
ls0
Ž 14 . where cos u s krrkr, Pl is Legendre polynomial, v0 is volume per lattice site; Fk l Ž r . denotes an orthogonal set of functions from which only the lowest term, Fk 0 Ž r . s Ž1rr .sin k Ž r y R ., is used below. Within the continual approximation, one has to change in the expression Ž11. from summation to integration over the region 0 F u F p , 0 F r F R and to use as an integrand the analytic extension of Eq. Ž14. obtained for r G R. Integration over u annihilates all summands in Ž14. with l / 0, and Eq. Ž11. takes the form < Zk < s
6Z0
Ž kR .
2
ž
1y
sin kR kR
/
,
R4a
Ž 15 .
Ž a is interatomic distance.. For kR < 1, < Z < turns to Z0 in accordance with Ž13.. In the opposite case, < Z <
4. Light absorption with generation of excitons localized near lattice defects For a deep enough potential well, W, the Schrodi¨ nger equation Ž7. has solutions with discrete energy levels lying below the band bottom. Such a discrete level a makes a contribution to absorption which can be estimated, proceeding from Ž8. and Ž9., by Eq. Ž12. with Za s Ca Ž r . d rr v0 .
H
Ž 17 .
For the lowest discrete state with a large localization radius r Žnot to be confused with the site state radius r ., Eq. Ž17. can be rewritten as Za f Ž 4pr3 . r 3rv0 .
Ž 18 .
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
The discrete spectrum for sure exists in two cases. Ži. Excitons localized near a pore in a rare-gas crystal. In rare-gas crystals ŽNe, Ar, Kr, Xe., there exist surface exciton bands the bottom of which is shifted by the value of 0.1 to 0.3 eV relative to the bottom of the corresponding bulk band. The wave vector of a surface exciton has an imaginary normal component k z and two real tangential components k x , k y . A similar excitonic state, localized near a pore, has nearly the same energy shift relative to the bulk band bottom, a radial multiplier with nearly the same penetration depth L s Ž2 < k z <.y1 and an angular multiplier – spherical function instead of the tangential multiplier, expŽ ik x x q ik y y ., of the true surface-exciton state. According to Ž17., only the lowest angular state makes a nonzero contribution to absorption characterized by the distortion size 3
Z f Ž 4pr3v0 . Ž R q L . y R 3 ,
Ž 19 .
with the penetration depth L G a. Žii. Excitons localized near a charged lattice defect. It is known w6;7x that a lattice defect in rare-gas crystals Žin particular, a small pore. can capture a light conduction-band electron which forms a broad cloud with radius R ch ) a. Higher-band excitons with a large polarizability can be localized near such charged defects. The radius of such a localized excitonic state can be estimated under condition Egy En < En y E loc ,
Ž 20 .
where En is the position of the n th band Žthe lowest band being numbered by n s 1., Eg is the dielectric gap width Ž En Eg for n `., and Eloc is the level of the localized exciton. Inequality Ž20. means that the electron and hole, forming an exciton, are weakly bound as compared to the binding of the hole with the negative charge. In the lowest approximation with respect to the electron–hole binding, the hole is localized on a hydrogen-like level near the negative charge. For estimations, let the charge be uniformly distributed within the sphere of the radius R ch . Then the
™
™
249
energy and radius of the localized state can be approximated with accuracy 10% Žin atomic units.: E loc f y
0.7
´ R ch
,
rs
0.8
´ < Eloc <
f 1.1 R ch for 6 -
m R ch ´
- 60.
Ž 21 .
Here m is the effective mass of holes Žassumed to be isotropic., e is the crystal permittivity. With allowance for Ž21., the distortion size Ž18. can be written Žin atomic units. as Zf
2.2
´ 3 < Eloc < 3 v0
f
6 R 3ch v0
.
Ž 22 .
As for rare-gas crystals, the estimation Ž21–22. is rather rough in view of a highly anisotropic character of the effective mass of holes w5x. For example, in the case of argon Ž e s 1.66, m ; 2 ., a charge with ˚ Ž1.6 times the interatomic distance. creradius 6 A ates an excitonic state with localization energy of ˚ and effective distortion size about 1 eV, radius 7 A Z f 40.
5. Comparison with experiment First it is useful to make a rough general estimation for the contribution of lattice defects to excitonic absorption in rare-gas crystals. According to Eqs. Ž16., Ž19., Ž22., the effective distortion size Z exceeds 10 even for a small vacancy cluster with radius a Žconsisting of 6 vacancies.. The contribution of such defects to absorption, proportional to Z 2 , exceeds at least by two orders of magnitude the absorption of impurity centers for the same concentration Ndef , atomic oscillator strength f and the absorption band half-width D v . For f f 1 and D v s 0.5 eV, such vacancy clusters in the concentration Ndef s 10 14 cmy3 give rise to a noticeable absorption coefficient exceeding 1 cmy1 . Structural defects of such a low concentration are present even in a perfect crystal. To illustrate the above inferences, we will analyze the luminescence excitation spectrum of an argon crystal of a good structure quality w7x presented in Fig. 1. This excitation spectrum, measured in the
250
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
Fig. 1. Two-site exciton luminescence excitation spectrum of solid argon w7x which mainly reproduces the absorption spectrum in the region of excitonic absorption. Bold line relates to a perfect crystal with a low concentration of vacancy clusters Ž Ndef ; 3= 10 14 cmy3 .. Thin line relates to the same sample after a strong irradiation sufficient to enhance Ndef by an order of magnitude. The dashed vertical lines indicate the bottom position for the lowest exciton band Ž E1 ., for the next higher band Ž E2 . and the dielectric gap Eg . Short lines labeled ‘‘S’’ show the bottom positions for surface excitons. The table attached to the figure divides the frequency axis into intervals and indicates for every of them the squared absorption size and the corresponding optical density of the sample before irradiation. The downfalls of the excitation spectrum in the region of low optical density Žbold line. disappear after irradiation.
luminescence band of two-site self-trapped excitons, mainly reproduces the spectrum of weak absorption since almost all excitations, generated by absorbed light, turn to two-site self-trapped excitons. In the figure, the excitation spectrum is shown by bold line for a perfect crystal and by thin line for the same crystal after a strong irradiation sufficient to produce a noticeable number of lattice defects Žvacancy clusters.. Abscissa axis is divided into spectral regions F1, F2, where absorbed light generates free excitons, and L1, L2, where excitons are generated in a localized state. The region L1 lies on the left-hand side from the lowest band bottom labeled as E1 Žspin-orbital splitting, insignificant for argon, is neglected.. Absorption in this region corresponds to generation of lowest-band excitons, localized near the surface of pores, and is characterized by distortion size Ž19.. In the region F1, covering the
energy extension Žabout 0.6 eV. of the lowest band, absorbed light produces free excitons with distortion size Ž15.. In the region F2, free excitons with distortion size Ž16. are created in the second band. In the region L2, where excitons of the second band are generated below its bottom E2 in localized states, the most essential contribution to absorption spectrum is made by charged defects with distortion size Ž22.. The excitation spectrum, shown in the figure, can be qualitatively explained under assumption that the crystal in the initial state Žbefore irradiation. contained rather small vacancy clusters with R s a Ž Z0 f 6. in a very low concentration Ndef s 3 = 10 14 cmy3 . The squared distortion size, estimated according to Ž19., Ž15., Ž16. and Ž22., is indicated in the table attached to the figure. In the second line of the table, optical density k d is estimated with the use of Eq. Ž12. for a crystal of the thickness d s 0.1 cm Žatomic oscillator strength is near to 0.8 and 0.2 for the lowest and second band, respectively.. The optical density, estimated in such a way in the spectral regions L1, F1, L2, F2, qualitatively explains the character of the excitation spectrum presented in the figure. The table attached to the figure is related to the perfect crystal before irradiation Žits excitation spectrum is shown by bold line.. In the regions L1 and L2, where the initial optical density exceeds unity, the excitation efficiency actually achieved saturation and, as seen from the figure, did not grow after the number of lattice defects was enhanced by irradiation Žnew photo-produced defects even somewhat diminished the excitation efficiency for the two-site exciton luminescence which can be assigned to the development of a competing relaxation channel w7x.. The picture is of opposite character in the regions F1 and F2 where the initial optical density is small Žespecially in the region F1., so that excitation efficiency is low and far from saturation. In these regions, excitation efficiency became several times greater after irradiation; this is indicative of the corresponding photoinduced increase Žby an order of magnitude. of the number of vacancy clusters. It should be emphasized that such a low concentration of lattice defects Ž; 10 14 or 10 15 cmy1 . is for sure insufficient to influence upon absorption or excitation spectrum through scattering of excitons
A.M. Ratnerr Physics Letters A 269 (2000) 245–251
Žthe corresponding free path length is of the macroscopic scale of 1 cm.. In conclusion it can be noted that a high sensitivity of absorption spectrum to lattice defects of insulator crystals can be used for the diagnostics of photoproduced defects Žvacancy aggregates. and for the study of their dose dependence, which is important, in particular, for enhancement of radiation hardness of scintillation crystals.
Acknowledgements The author is thankful to A.N. Ogurtsov and E.V. Savchenko for fruitful discussions. This work was partly supported by the grant INTAS 96-626.
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References w1x V.M. Agranovich, Theory of Excitons, Nauka, Moscow, 1968 Žin Russian.. w2x V.L. Bonch-Bruyevich, I.P. Zvyagin, R. Kaiper, A.G. Mironov, R. Enderlain, B. Esser, Electronic Theory of Unordered Semiconductors, Nauka, Moscow, 1981 Žin Russian.. w3x M. Wubs, J. Knoester, Chem. Phys. Lett. 284 Ž1998. 63. w4x I.M. Sokolov, A. Blumen, G. Oshanin, P. Argyrakis, in: M. Schreiber ŽEd.., Proc. 2nd Int. Conf. on Excitonic Processes in Condensed Matter, Dresden University Press, 1996. w5x A.M. Ratner, Phys. Rep. 269 Ž1996. 197. w6x A.N. Ogurtsov, E.V. Savchenko, J. Becker, M. Runne, G. Zimmerer, J. Lumin. 76 Ž1998. 478. w7x A.N. Ogurtsov, A.M. Ratner, E.V. Savchenko, V. Kisand, S. Vielhauer, J. Phys.: Condens. Matter 12 Ž2000. 2769. w8x L.D. Landau, E.M. Lifshits, Quantum Mechanics, Nauka, Moscow, 1963 Žin Russian..