Non-linear optical effects and transport phenomena of magnetic

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 38 (2005) 965–973

doi:10.1088/0022-3727/38/7/001

Non-linear optical effects and transport phenomena of magnetic semiconductors Pb1−xPrxTe near the semiconductor–metal phase transformation K Nouneh1,3 , I V Kityk2,3,5 , R Viennois1,6 , S Benet3 , K J Plucinski2 , S Charar1 , Z Golacki4 and S Paschen6 1

Groupe d’Etude des Semiconducteurs, CNRS-UMR 5650, Universit´e Montpellier II, Pl. Eug`ene Bataillon, 34095 Montpellier Cedex 5, France 2 Institute of Physics, Academy J.Dlugosz, Krajowej 36, Czestochowa, Poland 3 Laboratoire de Physique Appliqu´ee et Automatique, Universit´e de Perpignan, 52 Av. Paul Alduy, Perpignan, France 4 Institute of Physics, Polish Academy of Sciences, Pl.02-668 Warsaw, Poland 5 Institute of Physics, Academy of Czestochowa, Al.Armii Krajowej 13/15, Czestochowa, Poland 6 Max-Planck-Institue f¨ur Chemische Physik fester Stoffe, N¨othnitzer Str. 40, 01185, Dresden, Deutschland E-mail: [email protected]

Received 3 November 2004, in final form 4 January 2005 Published 17 March 2005 Online at stacks.iop.org/JPhysD/38/965 Abstract Measurements of transport and non-linear optical (NLO) properties of the new, synthesized semiconductor Pb0.9835 Pr0.0165 Te in the vicinity of low-temperature metal–semiconductor phase transformations were performed. A correlation between the temperature behaviour of transport properties near the phase transition and NLO susceptibilities of second- and third-order was observed. FTIR spectra show the substantial role played by the Pr3+ localized levels in the observed anomalies. Among the transport properties, the resistivity, Seebeck coefficient and specific heat ρ were measured. The presence of the minimum at about Tmin = 50 K in the temperature dependence of the resistivity ρ(T ) is due to the metal–semiconductor transition in Pb1−x Prx Te and the low-temperature upturn observed in the resistivity is well fitted by the Mott law. Substitution of PbTe by other rare earths shows the crucial role played by Pr in the observed dependences. Experimental temperature measurements using photo-induced NLO (pumped by a UV-laser as well as by second harmonics of the YAG–Nd lasers) and a probing YAG–Nd (at λ = 1.06 µm) laser, show the existence of two maxima in the photo-induced second harmonic generation (PISHG) at temperatures 17 and 30 K, the behaviours of which substantially depend on the wavelength of the pump beam. At the same time the third-order two-photon absorption possesses maxima near 50 K, i.e. at ρ about Tmin . Such discrepancies in the positions of the temperature maxima are caused by the difference in contributions of the photo-induced anharmonic phonons near the surfaces PISHG and the bulk. Varying the penetration depth of the photo-inducing light one can operate with the output NLO properties. (Some figures in this article are in colour only in the electronic version)

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1. Introduction Recently, the semi-magnetic semiconductors (SMSC), have been the subject of intensive exploration, for example the SMSC II–VI possessing transition metals [1]. The localized electronic d-states determine non-compensated magnetic momenta. However, recently, there have been investigations of the SCSM IV–VI possessing rare earths (REs) (with magnetism caused by the 4f levels screened by the 5s, 5p and 6s-states [1, 4]). Chalcogenides of RE (Pb–RE–X where X = Se, Te, S), like chalcogenides of plomb (Pb–X), possess a face centred cubic structure (NaCl-structure), point group m3m, and crystalline parameters close to that of PbX (PbTe = 6.5 Å, PrTe ≈ 6.4 Å) [2]. In this work, a new class of SMSCs, Pb1−x Prx Te, is studied where the magnetic ions are RE atoms never studied before in the SCSM family, particularly praseodymium. However, for comparison several other RE will be considered. It is well known that electronic properties of the majority of the RE–X are determined by 4f states of RE and by the double atomic configuration: 4f n 5d1 6s2 and 4f n+1 5d1 6s2 . In the case of Pr3+ , the electron shell responsible for the magnetic moment is 4 f2 . The 4 f shell is situated deeply inside the RE ion shells. So the spin–orbit interaction is higher than the crystal field splitting. As a consequence, one can expect the appearance of new physical properties [1] such as the existence of two exchange mechanisms: f–f interaction and sp–f interaction between the localized magnetic moments of spins and the valence electrons for the host lattice, respectively. The energies of the electronic terms depend predominantly on the Coulomb interaction between the localized f RE electrons of the RE solute [3]. However, the weaker coupling between the RE ions and host states may play a crucial role in the energy term positions of the RE ions [4]. Here, a crucial role is played by the magnetic states of the RE ions, particularly determined by the value of the non-compensated spin ferromagnetism. For instance, if one electron is removed from the f shell of the RE3+ , the resulting localized moment for the f-electrons is coupled through exchange interaction with the spin of the remaining delocalized valence electron. Generally, spin fluctuations give rise to a Kondo anomaly—strong coupling between the magnetic moment of the solute and quasi-free electrons if the concentration of charge carriers is sufficient. However, charge fluctuations, if significant, suppress Kondo anomalies and, accordingly, result in mixed-valent (heterovalence) states. The case of the Pr-doped semiconductor has never been studied earlier, except for the case of the optical properties of Pr3+ -doped GaN [5]. It is necessary to emphasize that the symmetry of the electronic ground state of Pr3+ is described as 1 singlet type or 3 non-magnetic doublet. This kind of fundamental occupied level may determine many amazing and interesting physical properties, particularly, properties very similar to those observed for Pr3+ -filled skutterudites [6]. As a consequence, one can expect that investigation of Pr3+ -doped semiconductors could also result in the observation of unusual and interesting physical properties related to the fundamental state of Pr3+ . So, in this paper, complex investigations of transport and non-linear optical (NLO) properties, which are extremely sensitive to the excited localized states, are carried out. 966

Now, let us discuss briefly the main physical properties of PrTe. PrTe is a paramagnetic and metallic compound with a RE crystal electric field (CEF) splitting of around 74 K between the 1 singlet ground state and the 4 triplet first excited level [7]. Praseodymium is known to be able to substitute on the Pb site in PbTe and, thus, because Pb has a valency of 2+, the Pr3+ impurity levels can form resonant states near the bottom of the conduction band, pinning the Fermi level and strongly affecting the transport properties to inject carriers into the conduction band. If the Pr3+ concentration is sufficient, one can expect a transformation from the semiconducting behaviour for pure PbTe to metallic-like behaviour for Pb1−x Prx Te. Below, it will be shown that metallic behaviour is observed above 50 K when the Pr3+ content is about 1.65 at% in weight.units. Some low-temperature anomalies are also observed in the electrical resistivity and thermopower but not in the heat capacity. After a discussion concerning the origin of these anomalies, which might be related to a phase transition, we will show below that they are related to the thermal anomalies of the photo-induced NLO signal detected in Pb1−x Prx Te, and the corresponding signal is relatively large compared to that for pure PbTe. Therefore, we will apply the methods of photo-induced non-linear optics [8–10], and particularly, photo-induced second harmonic generation (PISHG) for investigation at low temperatures, because this method is sensitive to the manifestation of electron–phonon anharmonic interactions during different kinds of phase transitions, including superconducting, antiferromagnetic, ferromagnetic etc [8]. Therefore, in this paper, we will show that this kind of experiment may also be useful in monitoring the metal–semiconducting phase transition.

2. Growth and chemical characterization of samples The PbPrTe single crystals investigated were grown by the Bridgman method. The synthesis was done by a direct reaction between the elements (Pb of 99.9999% purity, Te 99.9999% purity and Pr of 99.9% purity), taken at stoichiometric ratio. The particular components were put in a silica glass crucible with a conical tip and inner diameter of about 12 mm possessing graphitized walls, evacuated up to 10−5 Torr, and sealed off. The nominal composition was Pb0.9835 Pr0.0165 Te and the total weight of the load was about 50 g. Next, the tube was put in a vertical furnace 20 mm in diameter and 70 cm in length. The ampoule was kept for 72 h at a temperature of about 1000˚C for homogenization, and then cooled at a rate of about 20 mm day−1 . The process was interrupted when the ampoule went out of the furnace, reaching room temperature. For the Pr-doped single-crystal, a little segregation of Pr3+ was found in the sample. The purity of the samples was checked using x-ray diffraction (XRD) experiments, scanning electron microscopy and EDX experiments, which showed that the samples were single crystalline and mono-phase. High precision WDX microprobe experiments were carried out at the Max–Planck Institute in Dresden for the evaluation of the Pr concentration in the samples. In all the samples studied here, the Pr3+ concentration was varied in the side range. A similar technique was used for synthesis of other samples doped with other RE impurities.

Optical properties of semiconductors

Figure 1. Principal experimental set-up for two regime absorption and reflectivity for PISHG and TPA measurements. S: synchronizer for the power of the lasers; Pi=1–4 : Glan Thompson polarizer; specimens-magnetic camera for the specimens; PM: photomultipliers; BC:boxar; MN: grating monochromator; BS: beam-splitter; DL: Mi=1–6 : Mirror; delaying line.

3. Experimental details The electrical resistivity measurements were carried out using the standard four probe method. The contacts were attached to the material surface with silver paint. Thermo-power measurements between 2 and 400 K were performed exactly on the same sample and using a relaxation method similar to that described in footnote1 for a PPMS apparatus manufactured by ‘Quantum Design’. The heat capacity measurements between 0.4 and 300 K were performed by a relaxation method using a microcalorimeter and a PPMS apparatus from Quantum Design in magnetic fields up to 9 T. For the experiment we have used two types of photoinducing pump light beams. The first one was generated by a nanosecond nitrogen laser beam (λ = 0.377 µm; light power of about 2.5 MW per pulse with a pulse time duration varying between 1 and 10 ns). The second one used the second harmonic of the YAG–Nd laser working in the 30 ps pulse regime (λ = 530 nm) (see figure 1). All the PISHG measurements were done in the reflected light geometry because the effective layer thickness to which the illumination penetrated was equal to about 60 nm. The rotating polarizers P1 and P2 were supplied by a specific rotating mechanical equipment to operate by incident light polarization. The incident angle was varied within the range of 5–13˚ with respect to the sample’s surface normal. Such a wide range of angle variation was necessary to find a maximal output PISHG signal for every measured point to optimize the phase matching conditions. The diameter of the light beam spot was 1 Commercial apparatus for measuring thermal transport properties from 1.9 to 390.

varied within 0.85 to 1.24 mm. To exclude the influence of the surface instability to air, etc. all the measurements were done for the samples polished both in air as well as in vacuum up to 10−5 Torr. It was revealed that the temperature dependence of the PISHG was independent of the sample’s surface quality. Depending on the surface roughness, the light spot diameter was varied to achieve the maximal output PISHG signal. It was found that varying the roughness of surfaces only changed the absolute value of the PISHG but did not affect its spectral dependences. The fluorescence spectra lie in the near UV spectral range (do not overlap with the fundamental and doubled frequency signals) and we separated this parasitic background from the output PISHG using a grating monochromator with a spectral resolution of about 6 nm mm−1 . Because the maximal absorption value for both laser wavelengths is about 106 cm−1 , one can estimate that the effective depth of light penetration (skin effect) should be around 20–30 nm. So, one can suggest that we monitor processes near the surfaces. In that case, the role of the bulk effects is limited; however, near-the-surface states may reflect (are sensitive) general properties, which hold throughout the bulk sample. To eliminate the contribution of pure surface effects we have carried out separate measurements for the bulk NLO properties using CO2 IR laser beams which are transparent for the samples (λ = 10.6 µm). Moreover, this fact allows us to investigate NLO properties described both by third- and fourth-order tensors. The evaluations of the two-photon absorption (TPA) were done from intensity-dependent transparency for the CO2 (λ = 10.6 µm) fundamental pulse laser beam. The sample’s 967

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1.2x10 1.0x10 8.0x10

-3

Pb0.985 Pr0.015Te

-3

-4

-6.85 -3

1.0x10

[4.b]

[4.a]

Pb1-xPrx Te x = 1.65%

-4

1.0x10

Pb1-xPrx Te

x = 1.65%

-4

9.5x10

4.0x10

-4

10 K

ln (ρ)

6.0x10

ρ[Ω m]

ρ (Ω m)

-6.90 -3

-6.95

4.2K

-7.00

30 K

-4

9.0x10

-7.05

2.0x10

-4

-4

8.5x10

-7.10 10

0.0

0

50

T[K]

100

0.3

100

150

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0.4

0.5

0.6

1/T1/4 (K -1/4)

250

300

0.7

350

T (K) Figure 2. Temperature dependence of the resistivity ρ(T ) of Pb0.9835 Pr0.0165 Te. Inset: plot of ln(ρ) versus (1/T )1/4 (see the text for details).

temperature was monitored by a microthermobolometer chip with an accuracy up to 0.2 K. This was done simultaneously from both sides of the samples. The maximally achieved local heating due to power absorption did not exceed 0.8 K. Previously, we obtained almost the same value for the different metals and superconductors we investigated [8–10]. Moreover, the set-up (see figure 1) allowed us to scan the specimen’s surface with the probe and pump light beams. The beam’s profile sequences had a Gaussian-like shape with a half-width dispersion equal to about 78%. The laser power stability was kept within 0.1%. The generation of photoinducing (pumping) doubled-frequency Nd3+ : YAG laser (λ = 530 nm) or nitrogen lasers (λ = 337 nm) was temporarily synchronized with that of the fundamental YAG–Nd3+ laser (λ = 060 nm). This laser beam (spot diameter 0.2–13.6 mm; laser power 6–14 MW; pulse duration 1.3–8.5 ps) was used as the fundamental one for the PISHG. The pulse repetition time was synchronized in time for the photo-inducing and probing (fundamental) laser beams up to (100 ps). To determine the light polarization of the photo-inducing and probing beams, we have used polarizers with a degree of polarization of about 99.998(7)% in the considered spectral range. An electrooptically operated delaying line using a Li2 B4 O7 single crystal was carried out. It allowed us to vary the pump–probe delay time with a temporary resolution not worse than 0.56 ps. The maximal output PISHG signal was reliably observed only for the parallel polarizations of the pumping and fundamental laser beams, which was achieved with the help of a system of mirrors and polarizers shown in figure 1. When the non-collinearity of the beams was higher than 7˚, the output PISHG was drastically reduced (by at least two orders of magnitude). The nonhomogeneity of the output PISHG signal space distribution through the specimen surface was about 6%. The available precision of the PISHG measurements allowed us to determine the output PISHG signal with a relative error smaller than 0.8%. 968

4. Experimental results 4.1. Transport and thermoelectricity properties: Hopping conductivity region From figure 2 one can see that in Pb0.9835 Pr0.0165 Te, the electrical resistivity ρ(T ) decreases with increasing temperature ρ upto Tmin ∼ = 50(±10) K and, then, increases with temperature enhancement. A low-temperature upturn in ρ(T ) has already been observed by Yaraneri et al [11] in the case of PbTe alloying with Ge on the lead site. There are two possible reasons for such behaviour: the Kondo effect [12] or metal–semiconductor transition [13] or the introduction of magnetic impurities in metallic alloys (the ‘magnetic’ Kondo effect, [12] or even in a semiconductor heavily doped with Ce [14]), or the presence of some structural disorder effect [15, 16] (the ‘structural’ Kondo effect). If the low-temperature upturn in ρ(T ) were due to the Kondo effect, the temperature variation of the upturn should be described by a − ln T dependence [12]. Below 10 K, in the inset of figure 2, we can see that ρ(T ) follows a nearly −ln T dependence. If a ‘magnetic’ Kondo effect occurred in our compound, we would have a significant and easily observable magnetic contribution in the temperature dependence of the heat capacity CP (T ) [12] in addition to the lattice contribution, which should be equal to that of pure PbTe (from [17]). Because of the problems discussed above related the Kondo effect, we believe that the better and more natural explanation for the temperature variation of the electrical resistivity ρ(T ) of Pb0.9835 Pr0.0165 Te is a metal– semiconductor transition occurring at about 50 K. In the case of Pb0.9835 Pr0.0165 Te, the Pr3+ ions form an impurity band like in the case of Pb1−x Ybx Te alloys. Above 50 K, the conduction electrons are scattered inside this impurity band, whereas at lower temperatures (about 50 K), a hopping conduction with variable hopping length of the electrons occurs. This

Optical properties of semiconductors

FTIR [arb un ]

2

Such a temperature dependence is what is roughly observed in our sample, almost in the same temperature range (only up to 16 K), rather than the Mott law for ρ(T ) (see the insets in figures 2 and 4); particularly, at low temperatures, the hopping contribution to the Seebeck coefficient is dominant. This is in agreement with the hopping thermopower at low temperatures described by Zvyagin’s formula [19]:

1

Sh (T ) =

kB ∂(ln g(EF )) ξ KB (T0 T )1/2 , e ∂E

(2)

where g(EF ) and a are the density of states and localization radius at the Fermi level, respectively and ξ ∼ 0.1 is a numerical factor [20]. 0 120

4.2. Heat capacity 220

320

420

-1

f [cm ] Figure 3. FTIR spectra of PbPrTe at different temperatures: ♦, 25 K; X, 50 K; , 80 K. For comparison, we give the data for pure PbTe, (as ), which are almost temperature independent.

last observation is confirmed by the observation of the Mott law in ρ(T ) below 23.5 K (see inset in figure 2). Indeed, in the compounds for which the resistivity follows the Mott law, a hopping conduction [13] of the electron occurs with a temperature-dependent activation energy. Thus, in this case, we should have: n ρ(T ) = ρ0 e(T0 /T ) , (1) where T0 = β/(kB N(EF )a 3 ), and n is an exponent depending on the form of the density of states near the Fermi level. In the case of the Mott law, for which the density of states is assumed to be constant in the vicinity of the Fermi level, n = 41 . As we have seen in the inset of figure 2, ρ(T ) is described very well by equation (1) with n = 41 for T < 23.5 K, and if we modify the value of the n exponent, we cannot get a correct fit. The temperature range of the fitting is substantially broader and the fitting is much better than in the case of the − ln T temperature dependence. Thus, we can say that, below 23.5 K, we observe n = 41 without ambiguity in ρ(T ) for our sample. In order to evaluate the contribution of the Pr ions we have carried out measurements of the FTIR spectra at different temperatures, particularly below and above 50 K. From figure 3 one can clearly see that there appears a spectral maximum at a frequency of about 300 cm−1 . Comparison with PbTe (figure 3) unambiguously shows that the origin of the maximum is the Pr ions. Moreover, the minimum in the temperature dependence of the resistivity corresponds to the minimum of ρ(T ). So, one can say that a dominant role is played by the Pr ions in the observed temperature dependences and the presence or the proximity of the structural transition between the high-T cubic phase and the low-T rhombohedral phase in the solid solution Pb1−x Gex Te. In figure 4 we show the temperature dependence of the thermopower S(T ). It should be emphasized that in the regime of hopping conductivity with a variable hopping length the thermopower in Pb0.9835 Pr0.0165 Te has been predicted to have a square-root dependence on temperature (S(T ) ∝ T 1/2 ), and this has been confirmed experimentally [19, 20].

The heat capacity data below 10 K are plotted as CP /T versus T 2 in figure 5. Below 5 K, CP (T ) is well described by equation (4). This result shows that PbTe like Pb1−x Prx Te, has a very small and not very well determined value of γ , as can be expected for a semiconductor, and a linear part β ≈ 0.377 mJ mole−1 K−4 , with θD ≈ 168 K [17]. Therefore, no significant magnetic signal has been found in the heat capacity of Pb1−x Prx Te. For most materials at low temperatures, the heat capacity can be expressed as: C = γ T + βT 3 .

(3)

Here, γ is the Summerfield coefficient or electronic contribution to the specific heat and β is the lowest-order lattice contribution. Thus, a plot of CP /T versus T 2 would yield a straight line with slope β and intercept γ . CP = γ + βT 2 . T

(4)

From β (in mJ mole−1 K−4 ), we can evaluate the Debye temperature by
 1944γ 1/3 , (5) D = β where N is the number of atoms per molecule. However, as we can see from figure 5, there is no magnetic contribution below 10 K to CP (T ) because CP (T ) of Pb0.9835 Pr0.0165 Te collapses on CP (T ) for PbTe. The Summerfield coefficient also is almost equal to zero, as for pure PbTe, which implies that we do not have a sufficient number of charge carriers to observe the ‘magnetic’ Kondo effect, even if our material is heavily doped (x ≈ 1.65%). Moreover, as just mentioned in the introduction, the 1 singlet describes the ground state of Pr in PrTe [4] and one can assume the same thing in the case of Pr3+ -doped PbTe. In this case, we could not observe any ‘magnetic’ Kondo effect in Pr-doped PbTe. Magnetic susceptibility would be of interest for confirming that the 1 singlet state is the ground state of Pr3+ in Pr-doped PbTe. Therefore, this observation excludes a possibility that the low-T upturn in ρ(T ) is due to a ‘magnetic’ Kondo effect. For the Pb1−x Gex Te, Yaraneri et al [11] have been observed a Kondo effect. But in this case, the origin of the 969

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5

5

0 0

S( µV/K)

-5 -10

16K

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Pb1-xPrxTe x=1.65%

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2

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-30 -35 0

50

100

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250

300

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T (K)

CP /T (J/mole.K2)

Figure 4. Temperature dependence of the thermopower S(T ) of Pb0.9835 Pr0.0165 Te plotted as S(T ) versus T 3/2 . Inset : plot of S(T ) versus T 1/2 at low temperatures (see the text for details).

0.1

PbTe (ref.18) Pb1-xPrx Te (x=1.65%) Linear part Eq. 4

0.01

1E-3

1E-4 1

T 2 (K2)

10

100

Figure 5. Temperature dependence of specific heat for PbTe18 and Pb0.9835 Pr0.0165 Te.

dependence is the structural disorder [15, 16], and hence the role of the electron–electron interaction for the observed dependence is very crucial [3, 18]. Indeed, it was shown that in the system with strong disorder the interaction favours delocalization because electrons may help each other to overcome the random potential. This interaction can form a ‘soft’ gap in the density of states near the Fermi level, which is manifest in the anomalous-like temperature behaviour of the transport properties. In the particular case of Pb1−x Gex Te with low Ge concentration, the origin of the ‘structural’ Kondo effect is 970

primarily due to the small size of the Ge atoms compared to the lead atoms. Indeed, the small size of the Ge atoms implies that the Ge2+ ion has an off-site tunnelling motion, which is at the origin of the observed ‘structural’ Kondo effect in Pb1−x Gex Te. Yaraneri et al [11] have shown that the small size of the Ge2+ ion is a crucial factor for obtaining such a ‘structural’ Kondo effect at low temperature, because no Kondo effect has been observed in Pb1−x Snx Te, which is very similar to Pb1−x Gex Te except that the size of Sn2+ is substantially larger than that one of Ge2+ . Analogously, because Pr3+ has a size similar to Sn2+ , we suppose that the upturn at low temperature in ρ(T ) of Pb0.9835 Pr0.0165 Te is not due to the ‘structural’ Kondo effect. Earlier, a substantial magnetic signal in the specific heat was observed for Pb1−x Mnx Te with a maximum for T below 1 K [21]. However, in contrast to the case for the latter material, in Pb1−x Prx Te (present results) and in Pb1−x Ybx X [22] (where Yb is magnetic [22, 23] and X = S, Se or Te), the magnetic contribution is very weak compared to the contribution of the phonons. The signal found in Pb1−x Mnx Te (x = 0.024 and x = 0.056) has been attributed [21] to the bursting of Mn levels about 1 K by the sp–d exchange interaction, one based on a standard model in [24], although Escorne et al [25] have found the existence of a transition towards a spin-glass state in the same range of temperatures (below 0.25 and 1 K). If the model of Lusakowski et al were correct, we would observe a significant magnetic signal in Pb1−x REx X also (where RE is a magnetic rare earth) contrary to the present experimental data. In our opinion, the absence of any important magnetic signal in our samples confirms the prediction of Escorne et al [25], while at the same contradicting the model of Lusakowski et al [21], in explaining the magnetic contribution to specific heat in PbTe doped with magnetic ions.

Optical properties of semiconductors

Figure 6. Principal schema of physical mechanisms causing the photo-induced non-centrosymmetry.

4.3. Non-linear optics: PISHG, TPA and PIEOE One of the advantages of the photo-induced NLO experiments compared to transport ones consists in the possibility to perform measurements with macroscopically centrosymmetric materials, where generally optical SHG is forbidden by symmetry. At the same time, this method is particularly effective in the determination of the local non-centrosymmetry, which may be caused by electron–phonon or electron– paramagnon [26] local disturbances at low temperatures. Using the PISHG for macroscopically centrosymmetric (randomly disordered) materials is based on the possibility of photo-orientation of the excited state dipole momenta by a polarized photo-inducing beam. The latter due to interaction with locally non-centrosymmetric electron–boson sub-systems (like phonon or magnons) gives the output nonzero polarization of the matter existing during the short time (several tenths of picoseconds) of photo-inducing excitation. Generally, under the influence of power light beams, this excited system may be non-centrosymmetric due to the superposition of anharmonic electron–phonon interactions described by a third rank polar tensor. Particularly due to the excitation of a large number of phonons through piezoelectric and electrostricted interactions, there seems to be an additional possibility for operation with a sub-system of the excited dipole momenta (see figure 6). From figure 6 one can see that polarized pumping light causes the excited state dipoles to become oriented (photoalignment) leaving the ground states unchanged due to restrained mechanical stiffness. So we deal with interactions of the excited dipole moments which are temporarily occupied by photo-induced carriers with phonons. The role of the localized Pr levels is of great importance. The latter demonstrates substantial temperature dependence near the phase transition

temperature and critical points. So, one can expect the occurrence in the corresponding temperature dependences of the anomalies in the optical susceptibilities. Particularly, near the critical points, such temperature dependences could demonstrate several thermally singular dependences. The PISHG is strongly related to the excited dipole momenta sub-systems [8]. The estimations of temperature behaviour of the NLO susceptibilities were done using the following phenomenological formalism. The polarizability of the medium involves two terms—linear and non-linear [27]: Pi = PiL + PiNL = αij Ej(ω) + βij k Ej(ω) Ek(ω) + γij kl Ej(ω) Ek(ω) El(ω) , PiL = αij Ej(ω) ,

(6) (7)

PiNL = βij k Ej(ω) Ek(ω) + γij kl Ej(ω) Ek(ω) El(ω) , where αij , βij k , γij kl are microscopic susceptibilities (hyperpolarizabilites), which are related to the macroscopic susceptibility by the following equations: (ω) χij(ω) = L(ω) i Lj αij ,

(ω) (ω) (ω) χij(ω) k = Li Lj Lk βij k ,

(ω) (ω) (ω) (ω) χij(ω) kl = Li Lj Lk Ll γij kl ,  2  nz − 2 1 L(ω) = . z n2z + 3 ρ

(8)

Here, the indices i, j, k correspond to crystallographic components of the crystal; ρ is a density of the medium. χij(ω) k is the macroscopic second-order optical susceptibility responsible for the PISHG, and χij(ω) kl is the macroscopic thirdorder non-linear susceptibility responsible for the TPA. 971

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β [cm/GW]

15 10 5 0

5

30

55 T[K]

80

105

Figure 8. Temperature dependence of the TPA of Pb0.9835 Pr0.0165 Te—; pure PbTe— .

potential γij k = ∂ 3 U /(∂xi ∂xj ∂xk ), or Figure 7. Temperature dependent PISHG of Pb0.9835 Pr0.0165 Te caused by different pumping lasers: •—nitrogen laser; —doubled frequency harmonic of the Nd–YAG laser.

Using the two-band and three-band model approach, we can present the microscopic hyperpolarizability within a framework of two- and three-level models: j µ iµ , αij ∼ = Eg2

 j µ k µ iµ βij k ∼ , = Eg3

 j µ  k µ l µ iµ . γij kl ∼ = Eg4

(9)

 k,l = µ  (ex) Here, µ  i,j are transition dipole moments and µ k,l − (gr) µ  k,l are the differences between excited and ground state dipole moments. For the PISHG, an important term is the first-order hyperpolarizability βij k , which is crucially dependent on the excited optically aligned dipole moments. The latter is sensitive to the long-range phonon temperature dependences and defines the observed temperature dependences in the macroscopic susceptibility. Simultaneously, the conductivity of a free electron may be described as ne2 l¯ , (10) σ = V¯  where l¯ = τ¯ · V¯ is the free length path, V¯ = 3kβ T /m is the average velocity and n is the electron density. The dipole moment may be expressed as l · e = µ,  So, σ =

n · e · e · l¯ n·e · µ.  = V¯ 3mkβ T

(11)

The dipole moment has two contributions, µ  =µ  el + µ  ph , where µ  el is the dipole moment’s electronic part, which is less dependent on temperature, and µ  ph is the phonon part, which is more sensitive to the temperature as bosons are sensitive to the anharmonic electron–phonon interactions. The latter term is described by third-order derivatives of the anharmonic 972

U=

1 1 1 αij x 2 − βij k x 3 − γij kl x 4 . 2! 3! 4!

(12)

This anharmonic interaction gives an additional contribution near the critical points, which is described by the expression µ  ph ≈

γij k . (T − Tc )ξ

(13)

Here, Tc corresponds to the phase transition (critical) temperature in the case of superconductors, magnetically ordered compounds, etc [10, 27]. In our case, Tc should ρ correspond to the temperature Tmin where the metal– semiconductor transition occurs. Now, let us consider the experimental results of non-linear optics. In figure 7, we present the experimentally measured temperature dependence of the PISHG signals for Pb0.9835 Pr0.0165 Te obtained by two different lasers. We can see that for the nitrogen laser with a penetration length of about 20 nm the maximum of the PISHG signal appears below 35 K ρ (and therefore below Tmin ) and reaches two maxima around 17 and 30 K. In the case of the 530 nm pumping wavelength, where the penetrating wavelength is about 85 nm, the position of the maximum is substantially shifted towards higher temperatures and we do not observe a splitting. This fact may indicate different contributions of the near-the-surface and bulk states to the observed optical susceptibilities. We have also performed experiments for the pure PbTe sample corresponding to our reference sample. No PISHG signal was detected in PbTe. This results in agreement with its centrosymmetric structure and with the absence of any critical points in the pure PbTe sample. But we find a very small signal of TPA (see figure 8) compared to that found in Pr-doped PbTe. From figure 8, one can clearly see that ρ near 50 K (∼ =Tmin ) there exists a clear maximum for the TPA. Differences in the tensors causes the different temperature shapes of the dependences observed. This may be explained by the relation between the mobility of carriers and dipole moments, and is confirmed by the large difference between the PISHG, which is a NLO effect much more sensitive to the surface effects, and the TPA, which is more sensitive to the bulk. Comparing the temperature dependences obtained for the transport properties (see figures 2–5) one can clearly see that

Optical properties of semiconductors

the maximum observed in the bulk-sensitive NLO effect (TPA) ρ corresponds to Tmin and, therefore, to the temperature where the metal–semiconductor transition occurs. This is exactly the behaviour expected from the phenomenological theory of the NLO effect developed above to describe the effect of a phase transition on the temperature dependence of the NLO effect. It is important to note that the temperature resolutions obtained from the NLO susceptibilities obtained using the bulk-sensitive NLO methods (figures 7 and 8) are substantially higher than that obtained from direct transport experiments. TPA PIEOE Indeed, we find Tmax = 49(±1) K and Tmax = 51(±1) K, ρ compared with Tmin = 50(±10) K. This better resolution of the NLO effect may be a consequence of the crucial role played by the photoexcited dipole moments in the NLO effect (see figure 6). Their interaction with the ‘soft’ bosons (phonons and maybe also plasmons) would be substantially higher than for carriers determining transport effects, particularly mobility, thermoconductivity, etc. So, one can say that the photo-induced NLO methods may be considered promising for investigations of semiconductor–metal transitions. Another important issue concerns the possibility of separation of bulk-like and near-the-surface contributions. From figures 7 and 8 one can see that for near-the-surface states measured by the PISHG, we observe two temperature maxima shifted to at least 30 K towards low temperatures. So one can expect that the near-the-surface (30 nm in depth) phonons may be split near the surface due to differences in the response to the border conditions. At the same time, for the bulk effects, we see only one maximum at about 50 K, which may be a consequence of the superposition of these two contributions. Increase in the penetration depth due to the use of another pumping laser, shifts the position of the temperature maximum towards a temperature typical for the bulk-like materials.

5. Conclusions We report transport and heat capacity measurements of a Pr-doped semiconductor (here Pb1−x Prx Te with x = 0.0165) for the first time. No magnetic signal was found in the heat capacity. This observation permits us to interpret the presence of the minimum around 50 K in the temperature dependence of the electrical resistivity ρ(T ) as being due to the metal– semiconductor transition in Pb1−x Prx Te. The low-T upturn observed in ρ(T ) follows very well the Mott law for VRH conductivity. The low-T thermal variation of the thermopower S(T ) agrees with this observation. We also discussed the experimental temperature measurements of photo-induced non-linear optics of third- and second-order in order to better investigate the semiconductor– metal phase transformations near the temperature of the phase transition. The results found agree sufficiently well for the third-order NLO effect with the phenomenological theory developed and we have shown a higher sensitivity of the photo-induced methods to the phase transformation compared to the temperature dependence of the transport properties (ρ(T ), S(T )). Moreover, using photo-induced non-linear

optical methods as a tool for phase transition detection, we were able to separate bosonic and phononic contributions into bulk and near-the-surface states

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