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Near band-gap optical nonlinearities and bistability in Cd1−xMnxTe
Optical Materials 14 (2000) 161±170
www.elsevier.nl/locate/optmat
Near band-gap optical nonlinearities and bistability in Cd1ÿx MnxTe L. Kowalczyk *, B. Koziarska-Glinka, Le Van Khoi, R.R. Gaøaz zka, A. Suchocki Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02 668 Warsaw, Poland Received 3 May 1999; accepted 27 September 1999
Abstract Strong thermally induced self-focusing of laser beam near the band-edge in Cd1ÿx Mnx Te with 0 < x < 0.6 at room temperature has been investigated. Due to low thermal conductivity of the CdMnTe, it is possible to obtain large values of the nonlinear refractive index, even in the order of 10ÿ4 cm2 /W, without light-induced damage to the crystal. Large thermal nonlinearities at room temperature lead to the optical bistability. The results indicate that Cd1ÿx Mnx Te is a suitable material for cavityless, large contrast, thermally induced absorptive bistable operation in visible region in spite of slow response time. In contrast to self-induced absorption at room temperature, at low temperatures light-induced transparency has been observed in CdTe and Cd0:85 Mn0:15 Te, which has been ascribed to the Burstein±Moss eect in this material. This eect is much faster than the self-focusing due to its electronic nature. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 4265P; 4265J
1. Introduction CdMnTe belongs to the group of diluted magnetic semiconductors (DMS). This compound attracted a lot of research interest in recent years due to its magnetic and semiconducting properties [1]. One of the well-known possible application of CdMnTe are optical isolators utilizing an eect of large Faraday rotation observed in this compound. Its operational range could be tuned since the band-gap of the material depends strongly on the manganese contents. CdMnTe crystals also
* Corresponding author. Tel.: +48-22843 7001 ex. 2631; fax: +48-22843 0926. E-mail address: [email protected] (L. Kowalczyk).
exhibit interesting nonlinear optical properties, intensively studied in recent years, as thermal selffocusing, thermally induced optical bistability and the others. The positive feedback necessary for thermally induced bistability is provided by the local temperature rise in the illuminated spot and the temperature dependent absorption coecient of the material, therefore no Fabry±Perot cavity is needed for these systems. Such bistable elements may have some interesting device application, for example in all-optical modulators, due to the easiness of their fabrication especially if bistable operation is achieved at room temperature, visible wavelength and low critical power. Possibly, large optical nonlinearities in CdMnTe crystals could be also used for creation of optical thermal solitons in this material, which have been a subject of intensive studies in recent years [2].
0925-3467/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 9 9 ) 0 0 1 1 3 - 5
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In this paper, the investigations of near bandgap optical nonlinearities in CdMnTe are reported. They coexist independently with the magnetic eects observed in these crystals. The nonlinear eects have been studied in detail with the use of several optical methods, among them we have also used the Z-scan technique at low temperatures. The experimental details are given in Section 2 of the paper. The results of studies of self-focusing eect at room temperature are presented in Section 3.1. The values of photothermal focal length as well as the absorption coecient have been measured as a function of intensity of laser beam. From these measurements the values of nonlinear refractive index for Cd1ÿx Mnx Te have been determined as a few times 10ÿ4 cm2 /W for the wavelengths whose values of the linear absorption coecient of the material are equal to approximately 10 cmÿ1 . Due to small value of thermal conductivity of Cd1ÿx Mnx Te crystals, the local temperature rise in the illuminated spot is higher than for ZnSe. This results in the shortening of the photothermal focal length by about an order of magnitude as compared with ZnSe for the same value of absorption coecient. Unfortunately, the dynamics of the thermal bistable processes in Cd1ÿx Mnx Te is rather slow and dependent on the sample dimensions and the details of thermal contact with surrounding. Strong self-focusing observed in the examined crystals leads to cavityless, absorptive optical bistability in Cd1ÿx Mnx Te at room temperature, which is reported in Section 3.2. Thermally induced optical bistability has been observed under band-gap resonant conditions in several semiconductors: Si [3], GaAs [4], and in the visible region in ZnSe [5±8] and ZnS [6]. However, the value of critical power for bistability in bulk ZnSe investigated in Ref. [5] was about 3000 W/cm2 . The results of our studies show that the value of critical power for bistability in Cd1ÿx Mnx Te is 30 times lower than that for bulk ZnSe. The CdTe and CdMnTe crystals with relatively low concentration of Mn (up to 15%) exhibit extremely dierent behavior at low temperatures. Due to the eect of band-®lling (the Burstein± Moss eect), which occurs for a relatively low light intensity, an induced transparency is observed,
instead of induced absorption. This eect is very fast due to its electronic nature. The results of these studies are reported in Section 3.3. The band®lling has been found responsible for the increase of the luminescence intensity of certain transitions in CdTe/CdMnTe quantum wells and for their broadening [9]. It has been found to aect the nonlinear propagation of picosecond optical pulses in thin layers of CdTe at temperature 80 K [10]. In this paper we study the c.w. light-induced transparency, which is a very prominent ®ngerprint of this eect. 2. Experimental The CdTe and Cd1ÿx Mnx Te crystals were grown by the modi®ed Bridgman method. The room-temperature electron concentration of the CdTe n-type crystals was equal to 4.6 ´ 1014 cmÿ3 . The Cd1ÿx Mnx Te samples were semi-insulating. The samples were cut and polished to form slabs of thickness ranging from 0.5 to 3 mm. The absorption spectra of the crystal have been measured with the use of Hitachi±Perkin Elmer model 340 spectrophotometer. The self-focusing experiment was performed on samples (without antire¯ection coatings) at room temperature. The far-®eld pro®les of the laser beams with photon energies just below the absorption edge were observed on a screen after passing through the sample. The Coherent model 899 dye/Ti:sapphire laser pumped by the 20 W Inova 200 laser has been used. The Rhodamine 6G and DCM dyes or the Ti:sapphire laser have been used depending on the content of manganese in the examined samples in order to illuminate the crystal close to its bandgap. The bistability eect has been studied with the use of Coherent LABMASTER-E power meter. The dynamics of the bistability eect was recorded with the use of the EG&G PAR 4202 Signal Averager and stored in the computer. For the Z-scan measurements [11] the same lasers have been used. The laser beam has been expanded and subsequently focused by a lens. The samples were mounted on the translation table which allows to move them along the focused beam, passing through the focal point. For low-
L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
temperature measurements, the samples were mounted in the Oxford Instrument CF104 continuous-¯ow cryostat equipped with temperature controller. The intensity of the passing light after strong reduction with neutral-density ®lters was measured by the photomultipliers with S20 or S1 cathodes depending on the used laser wavelength. 3. Results and discussion 3.1. Self-focusing eect Fig. 1 shows the power dependence of the transmitted beam pro®les obtained by scanning a photodiode across a pattern 182 cm behind the sample for dierent incident powers. The beams in the absence of the sample and on transmission through the sample at low laser powers were nearly Gaussian in cross-section. As the incident power is increased, the far-®eld pro®le breaks up into a set of rings of ever increasing radius and number. The focal length of thermally induced lens was found by measuring image size. These results were also checked by direct observation of the focal
163
position. The reciprocal of the measured focal length as a function of the laser power density for Cd1ÿx Mnx Te with 0< x< 0.6 for dierent laser wavelength close to the band-gap and for ZnSe for the wavelength 514.5 nm is shown in Fig. 2. The focal strength as high as about 40 mÿ1 has been obtained for the sample with 60% of manganese without any damages to the surface of the sample. The self-focusing eect occurs when the refractive index of the nonlinear medium increases with the beam intensity. An intense laser beam with photon energies close to the band-energy and with transverse Gaussian distribution of intensity causes a radial gradient of temperature in the material and hence the change in the refractive index. In the case of the CdMnTe, due to positive value of the thermal derivative of the refractive index of the material [12], the central part of the beam sees a larger refractive index than the edge. Consequently, the beam is focused by itself. The focal length, f, of thermal lenses is given by Taghizadeh et al. [7]: p 2pK 1 f r q ; 1 dn a P n dT where r is the laser beam radius, n the refractive index, K the thermal conductivity, a the linear absorption coecient and P the input power.
Fig. 1. Diametric beam pro®les in the far ®eld (182 cm) behind Cd0:5 Mn0:5 Te sample for dierent incident laser intensities (k 595 nm): (a) without sample; (b) 4 W/cm2 ; (c) 10 W/cm2 . The asymmetry comes from the slight crystal imperfections.
Fig. 2. Examples of results of measurements of the thermal lens power as a function of incident laser light intensity for Cd1ÿx Mnx Te. The wavelengths used: x 0; k 854 nm; x 40; k 641 nm; x 50; k 595 nm; x 60; k 601 nm.
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The results presented in Fig. 2 can be explained qualitatively using Eq. (1). The thermal conductivity is much smaller for Mn-doped CdTe (K 0.075 W/cm deg) than for ZnSe (K 0.19 W/cm deg) [13] and it may decrease with content of Mn due to additional point impurity scattering. Due to smaller value of thermal conductivity, the focal temperature rise in the illuminated spot for CdMnTe is higher than for ZnSe at the same value of absorption coecient and input power. The thermo-optic coecient l=n dn=dt is associated with the temperature dependence of the absorption edge. The optical absorption of Cd1ÿx Mnx Te crystals was measured as a function of temperature and Mn content in Ref. [14]. It was shown there that for x < 0.5 direct interband transitions at the T point dominate the absorption spectrum. The energy of the absorption edge decreases with increasing of temperature. For x P 0.5, the fundamental absorption overlaps with the intrashell transitions within the Mn ions and the excitation of localized manganese 3d states becomes responsible for the absorption. The value of the nonlinear refractive index can be calculated using the model of laser beam-induced thermal lens [15]. The focal length, f, of the observed self-induced thermal lens is expressed by: f r02 =Dnd;
2
Fig. 2 shows linear dependence of thermal lens power, I=f , on laser intensity. The thermal lensing eect occurs in all examined samples for relatively low incident laser intensity. With the measured values of f, I, r, l and a, the nonlinear refractive index n2 is obtained from Eq. (5). They are presented in Fig. 3 for all measured samples. The values of nonlinear refractive index for the same value of the linear absorption coecient in all measured samples remain basically unchanged (within the error bars) up to a manganese concentration of about 50%. In the samples with manganese contents exceeding 50% a weak increase of n2 is observed. At these concentrations the fundamental absorption of the compound coincides with the intrashell d5 absorption of manganese ions, which may additionally contribute to the decrease of thermal conductivity of such samples. According to Eq. (5) the changes of the focal length at the same intensity of light depend on the absorption coecient of the sample. For that reason the measurements of the nonlinear refractive index n2 , presented in Fig. 3, are done for the same value of the absorption coecient in all measured samples. The dependence of the nonlinear refraction index on the absorption coecient is experimentally determined from the measurements of the focal length and the absorp-
where r0 is the half-width of the laser beam, Dn the refractive index change at the center of the Gaussian beam and d the eective thickness of the crystal de®ned as: d 1 ÿ exp ÿal=a;
3
where a is the absorption coecient of the crystal and l is the thickness of the sample. The refractive index change, Dn, is expressed as Dn n2 I;
4
where n2 is the nonlinear refraction index and I is the laser intensity. Substituting Eqs. (4) and (3) into Eq. (2), the relationship between the focal length f and the laser intensity is obtained as: r0 a : 5 f n2 I 1 ÿ exp ÿ al
Fig. 3. The values of the nonlinear refractive indices for various Cd1ÿx Mnx Te samples, determined from the thermal lens power dependence on the laser intensity. The results are given for the same value of the linear absorption coecient a 10 cmÿ1 .
L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
tion coecient as a function of wavelength. Determined this way the nonlinear refraction index for Cd1ÿx Mnx Te as a function of the linear absorption coecient is presented in Fig. 4. The nonlinear refraction index strongly increases with the increase of linear absorption coecient. The values of the nonlinear refraction index obtained in this paper are smaller than those obtained by Dai et al. [16] for Cd1ÿx Mnx Te for similar values of the absorption coecient and the incident light intensity. The results of the measurements described above have been con®rmed by the Z-scan measurements. This technique allows to measure independently the nonlinear refraction and nonlinear absorption by changing the aperture of the detector which records the light intensity passing through the sample when it is moved along the focused laser beam [11]. From the signal recorded with open aperture the nonlinear absorption can be deduced and the measurements with small aperture allow to obtain the value of the nonlinear refraction index n2 . Also the Z-scan gives instantaneously the sign of the nonlinearity [11]. The value of the nonlinear refraction, measured by the Z-scan technique, for small phase distortion and small aperture, is expressed as:
165
where DTpÿm is the dierence between the peak and valley values of the Z-scan transmittance, S is aperture linear transmittance, equal to: S 1 ÿ exp ÿ2ra2 =w2a ;
7
where ra is the aperture radius and wa denotes the beam radius at the aperture in the linear regime. The value of the phase distortion, DU0 , is equal to: DU0 k Dn d;
8
6
where k 2p=k is the wavevector of the light. An example of results obtained with the use of the Z-scan technique for the sample containing 15% of manganese is shown in Fig. 5. The dip in the transmission for the negative values of z followed by the increase of transmittance for z > 0, show that the refractive index changes are positive, as it has been observed in the self-focusing experiment. Obtained from the measurements a value of the nonlinear refractive index n2 , which is equal to about n2 2 10ÿ4 , is consistent with the results of the self-focusing eect. The open aperture Z-scan transmittance exhibits ¯at (within the error bars) behavior. This means that there is no nonlinear absorption up to power densities of about 100 W/cm2 . Above this light intensity the open aperture transmittance curves show appreciable dip at z 0, which is a ®ngerprint on the nonlinear absorption. This behavior, which eventually leads
Fig. 4. The nonlinear refractive index as a function of the linear absorption coecient for Cd0:5 Mn0:5 Te sample.
Fig. 5. The results of the Z-scan room temperature measurements for Cd0:85 Mn0:15 Te sample.
DTpÿm 0:406 1 ÿ S
0:25
jDU0 j;
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L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
Fig. 6. The results of the open aperture Z-scan measurements of CdTe crystal for higher laser intensities, measured in z 0.
to the thermal bistability, is presented in Fig. 6 for the CdTe sample. Similar results have been observed for the remaining CdMnTe crystals. 3.2. Bistability At a higher intensity of the laser beam, optical bistability was observed in all samples in which self-focusing eect was investigated. Transmitted laser power through the Cd0:6 Mn0:4 Te sample as a function of the incident laser intensity at 641 nm is presented in Fig. 7. The increase of laser light in-
Fig. 7. Experimental input±output characteristics for Cd0:6 Mn0:4 Te sample: (a) incident laser power up; (b) incident laser power down; (c) re¯ected light power. The lines are for guiding the eyes.
tensity above 430 mW leads to abrupt drop of sample transparency and switching from the transparent to an opaque state. The initial transparency is restored almost completely after a decrease of laser intensity below 360 mW. A hysteresis loop is formed this way. The type of hysteresis loop indicates, that the observed optical bistability is thermally induced. The switching is caused by positive feedback between the temperature rise in the illuminated spot and the temperature dependence of the absorption coecient. At the threshold the intensity of transmitted beam drops several times, but the intensity of the re¯ected light does not change appreciably, which is shown also in Fig. 7. This means that the eect of bistability is caused by the strong and abrupt increase of absorption in the sample. Another experiment was performed to investigate the magnitude and speed of the thermally induced tuning of the transmission of Cd0:6 Mn0:4 Te. A shutter was used to pulse the laser beam ``on'' for 6 s. Fig. 8 shows the resulting time dependence of the transmitted power for Cd0:6 Mn0:4 Te for various powers of incident light. It can be seen that for the small value of laser power the transmission is constant during the laser pulse (line c). The switching from the transparent to opaque state appears when the laser intensity is increased (lines b and a). From the results presented by line b in Fig. 8 one can estimate that when the switching occurs the absorption coecient increases by Da 1.2 cmÿ1 . Then from the temperature dependence of the absorption coecient, which was independently measured for the same sample, the
Fig. 8. Kinetics of the transmitted power for Cd0:6 Mn0:4 Te after turning on the light for dierent incident laser power: (a) 840 mW; (b) 420 mW; (c) 280 mW. The wavelength used: 641 nm.
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temperature rise in illuminated spot can be determined as DT 20 deg. The steady-state heating can be calculated assuming that the laser beam is a cylinder of radius r and heat is conducted through the CdMnTe to another cylindrical surface of radius R much larger than the beam [17]. This gives: DT P =2p Kl ln R=r0 :
9
Using DT 20 deg for the temperature rise in the beam, we obtain the value of thermal conductivity of Cd0:6 Mn0:4 Te as equal to K 0.025 W/cmK. Such small value of thermal conductivity is a reason of low critical power for thermal bistability in CdMnTe. The switch-down critical power 360 mW corresponds to power density of incident light 100 W/cm. This value is about 30 times smaller than the intensity necessary for switching-down in ZnSe [7]. 3.3. Optical nonlinearities in Cd1ÿx Mnx Te at low temperatures Observed at room temperature, the very large thermal nonlinearity close to the band-gap of Cd1ÿx Mnx Te is rather slow. In search for higher speed eects we performed Z-scan measurements at low temperatures. The examples of results of the open-aperture Z-scan measurements of the CdTe sample at 4.2 K are shown in Fig. 9. The results show that at low temperatures, below 100 K, we
Fig. 9. Results of the Z-scan open aperture measurements of CdTe crystal at T 4:24 K for two dierent incident light intensities (measured in z 0).
Fig. 10. Results of the Z-scan transmittance measurements of CdTe crystal at z 0 and z ÿ30 mm at T 4:2 K.
observe very strong light-induced transparency instead of nonlinear absorption, occurring at higher temperatures. The Z-scan curves are sharper for the smaller light intensity. That means that the self-induced transparency depends on the incident power. As it is shown in Fig. 10, the eect depends on the wavelength of the laser beam. The transparency in the focal point of the lens is up to about 20 times larger than far away from it. The absorption coecient changes by about 30 cmÿ1 for wavelength of 781.5 nm, which is presented in Fig. 11. The eect disappears for wavelengths longer than 785 nm for pure CdTe sample and for temperatures above 100 K.
Fig. 11. Changes of absorption coecient as a function of wavelength for CdTe crystal at T 4:2 K at z 0. Incident light intensity, measured in z 0 equal to 7.4 W/cm2 . The solid line is a computer ®t of the theory to the experimental data.
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Due to the band-®lling the absorption coecient for the energy of light larger than the bandgap energy diminishes. The detailed theory of the eect for the direct band-gap semiconductors was developed in Ref. [19]. According to this paper, the changes of the absorption coecient for the light of photon energy E are expressed by the formula: Da
Fig. 12. Results of the Z-scan measurements for Cd0:85 Mn0:15 Te crystal at T 4:2 K. Incident light intensity at z 0 equal to 1.4 W/cm2 , wavelength k 681:93 nm.
A similar eect, although not so eminent as in undoped CdTe, can be also observed for Cd1ÿx Mnx Te with x 0.15. The example of the open-aperture Z-scan measurements for this crystal at temperature T 4.2 K is shown in Fig. 12. We were unable to observe the self-induced transparency in the crystals with larger contents of manganese. We associate the observed eect of self-induced transparency to the band-®lling, that is the Burstein±Moss eect [18]. This eect comes from the fact the density of states in the conduction band is suciently low and a relatively small number of optically generated electrons can ®ll the band up to an appreciable depth, making the semiconductor degenerated. Since the lower energy states in the conduction band are ®lled, the electrons from the valence band require energies greater than the nominal band-gap to be optically excited into the conduction band. The condition of degeneracy can be described as: 3=2 h2 3n0 1; 10 2m kT 8p where m* is the eective mass, k the Boltzman constant, T the temperature and n0 the concentration of electrons in the conduction band. It is easier to ful®l this condition at low temperatures and in semiconductors with a small eective mass, as it is shown by the above formula.
Chh p E ÿ Eg fv Eah ÿ fc Ebh ÿ 1 E Clh p E ÿ Eg fv Eal ÿ fc Ebl ÿ 1; E
11
where Eg is the band-gap energy, the Chh and Clh (refer to heavy and light holes, respectively), are the coecients of the linear band-gap absorption: C p E ÿ Eg E Chh p Clh p E ÿ Eg E ÿ Eg E E
a0
12
and can be calculated from the ®t of the band-gap absorption to Eq. (12) using known values of the electron, me , heavy-hole, mhh , and light-hole, mlh , eective masses, respectively [19]. The fc (E) and fv (E) are the probabilities of a conduction and valence band state of energy E, respectively, being occupied by an electron. These probabilities are given by the Fermi±Dirac distribution functions: fc Ebh;bl f1 exp Ebh;bl ÿ EFc =kT gÿ1 ; 13 fc Eah;al f1 exp Eah;al ÿ EFv =kT gÿ1 ; where k is the Boltzman constant. The energies Exx are given by the expressions: me Eah;al Eg ÿ E ÿ Eg ; me mhh;lh Ebh;bl E ÿ Eg
mhh;lh me mhh;lh
:
14
The EFc andEFv are the carrier-dependent quasiFermi levels, estimated by the Nilsson approximation [20]:
L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
(
N N 64 0:05524 Nc Nc ) rÿ0:25 N N kT ; Nc Nc ( P P P 64 0:05524 EFv ÿ ln Nv Nv Nv rÿ0:25 ) P P kT ÿ Eg ; Nv Nv EF c
ln
N Nc
15
where N and P are the concentrations of electrons and holes, respectively, Nc and Nv are eective densities of states in the conduction and valence bands, respectively. The value of the band-gap energy and the value of the C constant have been estimated from the linear transmission data at z ÿ30 mm (Fig. 9), ®tting the formula 12 to the experimental data. Estimated this way the energy gap of CdTe at T 4:2 K was equal to Eg 1.5846 eV (782.48 nm). Using the values of eective masses, equal to: me 0:0963m0 ; mhh 0:8m0 ; and mlh 0:12m0 [21], and treating the values of N P and C as adjustable parameters we ®t Eq. (11) to the experimental data of changes of the absorption coecient under in¯uence of light. The best ®t, presented in Fig. 11 as a solid line, has been obtained for N P 3:5 1016 cmÿ3 . We consider the quality of the ®t as good. The deviations from the ®t, observed for the wavelengths longer than the band-gap energy, come from dierent from assumed by Eq. (12) dependence of the band-gap absorption on the energy of photons. This eect could be probably associated with the Urbach tail of the band-gap absorption. Substituting concentration of electrons N, derived from the ®t, to Eq. (10), the value of approx. 12 is obtained. This con®rms that the condition of degeneracy is well ful®lled and supports validity of our model. The average lifetime, s, of nonequilibrium electrons and holes upon light illumination can be roughly estimated as a ratio of the concentration of electrons and holes (N P), obtained from the ®t of Eq. (11), to the number of photons associated with the light intensity in the focal point. Esti-
169
mated this way the value of the average lifetime of nonequilibrium carrier in our CdTe samples is equal to about 40 ls at 4.2 K. Possibly, this value is even longer since we do not take into account the carrier diusion from the illuminated area. This value seems to be relatively large, although in the literature even longer recombination times in CdTe crystals have been reported due to carrier emission from shallow traps [22]. Also persistent photoconductivity, observed in CdTe and CdMnTe crystals, may increase the recombination times. Low temperature of measurements may contribute strongly to those eects. The results of the open-aperture Z-scan measurements in Cd0:85 Mn0:15 Te samples also exhibit bleaching of band-gap absorption (see Fig. 12), although the eect is not so pronounced as in undoped CdTe samples. This is associated most probably with an important reduction of the average lifetime of carriers in the appropriate bands and with the change of the absorption band-edge shape since the remaining parameters (except the value of the energy band-gap, which increases with the increase of manganese contents), as for example, the values of eective masses of carriers, do not change considerably between undoped CdTe and Cd0:85 Mn0:15 Te [23] crystals. The average lifetime undergoes probably further reduction in crystals with higher concentration of manganese, since we were unable to record the eect of band-®lling in these crystals even for stronger light intensities as used in previously described experiments. The other possible mechanisms of the nonlinearities in the semiconductor crystals as band-gap renormalization (shrinkage) due to screening of the carrier (mainly electrons due to their low eective mass) wavefunctions and the free carrier absorption should not contribute strongly to the nonlinearities observed in our crystals. These eects are important at higher concentration of carriers than used in our experiments [18]. Therefore we do not take them into account in our analysis. 4. Conclusions The thermally induced self-focusing at room temperature in Cd1ÿx Mnx Te for 0 < x < 0.6 leads to
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optical bistability with low critical power. The large strength of the self-focusing eect is due to very low thermal conductivity of Cd1ÿx Mnx Te (as compared with ZnSe, for example). The nonlinear refraction index for Cd1ÿx Mnx Te increases with manganese concentration, particularly for high content of Mn, where the fundamental absorption is obscured by transitions within Mn ions. Low light intensity to obtain cavityless, large contrast optical bistability and possibility of tuning the band-gap of the material with Mn concentration to required wavelength may be interesting for application of Cd1ÿx Mnx Te optical limiters and bistable devices in spite of relatively slow switching time. The switching times may be shortened by use of thin layers where the heat sink eect is reduced. Low-temperature measurements exhibit quite dierent properties of CdTe and CdMnTe crystals. Instead of self-focusing, optically induced transparency is observed that is associated with the Burstein±Moss eect. This is a much faster eect than thermal self-focusing due to its electronic nature. The theory of the Burstein±Moss eect ®ts very well to the optically induced transparency data. Very long recombination times are detected in pure CdTe samples in this experiment. A reduction of the recombination lifetime and changes of the shape of the fundamental band-to-band absorption are most probably responsible for the disappearance of that eect in samples with concentration of manganese exceeding 15%. Acknowledgements This work was partially supported by grant no. 2P03B 087 09 of the Polish Committee for Scienti®c Research.
References [1] R.R. Gaøaz zka, in: B.L.H. Wilson (Ed.), Proceedings of the 14th International Conference on Physics of Semiconductors, Edinburg 1978, Physics of Semiconductors 1978, Conference Series 43, p. 133. [2] N.N. Akhmediev, A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman & Hall, London, 1997. [3] H.J. Eichler, Optics Commun. 45 (1983) 62. [4] H.M. Gibbs, S.L.Mc. Call, T.N.C. Venkatesan, A.C. Gossard, A. Passner, W. Wiegman, Appl. Phys. Lett. 36 (1979) 451. [5] S.D. Smith, J.G.H. Mathew, M.R. Taghizadeh, A.C. Walker, B.S. Wherret, A. Hendry, Optics Commun. 51 (1984) 357. [6] G.L. Olbright, N. Peyhambrian, H.M. Gibbs, H.A.M. Macleod, F. Van Milligen, Appl. Phys. Lett. 45 (1984) 357. [7] M.R. Taghizadeh, J. Janossy, S.D. Smith, Appl. Phys. Lett. 46 (1985) 331. [8] A.K. Kar, B.S. Wherret, J. Opt. Soc. Am. B 3 (1986) 345. [9] W.Z. Shen, S.C. Shen, J. Appl. Phys. 80 (1996) 5941. [10] J.H. Collet, M. Pugnet, W.W. R uhle, L. Nardo, R. Sirvin, J. Appl. Phys. 68 (1990) 1815. [11] M. Sheik-Bahae, A.A. Said, T. Wei, D.J. Hagan, E.W. Van Stryland, IEEE J. Quant. Electronic 26 (1990) 760. [12] G.T. Johnston, Appl. Optics 16 (1977) 1796. [13] G.A. Slack, M. Aven, J.S. Prener, Physics and Chemistry of II±VI Compounds, North Holland, Amsterdam, 1967, p. 575. [14] N. Khoi, J. Gaj, Phys. Status Solidi B 83 (1977) K133. [15] G.B.A. Altshuler, V.S. Ermolayer, V. Yu Hramow, V. Zaulus, G. Liberts, Ferroelectrics 69 (1986) 67. [16] X.D. Dai, Y. Ito, Y.H. Ja, Australian J. Phys. 43 (1990) 303. [17] D.A.B. Miller, M.H. Mozolowski, A. Miller, Optics Commun. 27 (1978) 133. [18] E. Burstein, Phys. Rev. 93 (1954) 632. [19] B.R. Bennet, R. Soref, J.A. DEL Alamo, IEEE J. Quant. Electronics 26 (1990) 113. [20] N.G. Nilsson, Appl. Phys. Lett. 33 (1978) 653. [21] L.S. Dang, G. Neu, R. Romestain, Solid State Commun. 44 (1982) 1187. [22] C.J. Johnson, Mater. Res. Soc. Symp. Proc. 302 (1993) 463. [23] F. Long, P. Harrison, W.E. Hagston, J. Appl. Phys. 79 (1996) 6939.
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Near band-gap optical nonlinearities and bistability in Cd1ÿx MnxTe L. Kowalczyk *, B. Koziarska-Glinka, Le Van Khoi, R.R. Gaøaz zka, A. Suchocki Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02 668 Warsaw, Poland Received 3 May 1999; accepted 27 September 1999
Abstract Strong thermally induced self-focusing of laser beam near the band-edge in Cd1ÿx Mnx Te with 0 < x < 0.6 at room temperature has been investigated. Due to low thermal conductivity of the CdMnTe, it is possible to obtain large values of the nonlinear refractive index, even in the order of 10ÿ4 cm2 /W, without light-induced damage to the crystal. Large thermal nonlinearities at room temperature lead to the optical bistability. The results indicate that Cd1ÿx Mnx Te is a suitable material for cavityless, large contrast, thermally induced absorptive bistable operation in visible region in spite of slow response time. In contrast to self-induced absorption at room temperature, at low temperatures light-induced transparency has been observed in CdTe and Cd0:85 Mn0:15 Te, which has been ascribed to the Burstein±Moss eect in this material. This eect is much faster than the self-focusing due to its electronic nature. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 4265P; 4265J
1. Introduction CdMnTe belongs to the group of diluted magnetic semiconductors (DMS). This compound attracted a lot of research interest in recent years due to its magnetic and semiconducting properties [1]. One of the well-known possible application of CdMnTe are optical isolators utilizing an eect of large Faraday rotation observed in this compound. Its operational range could be tuned since the band-gap of the material depends strongly on the manganese contents. CdMnTe crystals also
* Corresponding author. Tel.: +48-22843 7001 ex. 2631; fax: +48-22843 0926. E-mail address: [email protected] (L. Kowalczyk).
exhibit interesting nonlinear optical properties, intensively studied in recent years, as thermal selffocusing, thermally induced optical bistability and the others. The positive feedback necessary for thermally induced bistability is provided by the local temperature rise in the illuminated spot and the temperature dependent absorption coecient of the material, therefore no Fabry±Perot cavity is needed for these systems. Such bistable elements may have some interesting device application, for example in all-optical modulators, due to the easiness of their fabrication especially if bistable operation is achieved at room temperature, visible wavelength and low critical power. Possibly, large optical nonlinearities in CdMnTe crystals could be also used for creation of optical thermal solitons in this material, which have been a subject of intensive studies in recent years [2].
0925-3467/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 9 9 ) 0 0 1 1 3 - 5
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In this paper, the investigations of near bandgap optical nonlinearities in CdMnTe are reported. They coexist independently with the magnetic eects observed in these crystals. The nonlinear eects have been studied in detail with the use of several optical methods, among them we have also used the Z-scan technique at low temperatures. The experimental details are given in Section 2 of the paper. The results of studies of self-focusing eect at room temperature are presented in Section 3.1. The values of photothermal focal length as well as the absorption coecient have been measured as a function of intensity of laser beam. From these measurements the values of nonlinear refractive index for Cd1ÿx Mnx Te have been determined as a few times 10ÿ4 cm2 /W for the wavelengths whose values of the linear absorption coecient of the material are equal to approximately 10 cmÿ1 . Due to small value of thermal conductivity of Cd1ÿx Mnx Te crystals, the local temperature rise in the illuminated spot is higher than for ZnSe. This results in the shortening of the photothermal focal length by about an order of magnitude as compared with ZnSe for the same value of absorption coecient. Unfortunately, the dynamics of the thermal bistable processes in Cd1ÿx Mnx Te is rather slow and dependent on the sample dimensions and the details of thermal contact with surrounding. Strong self-focusing observed in the examined crystals leads to cavityless, absorptive optical bistability in Cd1ÿx Mnx Te at room temperature, which is reported in Section 3.2. Thermally induced optical bistability has been observed under band-gap resonant conditions in several semiconductors: Si [3], GaAs [4], and in the visible region in ZnSe [5±8] and ZnS [6]. However, the value of critical power for bistability in bulk ZnSe investigated in Ref. [5] was about 3000 W/cm2 . The results of our studies show that the value of critical power for bistability in Cd1ÿx Mnx Te is 30 times lower than that for bulk ZnSe. The CdTe and CdMnTe crystals with relatively low concentration of Mn (up to 15%) exhibit extremely dierent behavior at low temperatures. Due to the eect of band-®lling (the Burstein± Moss eect), which occurs for a relatively low light intensity, an induced transparency is observed,
instead of induced absorption. This eect is very fast due to its electronic nature. The results of these studies are reported in Section 3.3. The band®lling has been found responsible for the increase of the luminescence intensity of certain transitions in CdTe/CdMnTe quantum wells and for their broadening [9]. It has been found to aect the nonlinear propagation of picosecond optical pulses in thin layers of CdTe at temperature 80 K [10]. In this paper we study the c.w. light-induced transparency, which is a very prominent ®ngerprint of this eect. 2. Experimental The CdTe and Cd1ÿx Mnx Te crystals were grown by the modi®ed Bridgman method. The room-temperature electron concentration of the CdTe n-type crystals was equal to 4.6 ´ 1014 cmÿ3 . The Cd1ÿx Mnx Te samples were semi-insulating. The samples were cut and polished to form slabs of thickness ranging from 0.5 to 3 mm. The absorption spectra of the crystal have been measured with the use of Hitachi±Perkin Elmer model 340 spectrophotometer. The self-focusing experiment was performed on samples (without antire¯ection coatings) at room temperature. The far-®eld pro®les of the laser beams with photon energies just below the absorption edge were observed on a screen after passing through the sample. The Coherent model 899 dye/Ti:sapphire laser pumped by the 20 W Inova 200 laser has been used. The Rhodamine 6G and DCM dyes or the Ti:sapphire laser have been used depending on the content of manganese in the examined samples in order to illuminate the crystal close to its bandgap. The bistability eect has been studied with the use of Coherent LABMASTER-E power meter. The dynamics of the bistability eect was recorded with the use of the EG&G PAR 4202 Signal Averager and stored in the computer. For the Z-scan measurements [11] the same lasers have been used. The laser beam has been expanded and subsequently focused by a lens. The samples were mounted on the translation table which allows to move them along the focused beam, passing through the focal point. For low-
L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
temperature measurements, the samples were mounted in the Oxford Instrument CF104 continuous-¯ow cryostat equipped with temperature controller. The intensity of the passing light after strong reduction with neutral-density ®lters was measured by the photomultipliers with S20 or S1 cathodes depending on the used laser wavelength. 3. Results and discussion 3.1. Self-focusing eect Fig. 1 shows the power dependence of the transmitted beam pro®les obtained by scanning a photodiode across a pattern 182 cm behind the sample for dierent incident powers. The beams in the absence of the sample and on transmission through the sample at low laser powers were nearly Gaussian in cross-section. As the incident power is increased, the far-®eld pro®le breaks up into a set of rings of ever increasing radius and number. The focal length of thermally induced lens was found by measuring image size. These results were also checked by direct observation of the focal
163
position. The reciprocal of the measured focal length as a function of the laser power density for Cd1ÿx Mnx Te with 0< x< 0.6 for dierent laser wavelength close to the band-gap and for ZnSe for the wavelength 514.5 nm is shown in Fig. 2. The focal strength as high as about 40 mÿ1 has been obtained for the sample with 60% of manganese without any damages to the surface of the sample. The self-focusing eect occurs when the refractive index of the nonlinear medium increases with the beam intensity. An intense laser beam with photon energies close to the band-energy and with transverse Gaussian distribution of intensity causes a radial gradient of temperature in the material and hence the change in the refractive index. In the case of the CdMnTe, due to positive value of the thermal derivative of the refractive index of the material [12], the central part of the beam sees a larger refractive index than the edge. Consequently, the beam is focused by itself. The focal length, f, of thermal lenses is given by Taghizadeh et al. [7]: p 2pK 1 f r q ; 1 dn a P n dT where r is the laser beam radius, n the refractive index, K the thermal conductivity, a the linear absorption coecient and P the input power.
Fig. 1. Diametric beam pro®les in the far ®eld (182 cm) behind Cd0:5 Mn0:5 Te sample for dierent incident laser intensities (k 595 nm): (a) without sample; (b) 4 W/cm2 ; (c) 10 W/cm2 . The asymmetry comes from the slight crystal imperfections.
Fig. 2. Examples of results of measurements of the thermal lens power as a function of incident laser light intensity for Cd1ÿx Mnx Te. The wavelengths used: x 0; k 854 nm; x 40; k 641 nm; x 50; k 595 nm; x 60; k 601 nm.
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The results presented in Fig. 2 can be explained qualitatively using Eq. (1). The thermal conductivity is much smaller for Mn-doped CdTe (K 0.075 W/cm deg) than for ZnSe (K 0.19 W/cm deg) [13] and it may decrease with content of Mn due to additional point impurity scattering. Due to smaller value of thermal conductivity, the focal temperature rise in the illuminated spot for CdMnTe is higher than for ZnSe at the same value of absorption coecient and input power. The thermo-optic coecient l=n dn=dt is associated with the temperature dependence of the absorption edge. The optical absorption of Cd1ÿx Mnx Te crystals was measured as a function of temperature and Mn content in Ref. [14]. It was shown there that for x < 0.5 direct interband transitions at the T point dominate the absorption spectrum. The energy of the absorption edge decreases with increasing of temperature. For x P 0.5, the fundamental absorption overlaps with the intrashell transitions within the Mn ions and the excitation of localized manganese 3d states becomes responsible for the absorption. The value of the nonlinear refractive index can be calculated using the model of laser beam-induced thermal lens [15]. The focal length, f, of the observed self-induced thermal lens is expressed by: f r02 =Dnd;
2
Fig. 2 shows linear dependence of thermal lens power, I=f , on laser intensity. The thermal lensing eect occurs in all examined samples for relatively low incident laser intensity. With the measured values of f, I, r, l and a, the nonlinear refractive index n2 is obtained from Eq. (5). They are presented in Fig. 3 for all measured samples. The values of nonlinear refractive index for the same value of the linear absorption coecient in all measured samples remain basically unchanged (within the error bars) up to a manganese concentration of about 50%. In the samples with manganese contents exceeding 50% a weak increase of n2 is observed. At these concentrations the fundamental absorption of the compound coincides with the intrashell d5 absorption of manganese ions, which may additionally contribute to the decrease of thermal conductivity of such samples. According to Eq. (5) the changes of the focal length at the same intensity of light depend on the absorption coecient of the sample. For that reason the measurements of the nonlinear refractive index n2 , presented in Fig. 3, are done for the same value of the absorption coecient in all measured samples. The dependence of the nonlinear refraction index on the absorption coecient is experimentally determined from the measurements of the focal length and the absorp-
where r0 is the half-width of the laser beam, Dn the refractive index change at the center of the Gaussian beam and d the eective thickness of the crystal de®ned as: d 1 ÿ exp ÿal=a;
3
where a is the absorption coecient of the crystal and l is the thickness of the sample. The refractive index change, Dn, is expressed as Dn n2 I;
4
where n2 is the nonlinear refraction index and I is the laser intensity. Substituting Eqs. (4) and (3) into Eq. (2), the relationship between the focal length f and the laser intensity is obtained as: r0 a : 5 f n2 I 1 ÿ exp ÿ al
Fig. 3. The values of the nonlinear refractive indices for various Cd1ÿx Mnx Te samples, determined from the thermal lens power dependence on the laser intensity. The results are given for the same value of the linear absorption coecient a 10 cmÿ1 .
L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
tion coecient as a function of wavelength. Determined this way the nonlinear refraction index for Cd1ÿx Mnx Te as a function of the linear absorption coecient is presented in Fig. 4. The nonlinear refraction index strongly increases with the increase of linear absorption coecient. The values of the nonlinear refraction index obtained in this paper are smaller than those obtained by Dai et al. [16] for Cd1ÿx Mnx Te for similar values of the absorption coecient and the incident light intensity. The results of the measurements described above have been con®rmed by the Z-scan measurements. This technique allows to measure independently the nonlinear refraction and nonlinear absorption by changing the aperture of the detector which records the light intensity passing through the sample when it is moved along the focused laser beam [11]. From the signal recorded with open aperture the nonlinear absorption can be deduced and the measurements with small aperture allow to obtain the value of the nonlinear refraction index n2 . Also the Z-scan gives instantaneously the sign of the nonlinearity [11]. The value of the nonlinear refraction, measured by the Z-scan technique, for small phase distortion and small aperture, is expressed as:
165
where DTpÿm is the dierence between the peak and valley values of the Z-scan transmittance, S is aperture linear transmittance, equal to: S 1 ÿ exp ÿ2ra2 =w2a ;
7
where ra is the aperture radius and wa denotes the beam radius at the aperture in the linear regime. The value of the phase distortion, DU0 , is equal to: DU0 k Dn d;
8
6
where k 2p=k is the wavevector of the light. An example of results obtained with the use of the Z-scan technique for the sample containing 15% of manganese is shown in Fig. 5. The dip in the transmission for the negative values of z followed by the increase of transmittance for z > 0, show that the refractive index changes are positive, as it has been observed in the self-focusing experiment. Obtained from the measurements a value of the nonlinear refractive index n2 , which is equal to about n2 2 10ÿ4 , is consistent with the results of the self-focusing eect. The open aperture Z-scan transmittance exhibits ¯at (within the error bars) behavior. This means that there is no nonlinear absorption up to power densities of about 100 W/cm2 . Above this light intensity the open aperture transmittance curves show appreciable dip at z 0, which is a ®ngerprint on the nonlinear absorption. This behavior, which eventually leads
Fig. 4. The nonlinear refractive index as a function of the linear absorption coecient for Cd0:5 Mn0:5 Te sample.
Fig. 5. The results of the Z-scan room temperature measurements for Cd0:85 Mn0:15 Te sample.
DTpÿm 0:406 1 ÿ S
0:25
jDU0 j;
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Fig. 6. The results of the open aperture Z-scan measurements of CdTe crystal for higher laser intensities, measured in z 0.
to the thermal bistability, is presented in Fig. 6 for the CdTe sample. Similar results have been observed for the remaining CdMnTe crystals. 3.2. Bistability At a higher intensity of the laser beam, optical bistability was observed in all samples in which self-focusing eect was investigated. Transmitted laser power through the Cd0:6 Mn0:4 Te sample as a function of the incident laser intensity at 641 nm is presented in Fig. 7. The increase of laser light in-
Fig. 7. Experimental input±output characteristics for Cd0:6 Mn0:4 Te sample: (a) incident laser power up; (b) incident laser power down; (c) re¯ected light power. The lines are for guiding the eyes.
tensity above 430 mW leads to abrupt drop of sample transparency and switching from the transparent to an opaque state. The initial transparency is restored almost completely after a decrease of laser intensity below 360 mW. A hysteresis loop is formed this way. The type of hysteresis loop indicates, that the observed optical bistability is thermally induced. The switching is caused by positive feedback between the temperature rise in the illuminated spot and the temperature dependence of the absorption coecient. At the threshold the intensity of transmitted beam drops several times, but the intensity of the re¯ected light does not change appreciably, which is shown also in Fig. 7. This means that the eect of bistability is caused by the strong and abrupt increase of absorption in the sample. Another experiment was performed to investigate the magnitude and speed of the thermally induced tuning of the transmission of Cd0:6 Mn0:4 Te. A shutter was used to pulse the laser beam ``on'' for 6 s. Fig. 8 shows the resulting time dependence of the transmitted power for Cd0:6 Mn0:4 Te for various powers of incident light. It can be seen that for the small value of laser power the transmission is constant during the laser pulse (line c). The switching from the transparent to opaque state appears when the laser intensity is increased (lines b and a). From the results presented by line b in Fig. 8 one can estimate that when the switching occurs the absorption coecient increases by Da 1.2 cmÿ1 . Then from the temperature dependence of the absorption coecient, which was independently measured for the same sample, the
Fig. 8. Kinetics of the transmitted power for Cd0:6 Mn0:4 Te after turning on the light for dierent incident laser power: (a) 840 mW; (b) 420 mW; (c) 280 mW. The wavelength used: 641 nm.
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167
temperature rise in illuminated spot can be determined as DT 20 deg. The steady-state heating can be calculated assuming that the laser beam is a cylinder of radius r and heat is conducted through the CdMnTe to another cylindrical surface of radius R much larger than the beam [17]. This gives: DT P =2p Kl ln R=r0 :
9
Using DT 20 deg for the temperature rise in the beam, we obtain the value of thermal conductivity of Cd0:6 Mn0:4 Te as equal to K 0.025 W/cmK. Such small value of thermal conductivity is a reason of low critical power for thermal bistability in CdMnTe. The switch-down critical power 360 mW corresponds to power density of incident light 100 W/cm. This value is about 30 times smaller than the intensity necessary for switching-down in ZnSe [7]. 3.3. Optical nonlinearities in Cd1ÿx Mnx Te at low temperatures Observed at room temperature, the very large thermal nonlinearity close to the band-gap of Cd1ÿx Mnx Te is rather slow. In search for higher speed eects we performed Z-scan measurements at low temperatures. The examples of results of the open-aperture Z-scan measurements of the CdTe sample at 4.2 K are shown in Fig. 9. The results show that at low temperatures, below 100 K, we
Fig. 9. Results of the Z-scan open aperture measurements of CdTe crystal at T 4:24 K for two dierent incident light intensities (measured in z 0).
Fig. 10. Results of the Z-scan transmittance measurements of CdTe crystal at z 0 and z ÿ30 mm at T 4:2 K.
observe very strong light-induced transparency instead of nonlinear absorption, occurring at higher temperatures. The Z-scan curves are sharper for the smaller light intensity. That means that the self-induced transparency depends on the incident power. As it is shown in Fig. 10, the eect depends on the wavelength of the laser beam. The transparency in the focal point of the lens is up to about 20 times larger than far away from it. The absorption coecient changes by about 30 cmÿ1 for wavelength of 781.5 nm, which is presented in Fig. 11. The eect disappears for wavelengths longer than 785 nm for pure CdTe sample and for temperatures above 100 K.
Fig. 11. Changes of absorption coecient as a function of wavelength for CdTe crystal at T 4:2 K at z 0. Incident light intensity, measured in z 0 equal to 7.4 W/cm2 . The solid line is a computer ®t of the theory to the experimental data.
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Due to the band-®lling the absorption coecient for the energy of light larger than the bandgap energy diminishes. The detailed theory of the eect for the direct band-gap semiconductors was developed in Ref. [19]. According to this paper, the changes of the absorption coecient for the light of photon energy E are expressed by the formula: Da
Fig. 12. Results of the Z-scan measurements for Cd0:85 Mn0:15 Te crystal at T 4:2 K. Incident light intensity at z 0 equal to 1.4 W/cm2 , wavelength k 681:93 nm.
A similar eect, although not so eminent as in undoped CdTe, can be also observed for Cd1ÿx Mnx Te with x 0.15. The example of the open-aperture Z-scan measurements for this crystal at temperature T 4.2 K is shown in Fig. 12. We were unable to observe the self-induced transparency in the crystals with larger contents of manganese. We associate the observed eect of self-induced transparency to the band-®lling, that is the Burstein±Moss eect [18]. This eect comes from the fact the density of states in the conduction band is suciently low and a relatively small number of optically generated electrons can ®ll the band up to an appreciable depth, making the semiconductor degenerated. Since the lower energy states in the conduction band are ®lled, the electrons from the valence band require energies greater than the nominal band-gap to be optically excited into the conduction band. The condition of degeneracy can be described as: 3=2 h2 3n0 1; 10 2m kT 8p where m* is the eective mass, k the Boltzman constant, T the temperature and n0 the concentration of electrons in the conduction band. It is easier to ful®l this condition at low temperatures and in semiconductors with a small eective mass, as it is shown by the above formula.
Chh p E ÿ Eg fv Eah ÿ fc Ebh ÿ 1 E Clh p E ÿ Eg fv Eal ÿ fc Ebl ÿ 1; E
11
where Eg is the band-gap energy, the Chh and Clh (refer to heavy and light holes, respectively), are the coecients of the linear band-gap absorption: C p E ÿ Eg E Chh p Clh p E ÿ Eg E ÿ Eg E E
a0
12
and can be calculated from the ®t of the band-gap absorption to Eq. (12) using known values of the electron, me , heavy-hole, mhh , and light-hole, mlh , eective masses, respectively [19]. The fc (E) and fv (E) are the probabilities of a conduction and valence band state of energy E, respectively, being occupied by an electron. These probabilities are given by the Fermi±Dirac distribution functions: fc Ebh;bl f1 exp Ebh;bl ÿ EFc =kT gÿ1 ; 13 fc Eah;al f1 exp Eah;al ÿ EFv =kT gÿ1 ; where k is the Boltzman constant. The energies Exx are given by the expressions: me Eah;al Eg ÿ E ÿ Eg ; me mhh;lh Ebh;bl E ÿ Eg
mhh;lh me mhh;lh
:
14
The EFc andEFv are the carrier-dependent quasiFermi levels, estimated by the Nilsson approximation [20]:
L. Kowalczyk et al. / Optical Materials 14 (2000) 161±170
(
N N 64 0:05524 Nc Nc ) rÿ0:25 N N kT ; Nc Nc ( P P P 64 0:05524 EFv ÿ ln Nv Nv Nv rÿ0:25 ) P P kT ÿ Eg ; Nv Nv EF c
ln
N Nc
15
where N and P are the concentrations of electrons and holes, respectively, Nc and Nv are eective densities of states in the conduction and valence bands, respectively. The value of the band-gap energy and the value of the C constant have been estimated from the linear transmission data at z ÿ30 mm (Fig. 9), ®tting the formula 12 to the experimental data. Estimated this way the energy gap of CdTe at T 4:2 K was equal to Eg 1.5846 eV (782.48 nm). Using the values of eective masses, equal to: me 0:0963m0 ; mhh 0:8m0 ; and mlh 0:12m0 [21], and treating the values of N P and C as adjustable parameters we ®t Eq. (11) to the experimental data of changes of the absorption coecient under in¯uence of light. The best ®t, presented in Fig. 11 as a solid line, has been obtained for N P 3:5 1016 cmÿ3 . We consider the quality of the ®t as good. The deviations from the ®t, observed for the wavelengths longer than the band-gap energy, come from dierent from assumed by Eq. (12) dependence of the band-gap absorption on the energy of photons. This eect could be probably associated with the Urbach tail of the band-gap absorption. Substituting concentration of electrons N, derived from the ®t, to Eq. (10), the value of approx. 12 is obtained. This con®rms that the condition of degeneracy is well ful®lled and supports validity of our model. The average lifetime, s, of nonequilibrium electrons and holes upon light illumination can be roughly estimated as a ratio of the concentration of electrons and holes (N P), obtained from the ®t of Eq. (11), to the number of photons associated with the light intensity in the focal point. Esti-
169
mated this way the value of the average lifetime of nonequilibrium carrier in our CdTe samples is equal to about 40 ls at 4.2 K. Possibly, this value is even longer since we do not take into account the carrier diusion from the illuminated area. This value seems to be relatively large, although in the literature even longer recombination times in CdTe crystals have been reported due to carrier emission from shallow traps [22]. Also persistent photoconductivity, observed in CdTe and CdMnTe crystals, may increase the recombination times. Low temperature of measurements may contribute strongly to those eects. The results of the open-aperture Z-scan measurements in Cd0:85 Mn0:15 Te samples also exhibit bleaching of band-gap absorption (see Fig. 12), although the eect is not so pronounced as in undoped CdTe samples. This is associated most probably with an important reduction of the average lifetime of carriers in the appropriate bands and with the change of the absorption band-edge shape since the remaining parameters (except the value of the energy band-gap, which increases with the increase of manganese contents), as for example, the values of eective masses of carriers, do not change considerably between undoped CdTe and Cd0:85 Mn0:15 Te [23] crystals. The average lifetime undergoes probably further reduction in crystals with higher concentration of manganese, since we were unable to record the eect of band-®lling in these crystals even for stronger light intensities as used in previously described experiments. The other possible mechanisms of the nonlinearities in the semiconductor crystals as band-gap renormalization (shrinkage) due to screening of the carrier (mainly electrons due to their low eective mass) wavefunctions and the free carrier absorption should not contribute strongly to the nonlinearities observed in our crystals. These eects are important at higher concentration of carriers than used in our experiments [18]. Therefore we do not take them into account in our analysis. 4. Conclusions The thermally induced self-focusing at room temperature in Cd1ÿx Mnx Te for 0 < x < 0.6 leads to
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optical bistability with low critical power. The large strength of the self-focusing eect is due to very low thermal conductivity of Cd1ÿx Mnx Te (as compared with ZnSe, for example). The nonlinear refraction index for Cd1ÿx Mnx Te increases with manganese concentration, particularly for high content of Mn, where the fundamental absorption is obscured by transitions within Mn ions. Low light intensity to obtain cavityless, large contrast optical bistability and possibility of tuning the band-gap of the material with Mn concentration to required wavelength may be interesting for application of Cd1ÿx Mnx Te optical limiters and bistable devices in spite of relatively slow switching time. The switching times may be shortened by use of thin layers where the heat sink eect is reduced. Low-temperature measurements exhibit quite dierent properties of CdTe and CdMnTe crystals. Instead of self-focusing, optically induced transparency is observed that is associated with the Burstein±Moss eect. This is a much faster eect than thermal self-focusing due to its electronic nature. The theory of the Burstein±Moss eect ®ts very well to the optically induced transparency data. Very long recombination times are detected in pure CdTe samples in this experiment. A reduction of the recombination lifetime and changes of the shape of the fundamental band-to-band absorption are most probably responsible for the disappearance of that eect in samples with concentration of manganese exceeding 15%. Acknowledgements This work was partially supported by grant no. 2P03B 087 09 of the Polish Committee for Scienti®c Research.
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