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Wannier excitons in liquid and solid krypton
Volume 122. number 6
WANNIER
CHEMICAL PHYSICS LETTERS
EXCITONS
P. LAPORTE,
IN LIQUID
AND
SOLID
18 Oclohrr
1985
The minimum conditions for the appearance of Wannier excitons in a liquid were formulated almost twenty years ago [I] but only in one pure liquid were such excitons observed and studied, namely liquid xenon [2-S] _Host-perturbed impurity levels in condensed rare gases were frequently regarded as trapped Wannier excitons [6], but the electronic excitations of a very large number of matrix-isolated impurities were understood without referring to Wannier excitons [7,8]. The search for Wannier excitons in pure liquids is also spurred by the fact that the occurrence of such excitons supports even the simplest and least sophisticated model of a Wannier exciton, namely that of an electron-hole pair, coupled by means of their Coulomb attraction in the polarizable medium. Liquid krypton is an obvious choice, since for it the “cyclotron product” [l] arc is roughly 100, w being the classical angular frequency of the electron-hole pair and ~c the electron mobility relaxation time *. Moreover, from the position of the photoconductivity
threshold at 11.55 eV [IO], the ~1= 2 Wannier exciton could be expected to appear well below the cut-off of a LiF window_ For solidified rare gases transmission experiments are
0330 Physics
0 Elsevier Publishing
Science Publishers Division)
restricted
to film thicknesses
in the order
of 100 A
due to the very strong exciton absorption. In the present paper rare gas liquids and solids are investigated in closed sample cells of macroscopic (of the order of cm) dimensions_ Therefore only reflection spectroscopy is applicable_ The near-normal incidence (NNI) reflection spectra
were
obtained-using
a stainless
steel
sample cell with a LiF front window. The two surfaces of the window were polished with an angle of So between them in order to avoid recording reflection from the front surface. The NNI spectrum for the liquid was measured in Saint Etienne, using a high-performance VUV photometric arrangement [I I ]: the solid spectrum was measured at HASYLAB, using synchrotron radiation at the HONORMI system [ I?]_ In order to increase the sensitivity to weak structures the NNI spectra were complemented by measuring oblique incidence (Of) reflection spectra at 70°. All 01 spectra
from mobility data by Miller et al. [9]_
0 009-2614/85/S (North-Holland
KRYPTON
J.L. SUBTIL
Received 2 Augusl 1985: in final form
* Estimated
27 December 1985
B.V.
525
I
Volume 122. number 6
CHEMICAL
PHYSICS LE-ITERS
were taken at HASYLAB, utilizing both the high degree of collimation and polarization of synchrotron radiation. A hemispherical lens made of LiF served as a front window. Details of the 01 cell which served mainly to determine optical constants will be published shortly [ 131. Preparing the solid samples in a closed cell instead of the more usual freezing the gas on a cold substrate eliminates much of the light scattering due to surface roughness and enables measurements at temperatures near the triple point, i.e. in a region where it is easier to grow good crystals: in fact. the samples were homogeneous and transparent. Fig. I represents the NNI reflection spectrum for a liquid sample near the triple point. Note the promito the nent peaks at ~10 and ~10.8 eV corresponding II = 1 f’(3/2) and the TV’= I r(l/2) “intermediate” ex-
I
I
Ial
T
Energy F&. 1. Liquid krypton reflectivities. experimental
data (T=
Lore&z fit of (a); (c)
-I
(eV1
(a) Near-normal incider.ce,
120 K; p = 1.74 X 10” cmm3); (b) 70° oblique incidence parallel polariza-
tion, experimental data;(d) 70” oblique incidence, par&e1 calculated rrom parameters used in (b).
polarization. 526
27 December 198.5
citons respectively. The further structure around 11.2 eV is not close to any atomic line; it will be shown below to be dub to the !I = 2 I’(3/2) Wumier exciton. The 01 spectrum of the liquid for polarization parallel to the plane of incidence is shown in fig. lc. The shapes of the bands are different now and in particular,, the band around 11.2 eV becomes much more prominent in the 01 spectrum_ The only way to analyze the reflection spectra quantitatively is a dispersion analysis assuming that a set of Lorentzian bands can describe the frequency dependence of the dielectric function el + ie2 of KT. In order to obtain the reflectivity at the window-Kr interface the optical properties of the window material [14] have to be taken into account as well. For liquid Kr the relationship between NNI and the 01 spectra is illustrated in figs. lb and Id, respectively_ Fig. lb is a NNI spectrum fitted to that of fig. la as described above. The peak photon energy E, the half width y (both in eV) and the oscilhtor strength fof each band appear in table I, including a broad extra band beyond the high-energy limit of the measurements; this band accounts for higher-en’ergy excitations and was indeed necessary to ensure a proper fit. In the analysis, a constant term E, = 1.67 was added to the sum of Lorentzians describing Ed. Fig. 1 d is the 70° 01 spectrum calculated from the same set of parameters; its similarity to the experimental 70” 01 spectrum indicates the consistency of the NNl and 01 measurements. A better correspondence could not be expected because of the limitations of the Lorentz model, the low intensity of the NNI spectrum (fig. la) in the 11.2 eV region and the incomplete polarization of the synchrotron radiation. A similar set of spectra is given in figs. 2a-2d for solid KT near the triple point. Again, the parameters of the Lorentzian fits to the NNI spectrum appear in table I_ In this case the constant term E, = 1.81 was used in the fit. The similarity of the experimental 01 reflection spectrum (2~) and that calculated from the dispersion analysis (2d) of the NNI spectrum is evident. Though the heights of the curves differ, the positions of the peaks at around 11.2 eV and beyond COTrespond within 0.03 eV with those in 2c. Because of this correspondence, one may confidently take the peak positions of the l2 spectra obtained by the fitting procedure as the correct positions of the absorption peaks. The absorption spectrum of solid KT is already
Volume
122, number 6
CHEMICAL
PHYSICS
LETl-ERS
27 December
1985
Table 1 fit of liquid or solid krypton reflectivity data The two low-energy peaks werefitted using two close oscillators: (A, El) and (C. D). For comparison results obtained in transmission espcknents on thin films [IS] are included. As expected for the solid the excitation energies are slightly lower at &&her temperatures -
Parameters for the Lorentzian
Oscillator
Liquid (T= 120 K, p = 1.74 X i022 cme3)
Solid (T= 113 K, p = 2.03 X 1O22 cmm3)
energy E Cev)
width y
osciuator
energy E
width 1
(eW
StiCllgthf
(eV)
(CV)
oscillator strength f
10.043 10.085
0.063 0.150
0.4 1 0.21
(7= 20 K) energy E (ev)
A
9.943
B
10.020
0.160 0.160
0.47 0.096
C
10.630
0.060
0.4 1
10.740
0.040
0.33
10.86
-
10.810
0.100
0.044
10.94 11.23 11.44
D
ll=2 n=3
11.205 _
0.200
0.045 -
11.190 11.433
0.100 0.06
0.066 0.005
high-energy contribution
12.2
1.4
2.2
12.5
1
2.7
known C
from
experiments
D
10
Fig. 2. Solid krypton reflectivities fg.
1 (T=
113 K;p
on thin films condensed
on-
is quite
differenr,
namely
the peak
can be regarded as a superposition of an atomic and an excitonic contribution as discussed in detail for Xc (see refs. [4,5]). We note that the peaks at 11.190 and 11.433 eV in the solid sample correspond to the II = 2 and II = 3 excitons, respectively [15] _This identities the 11.205 eV in the liquid as the II = 2 exciton. The results can be further analysed using the well-known effective mass formula for excitons:
OC
9
10.17 10.29
to a cold substrate (see, e.g., ref. [15]). However the data presented here are the first for a “hot”, highquality sample grown under optimum conditions in a closed cell. It is interesting to note that for both II = 1 excitons asymmetric lineshapes are observed_ This has been empirically taken into account by fitting each peak by two oscillators. In liquid Kr the first peak was also deconvoluted into two oscillators A. B. But in this case the reason
‘:’
1151
Solid film
E,, = EG - G/112, c = 13.6 12
(a). (b1, (cl. and (d) as in = 2.03 X 1O22 cm-31.
JJl*/E’,
(9
EC being the band gap and G the binding energy in eV, m* the reduced effective mass in terms of the free electron mass and E the dielectric constant, determined as the square of the refractive index 1~2,at the Bohr orbital frequency, and u a positive integer. In
Volume rare
122, number
gases
the
CHEMICAL
6
experimentally
observed
PHYSICS
n = 1 level
the rest of the series since the Bohr radius of this “intermediate” exciten is too small for the Wannier model to be reliably applicable. Using E2 = 1 l-1 9 eV and Ej = 11.433 eV for the solid we obtain EC = 11.63 f 0.03 eV and G = 1.75 + 0.30 eV. With E = 1.806 1161 one obtains )?I* = 0.42 f 0.07. These results are in very good accord with those obtained by absorption spectroscopy on thin krypton films deposited under stringent vacuum conditions [ 151. For the liquid one may take = 11.55 f 0.05 eV, where Epc is the photoEC ‘Epc conduction threshold [lo] _Combining this with E, = 11.205 eV yields G = 1.4 + 0.3 eV. Wth E = 1.687 [16] one obtains ttz* = 0.39 f 0.06. ‘Ne note that also in xenon [3] the effective mass in the liquid is smaller than in the solid. Obtaining estimates for the reduced effective mass in liquid krypton is of immediate relevance for the relationship between the energy V,, of a conduction electron and its zero Iield mobility [17] following a recent theory [IS]. The observation of Wannier excitons in rare gas liquids is in full accord with basic considerations on electron-hole interaction in these very simple disordered substances [I], while pointing to the need for a further theoretical effort. Considering binding energies, relaxation times and the LiF cut-off, liquid Kr may well be the only other liquid besides xenon in which Wannier excitons are experimentally observable. usually
deviates
From
[l] S.A. Rice and J. Jortner, J. Chem. Phys. 44 (1966)
528
4470.
1985
[Z] D. Beaglehole. Phys. Rev. Letters 15 (1965) 551. [3] U. Asaf and 1-T. Steinberger. Phys. Rev. El8 (1973) 16X [4] P. Laporte and 1-T. Steinlxrger, Phys. Rev. Al5 (1977) 2533. [S] P. Laporte, J.L. Subtil. U. Amf, LT. Steinberger and S. Wind. Phys. Rev. Letters 45 (1980) 2138. [6] J. Jortner. in: Vacuum‘ultraviolet radiation physics. eds. E.E. Koch. R. Haensel and C. Kunz (Pcrgamon/Vieweg, New YorklBraunnveig, 1974) p_ 283, and references therein. [7] J.-Y. Roncin and K. hioojani, Phys. Stat. Sol. 23 (1967) Kl; V-E. Bondybcy and L.E. Brus. AdvanEs in chemical physics, Vol. 41, cds. 1. Prigogine and S.A. Rice (Wiley, New York, 1980) p_ 268. [8] V. Saile. R. Rcininger and P. Laporte. to be published_ [ 91 L.S. Miller. S. Howe and WE Spear. Phys. Rev. 166 (1968) 871. [lo] R. Reininger, U. Amf. I.T. Steinberger, P. Laporte and V. Saile, Phys. Rev. D26 (1982) 6294. [ll] P. Laporte, JL. SubtiI,M. Bon and H. Daman~. Appl. Opt. 20 (1981) 2133. [ 121 V. Salle. P. GUrtlcr, E.E. Koch. A. Kozevnikov. hl. Skibowski and W. Steinmann. Appl. Opt. 15 (1976) 25.59. [ 131 J.L. Subtil. P. Laportc.
[14?
1151
1161 r171 I181
References
.27 Decembm
LE-I-I-ERS
R. Reintiger.
V. Sailc and
1-T. Steinberger. ;o be published. P. Laporte, J,L. Subtil, M. Courbon and L. Vincent, J. Opt. Sot. Am. 73 (1983) 1062. V. SaJle, W. Steinmann and E.E. Koch, in: Extended Abstracts. 5th International Conference on VUV Radiation Physics, VUV 5. MontpclLier. Prance. 5-9 September 1977. Vol. 1, eds. M.C. Castes. hi. Pouey and N. Poucy (CNRS, Paris. 1977) p. 199; V. Saile. Thesis, University of hlunich (1976). A.C. Sinnock. J. Phys. Cl3 (1980) 2375. R. Reininger, U. Auf. LT. Steinterger and S. BasaJc, Phyr Rev. B28 (1983) 4426. S. Basal and M.H. Cohen, Phys. Rev. B20 (1979) 3404.
WANNIER
CHEMICAL PHYSICS LETTERS
EXCITONS
P. LAPORTE,
IN LIQUID
AND
SOLID
18 Oclohrr
1985
The minimum conditions for the appearance of Wannier excitons in a liquid were formulated almost twenty years ago [I] but only in one pure liquid were such excitons observed and studied, namely liquid xenon [2-S] _Host-perturbed impurity levels in condensed rare gases were frequently regarded as trapped Wannier excitons [6], but the electronic excitations of a very large number of matrix-isolated impurities were understood without referring to Wannier excitons [7,8]. The search for Wannier excitons in pure liquids is also spurred by the fact that the occurrence of such excitons supports even the simplest and least sophisticated model of a Wannier exciton, namely that of an electron-hole pair, coupled by means of their Coulomb attraction in the polarizable medium. Liquid krypton is an obvious choice, since for it the “cyclotron product” [l] arc is roughly 100, w being the classical angular frequency of the electron-hole pair and ~c the electron mobility relaxation time *. Moreover, from the position of the photoconductivity
threshold at 11.55 eV [IO], the ~1= 2 Wannier exciton could be expected to appear well below the cut-off of a LiF window_ For solidified rare gases transmission experiments are
0330 Physics
0 Elsevier Publishing
Science Publishers Division)
restricted
to film thicknesses
in the order
of 100 A
due to the very strong exciton absorption. In the present paper rare gas liquids and solids are investigated in closed sample cells of macroscopic (of the order of cm) dimensions_ Therefore only reflection spectroscopy is applicable_ The near-normal incidence (NNI) reflection spectra
were
obtained-using
a stainless
steel
sample cell with a LiF front window. The two surfaces of the window were polished with an angle of So between them in order to avoid recording reflection from the front surface. The NNI spectrum for the liquid was measured in Saint Etienne, using a high-performance VUV photometric arrangement [I I ]: the solid spectrum was measured at HASYLAB, using synchrotron radiation at the HONORMI system [ I?]_ In order to increase the sensitivity to weak structures the NNI spectra were complemented by measuring oblique incidence (Of) reflection spectra at 70°. All 01 spectra
from mobility data by Miller et al. [9]_
0 009-2614/85/S (North-Holland
KRYPTON
J.L. SUBTIL
Received 2 Augusl 1985: in final form
* Estimated
27 December 1985
B.V.
525
I
Volume 122. number 6
CHEMICAL
PHYSICS LE-ITERS
were taken at HASYLAB, utilizing both the high degree of collimation and polarization of synchrotron radiation. A hemispherical lens made of LiF served as a front window. Details of the 01 cell which served mainly to determine optical constants will be published shortly [ 131. Preparing the solid samples in a closed cell instead of the more usual freezing the gas on a cold substrate eliminates much of the light scattering due to surface roughness and enables measurements at temperatures near the triple point, i.e. in a region where it is easier to grow good crystals: in fact. the samples were homogeneous and transparent. Fig. I represents the NNI reflection spectrum for a liquid sample near the triple point. Note the promito the nent peaks at ~10 and ~10.8 eV corresponding II = 1 f’(3/2) and the TV’= I r(l/2) “intermediate” ex-
I
I
Ial
T
Energy F&. 1. Liquid krypton reflectivities. experimental
data (T=
Lore&z fit of (a); (c)
-I
(eV1
(a) Near-normal incider.ce,
120 K; p = 1.74 X 10” cmm3); (b) 70° oblique incidence parallel polariza-
tion, experimental data;(d) 70” oblique incidence, par&e1 calculated rrom parameters used in (b).
polarization. 526
27 December 198.5
citons respectively. The further structure around 11.2 eV is not close to any atomic line; it will be shown below to be dub to the !I = 2 I’(3/2) Wumier exciton. The 01 spectrum of the liquid for polarization parallel to the plane of incidence is shown in fig. lc. The shapes of the bands are different now and in particular,, the band around 11.2 eV becomes much more prominent in the 01 spectrum_ The only way to analyze the reflection spectra quantitatively is a dispersion analysis assuming that a set of Lorentzian bands can describe the frequency dependence of the dielectric function el + ie2 of KT. In order to obtain the reflectivity at the window-Kr interface the optical properties of the window material [14] have to be taken into account as well. For liquid Kr the relationship between NNI and the 01 spectra is illustrated in figs. lb and Id, respectively_ Fig. lb is a NNI spectrum fitted to that of fig. la as described above. The peak photon energy E, the half width y (both in eV) and the oscilhtor strength fof each band appear in table I, including a broad extra band beyond the high-energy limit of the measurements; this band accounts for higher-en’ergy excitations and was indeed necessary to ensure a proper fit. In the analysis, a constant term E, = 1.67 was added to the sum of Lorentzians describing Ed. Fig. 1 d is the 70° 01 spectrum calculated from the same set of parameters; its similarity to the experimental 70” 01 spectrum indicates the consistency of the NNl and 01 measurements. A better correspondence could not be expected because of the limitations of the Lorentz model, the low intensity of the NNI spectrum (fig. la) in the 11.2 eV region and the incomplete polarization of the synchrotron radiation. A similar set of spectra is given in figs. 2a-2d for solid KT near the triple point. Again, the parameters of the Lorentzian fits to the NNI spectrum appear in table I_ In this case the constant term E, = 1.81 was used in the fit. The similarity of the experimental 01 reflection spectrum (2~) and that calculated from the dispersion analysis (2d) of the NNI spectrum is evident. Though the heights of the curves differ, the positions of the peaks at around 11.2 eV and beyond COTrespond within 0.03 eV with those in 2c. Because of this correspondence, one may confidently take the peak positions of the l2 spectra obtained by the fitting procedure as the correct positions of the absorption peaks. The absorption spectrum of solid KT is already
Volume
122, number 6
CHEMICAL
PHYSICS
LETl-ERS
27 December
1985
Table 1 fit of liquid or solid krypton reflectivity data The two low-energy peaks werefitted using two close oscillators: (A, El) and (C. D). For comparison results obtained in transmission espcknents on thin films [IS] are included. As expected for the solid the excitation energies are slightly lower at &&her temperatures -
Parameters for the Lorentzian
Oscillator
Liquid (T= 120 K, p = 1.74 X i022 cme3)
Solid (T= 113 K, p = 2.03 X 1O22 cmm3)
energy E Cev)
width y
osciuator
energy E
width 1
(eW
StiCllgthf
(eV)
(CV)
oscillator strength f
10.043 10.085
0.063 0.150
0.4 1 0.21
(7= 20 K) energy E (ev)
A
9.943
B
10.020
0.160 0.160
0.47 0.096
C
10.630
0.060
0.4 1
10.740
0.040
0.33
10.86
-
10.810
0.100
0.044
10.94 11.23 11.44
D
ll=2 n=3
11.205 _
0.200
0.045 -
11.190 11.433
0.100 0.06
0.066 0.005
high-energy contribution
12.2
1.4
2.2
12.5
1
2.7
known C
from
experiments
D
10
Fig. 2. Solid krypton reflectivities fg.
1 (T=
113 K;p
on thin films condensed
on-
is quite
differenr,
namely
the peak
can be regarded as a superposition of an atomic and an excitonic contribution as discussed in detail for Xc (see refs. [4,5]). We note that the peaks at 11.190 and 11.433 eV in the solid sample correspond to the II = 2 and II = 3 excitons, respectively [15] _This identities the 11.205 eV in the liquid as the II = 2 exciton. The results can be further analysed using the well-known effective mass formula for excitons:
OC
9
10.17 10.29
to a cold substrate (see, e.g., ref. [15]). However the data presented here are the first for a “hot”, highquality sample grown under optimum conditions in a closed cell. It is interesting to note that for both II = 1 excitons asymmetric lineshapes are observed_ This has been empirically taken into account by fitting each peak by two oscillators. In liquid Kr the first peak was also deconvoluted into two oscillators A. B. But in this case the reason
‘:’
1151
Solid film
E,, = EG - G/112, c = 13.6 12
(a). (b1, (cl. and (d) as in = 2.03 X 1O22 cm-31.
JJl*/E’,
(9
EC being the band gap and G the binding energy in eV, m* the reduced effective mass in terms of the free electron mass and E the dielectric constant, determined as the square of the refractive index 1~2,at the Bohr orbital frequency, and u a positive integer. In
Volume rare
122, number
gases
the
CHEMICAL
6
experimentally
observed
PHYSICS
n = 1 level
the rest of the series since the Bohr radius of this “intermediate” exciten is too small for the Wannier model to be reliably applicable. Using E2 = 1 l-1 9 eV and Ej = 11.433 eV for the solid we obtain EC = 11.63 f 0.03 eV and G = 1.75 + 0.30 eV. With E = 1.806 1161 one obtains )?I* = 0.42 f 0.07. These results are in very good accord with those obtained by absorption spectroscopy on thin krypton films deposited under stringent vacuum conditions [ 151. For the liquid one may take = 11.55 f 0.05 eV, where Epc is the photoEC ‘Epc conduction threshold [lo] _Combining this with E, = 11.205 eV yields G = 1.4 + 0.3 eV. Wth E = 1.687 [16] one obtains ttz* = 0.39 f 0.06. ‘Ne note that also in xenon [3] the effective mass in the liquid is smaller than in the solid. Obtaining estimates for the reduced effective mass in liquid krypton is of immediate relevance for the relationship between the energy V,, of a conduction electron and its zero Iield mobility [17] following a recent theory [IS]. The observation of Wannier excitons in rare gas liquids is in full accord with basic considerations on electron-hole interaction in these very simple disordered substances [I], while pointing to the need for a further theoretical effort. Considering binding energies, relaxation times and the LiF cut-off, liquid Kr may well be the only other liquid besides xenon in which Wannier excitons are experimentally observable. usually
deviates
From
[l] S.A. Rice and J. Jortner, J. Chem. Phys. 44 (1966)
528
4470.
1985
[Z] D. Beaglehole. Phys. Rev. Letters 15 (1965) 551. [3] U. Asaf and 1-T. Steinberger. Phys. Rev. El8 (1973) 16X [4] P. Laporte and 1-T. Steinlxrger, Phys. Rev. Al5 (1977) 2533. [S] P. Laporte, J.L. Subtil. U. Amf, LT. Steinberger and S. Wind. Phys. Rev. Letters 45 (1980) 2138. [6] J. Jortner. in: Vacuum‘ultraviolet radiation physics. eds. E.E. Koch. R. Haensel and C. Kunz (Pcrgamon/Vieweg, New YorklBraunnveig, 1974) p_ 283, and references therein. [7] J.-Y. Roncin and K. hioojani, Phys. Stat. Sol. 23 (1967) Kl; V-E. Bondybcy and L.E. Brus. AdvanEs in chemical physics, Vol. 41, cds. 1. Prigogine and S.A. Rice (Wiley, New York, 1980) p_ 268. [8] V. Saile. R. Rcininger and P. Laporte. to be published_ [ 91 L.S. Miller. S. Howe and WE Spear. Phys. Rev. 166 (1968) 871. [lo] R. Reininger, U. Amf. I.T. Steinberger, P. Laporte and V. Saile, Phys. Rev. D26 (1982) 6294. [ll] P. Laporte, JL. SubtiI,M. Bon and H. Daman~. Appl. Opt. 20 (1981) 2133. [ 121 V. Salle. P. GUrtlcr, E.E. Koch. A. Kozevnikov. hl. Skibowski and W. Steinmann. Appl. Opt. 15 (1976) 25.59. [ 131 J.L. Subtil. P. Laportc.
[14?
1151
1161 r171 I181
References
.27 Decembm
LE-I-I-ERS
R. Reintiger.
V. Sailc and
1-T. Steinberger. ;o be published. P. Laporte, J,L. Subtil, M. Courbon and L. Vincent, J. Opt. Sot. Am. 73 (1983) 1062. V. SaJle, W. Steinmann and E.E. Koch, in: Extended Abstracts. 5th International Conference on VUV Radiation Physics, VUV 5. MontpclLier. Prance. 5-9 September 1977. Vol. 1, eds. M.C. Castes. hi. Pouey and N. Poucy (CNRS, Paris. 1977) p. 199; V. Saile. Thesis, University of hlunich (1976). A.C. Sinnock. J. Phys. Cl3 (1980) 2375. R. Reininger, U. Auf. LT. Steinterger and S. BasaJc, Phyr Rev. B28 (1983) 4426. S. Basal and M.H. Cohen, Phys. Rev. B20 (1979) 3404.