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Four-photon transitions in ZnS
Solid State Communications,Vol. 16, Pp. 1109—1111, 1975.
Pergamon Press.
Printed in Great Britain
FOUR-PHOTON TRANSITIONS IN ZnS* I.M. Catalano, A. Cingolani and A. Minafra Istituto di Fisica dell’Universitâ, Bar Italy (Received 7 January 1975 by F. Bassani)
The non-linear photoconductivity in ZnS has been studied by means of threephoton and four-photon excitation with a ruby and neodymium laser, respectively. By using the experimental results obtained at the two different wavelengths and the previous value of three photon non linear cross section ‘y~,a quantitative determination of ‘14 is obtained. The results are compared with three different theoretical calculations: semiclassical perturbation theory, quantum electrodynamic theory and Keldysh’s theory.
THE THEORETICAL calculation of the multiphoton transition rates in solids has been approached by three different methods. The first one,1 also from the historical standpoint, is the perturbative calculation which starts from a semiclassical Hamiltonian. These perturbative calculations agree very well with experimental data, particularly for two- and three-photon direct transitions24 and for two photon—phonon-assisted ones.5 On the other hand the Keldysh’s high frequencylimit, time-dependent theory6 is very useful in the case of multiphoton transitions in which a large numben of photons is involved, but usually gives transition rates which are several orders of magnitude smaller than experimental data in the case of a few photon processes (e.g. two-photon ones).
This method is the only one which has enough sensitivity for giving quantitative results for high order processes. In the case of four-photon absorption, an order of magnitude extimate, obtained by perturbative calculation, gives a free carrier density of l0~ electrons/cm3 at 100MW/cm2, which is readily detectable by means of charge measurements. It is interesting to note that Keldysh theory foresees a free carrier density 1 03 electrons/cm3 at the same intensity, which would be hardly measured even by this method. ‘~
‘-‘
All measurements were carried out on an undoped monocrystal of ZnS (energy gap at 77°K,Eg = 3.8 eV), cut in the shape of a rectangular prism, with dimensions of several millimeters.
More recently a full quantum treatment has been developed for n-photon processes,7 whose results the present authors have used for analyzing the experiment on three-photon absorption in CdS.4 In the present work the indirect quantitative method based on non-linear photoconductivity (NLP) has been applied to the measurement of the four-photon non-linear cross-section ~ in ZnS. This is obtained by comparing the photocurrent induced by three- and four-photon absorption using two different wavelengths. *
Work partially supported by CNR. 1109
The sample was fixed to the copper cold finger of a vacuum cryostat operating at 80°K.The threephoton photocurrent was excited by a Q-switched ruby laser (3hw = 5.34 eV) with 20 nsec pulse duration and 200 MW maximum peak power. The fourphoton photocurrent was excited by a Q-switched neodymium laser (4hw = 4.68 eV) with the same pulse duration and same hundreds MW peak power. The pulse energy was monitored by using a beam splitter and a silicon photodiode. The monitor system was calibrated with a thermocouple calorimeter.
1110
FOUR-PHOTON TRANSITIONS IN ZnS
r
/
.92
~
rUby I~,er
neodymium
~
Q(4)
IR(0~On~~e~
IR i Nd
where
74’Nd
3 73 ‘Rb are the three and four photon
Q(3) and Q(4)
total generated charge. One gets then, from Fig. I 73
I
I
I
I
id _____________________________________ id’ io’~ icf I,~pho~o~2’~ 1.
2 —
a
FIG.
Vol. 16, No.9
3 x
l0~
cm2 sec.
(2)
The three photon non linear cross section y~of ZnS has been determinated previously by non-linear luminescence measurements.9 and combining this result with the one of equation (2), For ruby laser light, ‘y~= 0.7 x 10_80 cm6 sec2 one gets the four photon non-linear cross section for ZnS at the frequency of neodymium y~= 2 x l0~1 cm8 sec3.
Q vsI 0 plot for ZnS: (a) ruby excitation; (b) neodymium excitation.
Theelsewhere8 set up for NLP measurements described and basically is a Q has vs jobeen measurement, where Q is the total charge generated by a single laser pulse and 1o the incident photon flux (corrected for reflection losses). For an n-photon process:
Theoretically, y~has been calculated either by semiclassical perturbation theory using a two band model, or by using quantum electrodynamics. Accord10 the four photon non linear ing to the first treatment, cross which has made the using Hartree— Fock section, approximation andbeen including usually neglected A2 term, is: =
2b0
3
~.J2 ir3h4 N(hw)8
line through the experimental points has a slope three, which is consistent with a three-photon absorption process; the other the slope fourphoton for neodymium on excitation is hand, an evidence of aoffour process. From these experimental results it is possible to determine the ratio between the non-linear cross section y~and ~3. In fact by equation (1) it is obtained:
m~2 P m’1m2 ~
2
C
Q=K 110 (1) where the “constant” k depends, among other things, on the transport parameters of the crystals and its geometrical characteristics. The analytical expression of K has been calculated in reference 8. Figure 1 shows the results of the non-linear photocurrent measurements made with ruby (a) and neodymium (b) excitation in ZnS at 80°K.In the case of ruby excitation the straight
e8 C~’~
312 [i
X
________
8 1flvm4hc~.)_Eg~ m~ hw (4hw E~)2~ (3) hw
—
(4hwEg)
+ 16 (mcvm\i 2 21 m~
—
)
where Eg is the fundamental energy gap, the dielectric constant and IP~,12 the square of the matrix element of the momentum operator; m~ 1,is the reduced mass of valence and conduction bands. The other symbols have the usual meaning. Assuming the effective mass 3 equation Cv of reference 2 6.2 x l0_38 erg gr, (3) gives 7~=10,2.1Px 10_Ill cm8 sec3. The quantum treatment7 uses a three band model with a single valence and two conduction band c’ and c and the resulting formula is: 2”2ir3e8h4 IP~,12 74 = Nc4(hw)’°(ji/m)312e2m712 X —
—~
X
32 ~(4hW
~
~g)7/~
—
64
X
Vol. 16, No.9
FOUR-PHOTON TRANSITIONS IN ZnS
_______________ ~56 —~. P2 + 8h~+ ~ (4hw Eg)512 hw 1— m Jl5 hw 0 ________
—
+
_~_
(-~-~2
675 \mJ
(4hw
—
2]
+
we can that experimental data are inConcluding, fair agreement withsay perturbative semiclassical treatment. The numerical discrepancies which is observed in the case of quantum calculation can be due to the use of the two band model. On the other
E,)31’2(h~))2x
m2
I1’~~d1I
1111
X 15 h~~8hWa+ 9~I I 2~2) where j.i = m~’+ mc,’, ~ is the energy difference between the minimum of conduction band and the intermediate band c’ and c and a = ~2 — (hw)2 Assuming the energy level c’ at 8.9 eV,’~equation (4) gives = 3.2 x 8 sec3. The Keldysh theory cm (see formula (41)10~H2 of reference 6) gives for the fourphoton non linear cross section the value: ~ = 2 x l0’17cm8 sec3.
hand valid model the Keldysh transitions theory is in definitively which a small rulednumber out as of photons isfor involved. The anomalous behaviour of a experiments in the presence of a magnetic field (twophoton magneto conductivity processes’2) for which the Keldysh’s theory gives reasonable results also in the case of a low order process, is not yet well understood.
REFERENCES 1. 2.
BRAUNSTEIN R. and OCKMAN N., Phys. Rev. 134A, 499 (1964); BASSANI F. and HASSAN A.R., Nuovo Cimento 7B; 313 (1972); YEE J.H.,J. Phys. Chem. Solids 33, 643 (1972). BASOV N.G., GRASYUK A.Z., AUBAREW I.G., KATULIN V.A. and KORKHIN O.N., Soy. Phys. JETP 23, 366 (1966).
3.
CATALANO I.M., CINGOLANI A. and MINAFRA A.,Phys. Rev. 9,707(1974).
4.
6.
ASJKINADZE B.M., RYVKIN S.M. and YAROSHETSKH I.D., Soy. Phys. Semicond. 2, 1285 (1969); CATALANO I.M., CINGOLANI A. and MINAFRA A., Opt. Commun., 11,254(1974). YEE J.H. and CHAN H.H.M., Opt. Commun. 10,56(1974); CATALANO l.M., CINGOLANI A. and MINAFRA A., Solid State Commun. (to be published). KELDYSH L.V., Soy. Phys. JETP 20, 1307 (1965).
7.
KOVARSKII V.A. and PERLIN E.Yu, Phys. Status Solidi (b) 45,47 (1971).
8.
CINGOLANI A., FERRERO F., MINAFRA A. and TRIGGIANTE D.,Nuovo Cimento 4B, 217 (1971).
9. 10.
CATALANO I.M., CINGOLANI A. and MINAFRA A., Opt. Commun. 7,270(1973); CATALANO I.M., CINGOLANI A. and MINAFRA A.,Phys. Rev. B8, 1488 (1973). YEEJ.H.,Phys.Rev. B3, 355 (1971).
11. 12.
COHEN M.L. and BERGSTRESSER P.K., Phys. Rev. 141, 789 (1966). WElLER M.W., BIERING R.W. and LAX B.,Phys. Rev. 184,709 (1969).
5.
Pergamon Press.
Printed in Great Britain
FOUR-PHOTON TRANSITIONS IN ZnS* I.M. Catalano, A. Cingolani and A. Minafra Istituto di Fisica dell’Universitâ, Bar Italy (Received 7 January 1975 by F. Bassani)
The non-linear photoconductivity in ZnS has been studied by means of threephoton and four-photon excitation with a ruby and neodymium laser, respectively. By using the experimental results obtained at the two different wavelengths and the previous value of three photon non linear cross section ‘y~,a quantitative determination of ‘14 is obtained. The results are compared with three different theoretical calculations: semiclassical perturbation theory, quantum electrodynamic theory and Keldysh’s theory.
THE THEORETICAL calculation of the multiphoton transition rates in solids has been approached by three different methods. The first one,1 also from the historical standpoint, is the perturbative calculation which starts from a semiclassical Hamiltonian. These perturbative calculations agree very well with experimental data, particularly for two- and three-photon direct transitions24 and for two photon—phonon-assisted ones.5 On the other hand the Keldysh’s high frequencylimit, time-dependent theory6 is very useful in the case of multiphoton transitions in which a large numben of photons is involved, but usually gives transition rates which are several orders of magnitude smaller than experimental data in the case of a few photon processes (e.g. two-photon ones).
This method is the only one which has enough sensitivity for giving quantitative results for high order processes. In the case of four-photon absorption, an order of magnitude extimate, obtained by perturbative calculation, gives a free carrier density of l0~ electrons/cm3 at 100MW/cm2, which is readily detectable by means of charge measurements. It is interesting to note that Keldysh theory foresees a free carrier density 1 03 electrons/cm3 at the same intensity, which would be hardly measured even by this method. ‘~
‘-‘
All measurements were carried out on an undoped monocrystal of ZnS (energy gap at 77°K,Eg = 3.8 eV), cut in the shape of a rectangular prism, with dimensions of several millimeters.
More recently a full quantum treatment has been developed for n-photon processes,7 whose results the present authors have used for analyzing the experiment on three-photon absorption in CdS.4 In the present work the indirect quantitative method based on non-linear photoconductivity (NLP) has been applied to the measurement of the four-photon non-linear cross-section ~ in ZnS. This is obtained by comparing the photocurrent induced by three- and four-photon absorption using two different wavelengths. *
Work partially supported by CNR. 1109
The sample was fixed to the copper cold finger of a vacuum cryostat operating at 80°K.The threephoton photocurrent was excited by a Q-switched ruby laser (3hw = 5.34 eV) with 20 nsec pulse duration and 200 MW maximum peak power. The fourphoton photocurrent was excited by a Q-switched neodymium laser (4hw = 4.68 eV) with the same pulse duration and same hundreds MW peak power. The pulse energy was monitored by using a beam splitter and a silicon photodiode. The monitor system was calibrated with a thermocouple calorimeter.
1110
FOUR-PHOTON TRANSITIONS IN ZnS
r
/
.92
~
rUby I~,er
neodymium
~
Q(4)
IR(0~On~~e~
IR i Nd
where
74’Nd
3 73 ‘Rb are the three and four photon
Q(3) and Q(4)
total generated charge. One gets then, from Fig. I 73
I
I
I
I
id _____________________________________ id’ io’~ icf I,~pho~o~2’~ 1.
2 —
a
FIG.
Vol. 16, No.9
3 x
l0~
cm2 sec.
(2)
The three photon non linear cross section y~of ZnS has been determinated previously by non-linear luminescence measurements.9 and combining this result with the one of equation (2), For ruby laser light, ‘y~= 0.7 x 10_80 cm6 sec2 one gets the four photon non-linear cross section for ZnS at the frequency of neodymium y~= 2 x l0~1 cm8 sec3.
Q vsI 0 plot for ZnS: (a) ruby excitation; (b) neodymium excitation.
Theelsewhere8 set up for NLP measurements described and basically is a Q has vs jobeen measurement, where Q is the total charge generated by a single laser pulse and 1o the incident photon flux (corrected for reflection losses). For an n-photon process:
Theoretically, y~has been calculated either by semiclassical perturbation theory using a two band model, or by using quantum electrodynamics. Accord10 the four photon non linear ing to the first treatment, cross which has made the using Hartree— Fock section, approximation andbeen including usually neglected A2 term, is: =
2b0
3
~.J2 ir3h4 N(hw)8
line through the experimental points has a slope three, which is consistent with a three-photon absorption process; the other the slope fourphoton for neodymium on excitation is hand, an evidence of aoffour process. From these experimental results it is possible to determine the ratio between the non-linear cross section y~and ~3. In fact by equation (1) it is obtained:
m~2 P m’1m2 ~
2
C
Q=K 110 (1) where the “constant” k depends, among other things, on the transport parameters of the crystals and its geometrical characteristics. The analytical expression of K has been calculated in reference 8. Figure 1 shows the results of the non-linear photocurrent measurements made with ruby (a) and neodymium (b) excitation in ZnS at 80°K.In the case of ruby excitation the straight
e8 C~’~
312 [i
X
________
8 1flvm4hc~.)_Eg~ m~ hw (4hw E~)2~ (3) hw
—
(4hwEg)
+ 16 (mcvm\i 2 21 m~
—
)
where Eg is the fundamental energy gap, the dielectric constant and IP~,12 the square of the matrix element of the momentum operator; m~ 1,is the reduced mass of valence and conduction bands. The other symbols have the usual meaning. Assuming the effective mass 3 equation Cv of reference 2 6.2 x l0_38 erg gr, (3) gives 7~=10,2.1Px 10_Ill cm8 sec3. The quantum treatment7 uses a three band model with a single valence and two conduction band c’ and c and the resulting formula is: 2”2ir3e8h4 IP~,12 74 = Nc4(hw)’°(ji/m)312e2m712 X —
—~
X
32 ~(4hW
~
~g)7/~
—
64
X
Vol. 16, No.9
FOUR-PHOTON TRANSITIONS IN ZnS
_______________ ~56 —~. P2 + 8h~+ ~ (4hw Eg)512 hw 1— m Jl5 hw 0 ________
—
+
_~_
(-~-~2
675 \mJ
(4hw
—
2]
+
we can that experimental data are inConcluding, fair agreement withsay perturbative semiclassical treatment. The numerical discrepancies which is observed in the case of quantum calculation can be due to the use of the two band model. On the other
E,)31’2(h~))2x
m2
I1’~~d1I
1111
X 15 h~~8hWa+ 9~I I 2~2) where j.i = m~’+ mc,’, ~ is the energy difference between the minimum of conduction band and the intermediate band c’ and c and a = ~2 — (hw)2 Assuming the energy level c’ at 8.9 eV,’~equation (4) gives = 3.2 x 8 sec3. The Keldysh theory cm (see formula (41)10~H2 of reference 6) gives for the fourphoton non linear cross section the value: ~ = 2 x l0’17cm8 sec3.
hand valid model the Keldysh transitions theory is in definitively which a small rulednumber out as of photons isfor involved. The anomalous behaviour of a experiments in the presence of a magnetic field (twophoton magneto conductivity processes’2) for which the Keldysh’s theory gives reasonable results also in the case of a low order process, is not yet well understood.
REFERENCES 1. 2.
BRAUNSTEIN R. and OCKMAN N., Phys. Rev. 134A, 499 (1964); BASSANI F. and HASSAN A.R., Nuovo Cimento 7B; 313 (1972); YEE J.H.,J. Phys. Chem. Solids 33, 643 (1972). BASOV N.G., GRASYUK A.Z., AUBAREW I.G., KATULIN V.A. and KORKHIN O.N., Soy. Phys. JETP 23, 366 (1966).
3.
CATALANO I.M., CINGOLANI A. and MINAFRA A.,Phys. Rev. 9,707(1974).
4.
6.
ASJKINADZE B.M., RYVKIN S.M. and YAROSHETSKH I.D., Soy. Phys. Semicond. 2, 1285 (1969); CATALANO I.M., CINGOLANI A. and MINAFRA A., Opt. Commun., 11,254(1974). YEE J.H. and CHAN H.H.M., Opt. Commun. 10,56(1974); CATALANO l.M., CINGOLANI A. and MINAFRA A., Solid State Commun. (to be published). KELDYSH L.V., Soy. Phys. JETP 20, 1307 (1965).
7.
KOVARSKII V.A. and PERLIN E.Yu, Phys. Status Solidi (b) 45,47 (1971).
8.
CINGOLANI A., FERRERO F., MINAFRA A. and TRIGGIANTE D.,Nuovo Cimento 4B, 217 (1971).
9. 10.
CATALANO I.M., CINGOLANI A. and MINAFRA A., Opt. Commun. 7,270(1973); CATALANO I.M., CINGOLANI A. and MINAFRA A.,Phys. Rev. B8, 1488 (1973). YEEJ.H.,Phys.Rev. B3, 355 (1971).
11. 12.
COHEN M.L. and BERGSTRESSER P.K., Phys. Rev. 141, 789 (1966). WElLER M.W., BIERING R.W. and LAX B.,Phys. Rev. 184,709 (1969).
5.