Crystal Structure of Nanostructured PbS Films at ... - Springer Link

ISSN 10637834, Physics of the Solid State, 2009, Vol. 51, No. 11, pp. 2375–2383. © Pleiades Publishing, Ltd., 2009. Original Russian Text © S.I. Sadovnikov, A.A. Rempel, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 11, pp. 2237–2245.

LOWDIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Crystal Structure of Nanostructured PbS Films at Temperatures of 293–423 K S. I. Sadovnikov* and A. A. Rempel Institute of Solid State Chemistry, Ural Division, Russian Academy of Sciences, ul. Pervomaіskaya 91, Yekaterinburg, 620041 Russia * email: [email protected] Received December 29, 2008; in final form, February 27, 2009

Abstract—The crystal structure of lead sulfide films prepared by the hydrochemical deposition has been stud ied by Xray diffraction. The thickness of the films synthesized is ~100 nm, the size of coherent scattering regions is ~70 nm, and the value of microstrains is ~0.20%. It is established, for the first time, that the as synthesized PbS films and the same films annealed in the temperature range 293–423 K have a cubic crystal structure (space group Fm3m) different from the B1type structure. In the crystal lattice of the structure revealed, sulfur atoms are located not only in the 4(b) positions but also in the 8(c) positions. The occupancies of the 4(b) and 8(c) positions by the S atoms are ~0.84 and 0.08, respectively. PACS numbers: 61.46.w, 64.70.Nd, 68.55.a, 81.15.Lm DOI: 10.1134/S1063783409110298

1. INTRODUCTION Under standard conditions, lead sulfide is a nar rowbandgap semiconductor (band gap is 0.4 eV; direct transitions) and has a cubic B1 structure. Owing to the narrow band gap and a high photosensitivity in the infrared range, lead sulfide and coatings based on it are well studied and used as temperaturesensitive sensors, detectors in the infrared spectral range (from 850 to 3100 nm), photomultipliers, photoresistors, and selective sensors based on lead ions [1–3]. As the leadsulfide grain sizes decrease to several tens of nanometers and lower, its properties are sub stantially changed. This fact provided an enhanced interest in PbS, since films based on nanocrystalline lead sulfide can markedly differ from the thick coarse grained films. The use of nanocrystalline PbS gives possibilities of improving existing devices and opens prospects of designing more sensitive IR sensors, detectors, and pickups. It is known that the transition of sulfides to nano structured state is accompanied by not only a transfor mation of their properties induced by the size effects, but also structural changes. For example, cadmium sulfide as single crystals and coarsegrained powders have the crystal structure of the hexagonal wurtzite (type B4) [4] and cubic sphalerite (type B3) [5, 6]. However, the CdS structures in thin films, as well as in nanodisperse and ultradisperse powders, do not coin cide with the crystal structures of the two modifica tions. Recently, it was shown [7, 8] that CdS nanopar ticles have a specific disordered structure with a ran dom alternating of closepacked atomic planes, and

the mean lattice of such a disordered closepacked structure is described by space group P6/mmm. Information on the crystal structure of leadsulfide films is ambiguous. It is generally believed that PbS films have the B1type cubic structure (space group Fm3m ). The study of PbS films performed in [9] showed that, as the temperature increases to 375 K, the relative intensity of the (220) diffraction reflection increases as compared to that of the (200) reflection. This suggested that, in the PbS film at 375 K, a weak phase transition from the B1type structure to the B3 type cubic structure (space group F43m) occurs; in this case, at a temperature of 300 K, the phase with the B3 structure is metastable and can coexist with the phase having the B1type structure. The assumption on the PbSB1 PbSB3 transition at a temperature of 375 K implies that the phase with the B3 structure is higher temperature compared to the phase with the B1 struc ture and, thus, it must remain on heating. However, even at a temperature of 475 K, the ratio of the inten sities of the (200) and (220) reflections is reversed, which is in contradiction with the assumption made in [9]. Moreover, the authors of [9] assumed that a part of the Pb atoms occupy the 4(b) position with the coor dinates (1/2 1/2 1/2), but such positions are generally absent in the B3type structure, however, they exist in space group F43m . To conserve the sulfide composi tion, the occupancies of the 4(a) and 4(b) positions were taken equal to 0.75 and 0.25, respectively. Thus, in [9], the PbSfilm structure is cubic but it does not belong to the B3 type, as was mistakenly assumed in [9].

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SADOVNIKOV, REMPEL

The B1 and B3type structures are cubic and give the same set of reflections in a diffraction experiment; thus, with the lattice periods being equal, the elucida tion of the existence of one or other structure or of both the structures is possible only by a quantitative analysis of the ratio of the reflection intensities. It is a very complex problem which needs the measurements of Xray diffraction patterns with a high accumulation of the signal and high resolution. Additional hamper ing factor is the broadening of the reflections due to a small crystallite size in the films. In this connection, in this work, we performed the in situ studies of the effect of temperature on a change in the structure of nanocrystalline PbS films with inclusion of the value of microstrains and crystallite size in the films. 2. SAMPLE PREPARATION AND EXPERIMENTAL TECHNIQUE The lead sulfide films were chemically deposited from aqueous solutions on glass substrates. The chem ical deposition of sulfides differs from other film syn thesis methods (electrodeposition, spray pyrolysis, and microwave heating) in that it does not require complex experimental equipment and allows one to obtain highquality coatings with a large area. The hydrochemical deposition of PbS was realized using the reaction between watersoluble lead acetate Pb(OAc)2 and diamide of thiocarbonic acid (NH2)2CS which plays a part of sulfidizer in an alkaline solution containing sodium citrate Na3Cit and hydroxide NaOH. As a result of hydrolysis of N2H4CS, sulfide ions appear in the solution, and the sulfide formation and deposition begin [10].The deposition reaction was carried out in a TZhTS01 liquid thermostat in aque ous alkaline solutions of the Pb(OAc)2–Na3Cit– NaOH–(NH2)2CS at pH = 12 and T = 325 K. The value of pH was measured by an Eutech Instruments SyberScan 2100 pH/Ion Meter with the CyberComm Pro DAS Software. The initial concentrations of the reagents were as follows: [Pb(AcO)2] = 0.005 mol/l, [(NH2)2CS] = 0.025 mol/l, and [Na3Cit] = 0.025 mol/l. The ~100nmthick lead sulfide films prepared were annealed in air at a temperature from 293 to 423 K with a temperature step of 30–40 K. The con trol of the chemical composition of the film shows that it is not changed after annealing, i.e. no oxidation occurred. After heating to 423 K, the film was cooled in air to 293 K, then, it was again heated to 393 K to state the reproducibility of the results. During the annealing experiments, a change in the structure of the nanocrystalline PbS film was controlled using an in situ Xray diffraction analysis. The Xray diffraction measurements of the films were performed using a Philips X’Pert automated diffractometer (CuK α1, 2

radiation) in the range of 2θ angles from 18° to 90° with a step Δ(2θ) = 0.016 deg/s and large exposure time ~500 s in each point. During the experiment, the voltage and current were 40 kV and 35 mA, respec tively. The Philips X’Pert diffractometer was equipped with a X’Celerator positionsensitive high speed sector detector which is an integral device of several parallel detectors [11, 12]. Owing to this, the X’Celerator detector measures the reflection inten sity in the 7.2°wide range of the 2θ angles, not in a point, as a common proportional counterdetector does. For example, if the measurement starts from 18°, at the initial moment, the detector captures the angle range 14.4°–21.6° and begins to move to a range of larger angles. By the time when the detector scans the reflection intensity in the angular range from 18.0° to 25.2°, the exposure time in the 18° point with a step of 0.016° is even 450 s. As a result, the duration of measuring the Xray diffraction of the film was decreased approximately by a factor of 100 (from 600– 700 h as a common detector is used to 7–8 h with the X’Celerator detector) without any loss of the resolu tion quality. The determination of the phase composi tion of the samples and crystallattice parameters of various phases and the finished refinement of the film structures were performed using an X’Pert Plus pro gram [13]. The profiles of the Xray diffraction reflec tions were described using the pseudoVoigt function 2 –1

( θ – θ0 ) V ( θ ) = cA 1 + 2 θL

(1) 2

( θ – θ0 ) + ( 1 – c )A exp – , 2 2θ G which is a superposition of the Lorentzian function (first term) and Gaussian function (second term). In relationship (1), c is the relative contribution of the Lorentzian function to the total reflection intensity, θL and θG are the parameters of the Lorentz and Gauss distributions, respectively; A is the normalization fac tor of intensity, and θ0 is the position of the pseudo Voigt function maximum. As the Xray diffraction spectrum was measured using the radiation with two wavelengths (Cu K α1 and Cu K α2 ), each of the reflec tions was a doublet which described using the two pseudoVoigt functions (1). If in the pseudoVoigt function θL = θG = b and θ0 = 0, the exact value of the full width of the reflec tion at the half maximum (FWHM) is determined at any value of the parameter b by solution of nonlinear equation (1) when V(θ) = A/2. Eq. (1) has no any exact solution; however, as shown in [14, 15], a good approximation of the solution is the relationship 2

FWHM ( 2θ ) = b ( 2.355 – 0.276c – 0.079c ).

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CRYSTAL STRUCTURE OF NANOSTRUCTURED PbS FILMS

2377

200

30.0

30.4

30.8

400

331 420

422

β*(2θ), nm–1

29.6 311 222

220

200

111

0.06

0.04

0.02

1 2 3

Counts

293 K (after quenching from 393 K) 393 K

0

2

4 6 s = 2(sinθ)/λ, nm–1

8

293 K (after quenching from 423 K) Fig. 2. Dependences of the reduced broadening β*(2θ) of diffraction reflections on the scattering vector s: (1) initial nanocrystalline lead sulfide film, (2) film at a temperature of 423 K, and (3) film after cooling to 293 K. Straight lines show the linear approximation of the β*(s) dependence.

423 K 383 K 358 K 323 K asprepared 30

40

50 2θ, deg

60

70

80

Fig. 1. Xray diffraction patterns of the nanocrystalline film of sulfide PbS prepared in situ at different annealing temperatures. The inset shows the systematic shift of the (200) reflection with an increase in the temperature. The films were deposited onto the glass substrate (Cu K α 1, 2

radiation). Xray diffraction patterns are given on the same scale.

The experimental Xray diffraction reflections of the films are markedly broadened. The broadening was determined by comparison of the width of each reflection with the instrumental width of this reflec tion or, in other words, with the resolution function FWHMR of the Xray diffractometer β ( 2θ ) =

2

2

( FWHM exp ) – ( FWHM R ) .

(3)

The resolution function 2

FWHM R ( 2θ ) = ( u tan θ + v tan θ + w )

1/2

(4)

for the Philips X’Pert Xray diffractometer was deter mined during a special Xray diffraction experiment on cubic lanthanum hexaboride LaB6 (NIST Standard Reference Powder 660a) with the lattice parameter a = 0.41569162 nm. In the wellannealed and homoge neous coarsegrained LaB6 powder with an average particle size of 5 μm, there are no reasons causing the physical broadening of the Xray diffraction reflec PHYSICS OF THE SOLID STATE

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tions (small particle size, microstrains, inhomogeneity [16]) and there is only the instrumental broadening of the diffraction reflections. An approximation of the experimental FWHMR(2θ) dependence gives the parameters of the instrumental function of the angular resolution of the Philips X’Pert diffractometer as fol lows: u = 0.0023, v = 0.0075, and w = 0.0070. 3. MICROSTRUCTURE OF THE PbS FILMS The Xray diffraction patterns of the PbS nano structured film measured in situ at various tempera tures are shown in Fig. 1. The comparison of the experimental Xray diffraction patterns with the instrumental resolution function (4) shows that all the observed diffraction reflections are noticeably broad ened. The broadening β(2θ) was found by relation ship (3). With known value of β(2θ), the average size of the coherent scattering region ( 〈 D〉 ) and microstrain ε can be estimated using the Williamson–Hall method [17, 18]. To do this, the dependence of the reduced broadening β*(2θ) = [β(2θ)cosθ]/λ on the scattering vector length s = 2(sinθ)/λ is constructed. The average size of the coherent scattering region, which can be considered, as a first approximation, as the particle size, is found as 〈 D〉 = 1/[β*(2θ)]s = 0, extrapolating the β*(s) dependence to the value s = 0. The micros train is determined from the slope angle ϕ of the β*(s) dependence as ε = [(tanϕ)/2] × 100% [14–16]. Figure 2 shows the dependence of the reduced broadening β*(2θ) on the scattering vector length s of the initial lead sulfide film, the same film at a temper ature of 423 K, and after decrease in temperature to 293 K. Taking into account the error of determination of FWHMexp, the reduced broadening was calculated accurate to ±0.005, and the error of determination of 2009

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SADOVNIKOV, REMPEL

0.596 PbS film, 293K, a = 0.59395 nm PbS film, 423K, a = 0.59513 nm

0.595

a, nm

a, nm

0.594 0.592 0.590

0.594 1 2 3 4 0.593 280

0.588

1.0 1.5 2.0 0.5 0.5[cos2θ/sinθ + cos2θ/θ]

0

400

Fig. 4. Influence of the temperature on the lattice param eter a of the nanocrystalline lead sulfide film: (1) variation in the parameter a as the temperature increases from 293 to 423 K, (2) the lattice parameter measured at 293 K after cooling of the film and the temperature of the test heating equal to 393 K, (3) lattice parameter measured at 293 K after cooling from the test heating temperature, and (4) lattice parameter measured at 293 K within six months after annealings of the film.

the average particle size is ±10 nm at all temperatures under study. It is seen from Fig. 2 that the average par ticle size increases with temperature from 70 nm (ini tial state) to 175 nm (423 K). As the temperature decreases from 423 to 293 K, the particle size is unchanged; however, it somewhat increases during the subsequent heating (Table 1). The microstrain is ε = 0.20 ± 0.05% in the initial nanocrystalline PbS film and increases to ε = 0.30 ± 0.05% at 383 K. The subsequent change in tempera ture does not influence the microstrains. It is known that the microstrains in coarsecrystalline materials decrease as the annealing temperature increases. The conservation of the unchanged microstrains in the PbS nanostructured film means that the film anneal Table 1. Changes in the particle sizes of the nanostructured PbS film at temperatures of 293–423 K

293 323 358 383 423 293 393 293

360 T, K

2.5

Fig. 3. Determination of the lattice parameter of the nanocrystalline films of the lead sulfide at temperatures of 293 and 423 K by the extrapolation to an angle of 90° using the Nelson–Riley function 0.5[cos2θ/sinθ + cos2θ/θ]. Dashed lines show the 95% confidence interval of deter mination of the temperature dependence a(T).

T, K

320

〈D〉, nm 70 100 135 170 175 180* 200** 200***

*Particle size after the decrease in the temperature from 423 to 293 K. **Particle size after the repeated increase in the temperature from 293 to 393 K. ***Particle size after cooling from 393 to 293 K.

ing in the temperature range under study does not influence the source of microstrains which is most likely related to the specific features of the structure. In what follows, we discuss these specific features. 4. SPECIFIC FEATURES OF THE CRYSTAL STRUCTURE OF THE PbS FILMS The approximation of the Xray diffraction reflec tions by function (1) allowed us to exactly determine the position of maximum θ0 of each reflection. According to the Wulff–Bragg equation, from θ0 found, we calculated the interplanar distances dhkl and, for each of the reflections, the cubic crystal lattice 2

2

2

parameter a = d hkl h + k + l . The systematic error of determination of the parameter a associated with nonideality of adjusting the sample during measuring of the Xray diffraction pattern was leveled using the Nelson–Riley extrapolation function 0.5[cos2θ/sinθ + cos2θ/θ] [19]. The extrapolation to the angle θ = 90° shows that the lead sulfide has the lattice parameter a = 0.59395 ± 0.00015 nm at room temperature and a = 0.59513 ± 0.00015 nm at 423 K. (Fig. 3). The change in the lattice parameter on heating shifts the maxima of the Xray diffraction reflections. As an example, the inset to Fig. 1 shows the change in the position of the (200) Xray diffraction reflection of the film of the nanocrystalline lead sulfide as temperature increases. Thus, the lead sulfide has, at a temperature of 423 K, the lattice parameter which is approximately 1.2 pm larger than that at room temperature. Figure 4 shows the temperature dependence of the lattice parameter: taking into account the measure ment error, the a(T) dependence can be taken linear


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over the temperature range from 293 to 423 K. According to [20], the thermal expansion coefficient of the polycrystalline lead sulfide at 300 K is α = 20 × 10–6 K–1, and the same coefficient α calculated for the PbS film in this work is 14 × 10–6 K–1. The difference in the values of the coefficients α is most likely due to the measurement method: the coefficient α available from the literature was obtained by a direct measure ment of the thermal expansion of a PbS sample; in this work, the value of α is found by a change in the lattice parameter. As is seen from Fig. 4, after heating to 423 K and cooling to 293 K, the lattice parameter decreases to a = 0.59326 nm which is markedly less then the lattice parameter a = 0.59395 nm of the initial film. The sub sequent (after cooling to 293 K) control increase in temperature to 393 K increases the lattice parameter to a = 0.59492 nm, i.e., by Δa = 1.7 pm. Such a large change in the lattice parameter of PbS as the temper ature increases was revealed for the first time. Note that the lattice parameter of the initial nanocrystalline film of lead sulfide measured after six months after the study of its thermal stability, increases from 0.59326 to 0.59395 nm (Fig. 4), i.e., to the value equal to the lat tice parameter of the initial assynthesized nanocrys talline film. This allows the assumption that the initial state of the film is the equilibrium state at a tempera ture of 293 K. To elucidate the reasons of the observed changes in the lattice parameter, we performed additional analy sis of the data on the intensity of the Xray diffraction reflections that permits the estimate of the possibility of the phase transition from the cubic B1type crystal structure to the cubic B3type structure or to an inter mediate state, in which the proportion of PbS with the B1type structure is y and the proportion of PbS with the B3type structure is (1 – y). In the general case, the measured intensity of the ith structure reflection (hkl) is [21–23] 2

I i = KF hkl P hkl PLG ( θ )f T ,

and, in the trigonometric form, has a general expres sion 2

F hkl = [ f j cos [ 2π ( x j h + y j k + z j l ) ] ] +

2

F B1 = { f Pb [ 1 + cos π(h + k) + cos π(h + l) + cos π ( k + l ) ] + f S [ cos π ( h + k + l ) 2

In the unit cell of PbS with the B3 structure (space group F43m ), four lead atoms occupy positions 4(a) with the same coordinates which they occupy in the B1 structure, namely, (000), (1/2 1/2 0), (1/2 0 1/2), and (0 1/2 1/2) and four sulfur atoms occupy positions 4(c) with the coordinates (1/4 1/4 1/4), (3/4 3/4 1/4), (3/4 1/4 3/4), and (1/4 3/4 3/4). According to the foregoing, the structural factor of lead sulfide with the B3 structure is given by ⎧ 2 F B3 = ⎨ f Pb [ 1 + cos π ( h + k ) + cos π ( h + l ) ⎩ π + cos π ( k + l ) ] + f S cos ( h + k + l ) 2 π π + cos ( 3h + 3k + l ) + cos ( 3h + k + 3l ) 2 2 2

⎫ + cos π ( h + 3k + 3l ) ⎬ 2 ⎭

2

The structural factor F hkl appearing in relation ship (5) is the square of the structural amplitude i

j

j

j

2 π π + f S sin ( h + k + l ) + sin ( 3h + 3k + l ) 2 2 2 π π + sin ( 3h + k + 3l ) + sin ( h + 3k + 3l ) . 2 2

(6)

j


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+ cos πh + cos πk + cos πl ] } .

is variable

∑ f exp [ –i2π ( x h + y k + z l ) ]

(7)

where xj, yj, and zj are the coordinates of the jth atom, and fj is the atomic scattering factor. Summation in relationships (6) and (7) is performed over atoms of the unit cell of the crystal structure under consider ation. The unit cell basis of lead sulfide with the B1 struc ture (space group Fm3m contains eight atoms among them four Pb atoms in positions 4(a) with the coordi nates (000), (1/2 1/2 0), (1/2 0 1/2), and (0 1/2 1/2) and four S atoms in positions 4(b) with the coordinates (1/2 1/2 1/2), (0 0 1/2), (0 1/2 0), and (1/2 0 0). If the atomic scattering factors of lead and sulfur atoms are fPb and fS, respectively, the structural factor of the B1 type structure calculated by relationship (7) has the form

where K is the instrument constant, is the struc tural factor, Phkl is the multiplicity factor, PLG(θ) is the angular factor of the intensity, and fT is the tempera ture factor. As the B1 and B3type structures and the assumed intermediate structure are cubic, their coefficients K, Phkl, PLG(θ), and fT are the same at other conditions

F hkl =

∑

f j sin [ 2π ( x j h + y j k + z j l ) ] ,

j

2 F hkl

being identical, and only the factor quantity.

2

2

(5)

2 F hkl

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If the film is twophase and the relative content in the film of the phase with the B1 structure is y and the phase with the B3 structure is (1 – y), the intensity of arbitrary reflection Ii is a superposition of the intensi ties of like reflections of the B1 and B3 structures. In such a twophase film, the sulfur atoms occupy the nonmetallic positions of the B1 structure with the probability of y and the nonmetallic positions of the B3 structure with the probability (1 – y). In this case, the structural factor is the superposition of the struc tural factors of the B1 and B3 phases and has the form ⎧ 2 F B1 + B3 = ⎨ f Pb [ 1 + cos π ( h + k ) + cos π ( h + l ) ⎩

∑

+ cos π ( k + l ) ] + yf S [ cos π ( h + k + l ) + cos πh + cos πk + cos πl ] π π + ( 1 – y )f S cos ( h + k + l ) + cos ( 3h + 3k + l ) (10) 2 2 2

⎫ π π + cos ( 3h + k + 3l ) + cos ( h + 3k + 3l ) ⎬ 2 2 ⎭

⎧ + ⎨ ( 1 – y )f S sin π ( h + k + l ) + sin π ( 3h + 3k + l ) 2 2 ⎩ 2

⎫ π π + sin ( 3h + k + 3l ) + sin ( h + 3k + 3l ) ⎬ . 2 2 ⎭ The atomic scattering factors of lead fPb and sulfur fS were calculated using the relationship f =

2

2

( f 0 + Δf ') + ( Δf '') ,

(11)

in which f0 is determined by the relationship sin θ⎞ = f 0 ≡ f ⎛ ⎝ λ ⎠

4

b 1 sin θ

⎞ + c. ∑ a exp ⎛⎝ – ⎠ λ i

2

intense and most suitable for comparison are the (111), (200), and (220) reflections. Thus, if, as the temperature is changed at other conditions being identical, the relative intensity of the (111) reflection increases with the relative intensity of the (220) reflec tion being unchanged, this demonstrates an increase in the content of the phase with the B3 structure in the film. The determination of the phase composition and crystal lattice parameters of possible cubic phases and final refinement of the structure of the PbS film corre sponding to various temperatures from 293 to 423 K were carried out using the X’Pert Plus program [13]. To estimate the validity of the structural models, we used the Rietveld factor of reliability [25] Rl = N I – I calc ( i ) / Ni + 1 I exp ( i ) , where Iexp(i) and i = 1 exp ( i ) Icalc(i) are the experimental and calculated intensities of the ith reflection, respectively. The minimization of the experimental Xray diffraction patterns in an approximation of twophase film gives y = 0.90 ± 0.02 and a better convergence RI(B1 + B3) = 0.04 than the minimization in an approximation in which the film contains the only phase with either the B1 or B3 struc ture (RI(B1) = 0.05 and RI(B3) = 0.12, respectively). Along with this, it follows from the minimization that the parameters of the phases with the B1 and B3 struc tures are absolutely equal. Physically, this is unlikely and indicates that the PbS film is singlephase, but its structure is similar to the B1 and B3 structures but it differs from them. The equilibrium structure of lead sulfide is the cubic B1type structure with space group Fm3m , and a dominant phase of the PbS film in the twophase model is also the phase with the B1 structure. Thus, we can assume that the real structure of the PbS film, as before, belongs to space group Fm3m , but its sulfur atoms are placed not only in the octahedral interstitial positions (in the 4(b) positions) but also in tetrahedral interstitial sites (in the 8(c) positions (Fig. 5). In such a structure of the PbS film, the occupation probability of positions 4(b) and 8(c) with S atoms are y and (1 – y)/2, respectively. The 8(c) positions of the cubic crys talline lattice with space group Fm3m have the follow ing coordinates: (1/4 1/4 1/4), (3/4 3/4 1/4), (3/4 1/4 3/4), (1/4 3/4 3/4), (3/4 3/4 3/4), (3/4 1/4 1/4), (1/4 3/4 1/4), and (1/4 1/4 3/4). Taking into account the coordinates of the 4(a) positions occupied with Pb atoms and the 4(b) and 8(c) positions occupied with S atoms with the proba bilities y and (1 – y)/2, respectively, the structure amplitude of the proposed cubic (space group Fm3m ) phase is

(12)

i=1

In relationships (11) and (12), the dispersion correc tions Δf ' and Δf '' for the calculation of the atomic scat tering function of the Xray CuKα radiation and the coefficients ai, bi, and c of lead and sulfur are taken from the International Tables for the XRay Crystal lography [24]. It follows from relationships (8)–(10) that the 2 structural factors F hkl of the (220), (400), and (422) reflections in the B1 and B3type structures are the same; the structural factors of the (111), (311), and (331) reflections in the B3 structure are larger than those in the B1 structure, and the structural factors of the (200), (222), and (420) reflections are smaller than those in the B1 structure. As seen from Fig. 1, the most

∑

F = f Pb { 1 + exp [ – iπ ( h + k ) ] + exp [ – iπ ( h + l ) ] + exp [ – iπ ( k + l ) ] } + yf S { exp [ – iπ ( h + k + l ) ]


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10000

(422)

20000

(331) (420)

30000

(400)

T = 293 K (220)

Counts

40000

(311) (222)

50000

2381

(111) (200)


0 –10000

Iobs – Icalc RI = 0.011

50000 T = 393 K

Counts

40000

0 –10000

(13)

50 60 2θ, deg

70

80

1, 2

+ exp [ – iπ ( h + 3k + 3l )/2 ]

2

⎫ + cos π ( h + 3k + l ) + cos π ( h + k + 3l ) ⎬ 2 2 ⎭

+ exp [ – iπ ( 3h + 3k + 3l )/2 ] + exp [ – iπ ( h + k + 3l )/2 ] + exp [ – iπ ( h + 3k + l )/2 ]

(14)

⎧1 – y π π + ⎨ f S sin ( h + k + l ) + sin ( 3h + 3k + l ) 2 2 2 ⎩

+ exp [ – iπ ( 3h + k + l )/2 ] }. With the known structure amplitude (13), we can easily obtain the structural factor F 2 of the cubic (space group Fm3m ) phase of the PbS film

π π + sin ( 3h + k + 3l ) + sin ( h + 3k + 3l ) 2 2

2

F = { f Pb [ 1 + cos π ( h + k ) + cos π ( h + l )

π π + sin ( 3h + 3k + 3l ) + sin ( 3h + k + l ) 2 2

+ cos π ( k + l ) ] + yf S [ cos π ( h + k + l )

2

⎫ π π + sin ( h + 3k + l ) + sin ( h + k + 3l ) ⎬ . 2 2 ⎭

+ cos πh + cos πk + cos πl ] 1–y π π + f S cos ( h + k + l ) + cos ( 3h + 3k + l ) 2 2 2 π π + cos ( 3h + k + 3l ) + cos ( h + 3k + 3l ) 2 2 π π + cos ( 3h + 3k + 3l ) + cos ( 3h + k + l ) 2 2 Vol. 51

40

Fig. 6. Experimental (crosses) and calculated (solid line) Xray diffraction patterns of the PbS films prepared in situ at 293 and 393 K. For clarity, only each third experimental point is shown. The differences (Iobs – Icalc) between the experimental and calculated Xray diffraction patterns are shown in the bottom parts. Cu K α radiation.

+ [ ( 1 – y )f S /2 ] { exp [ – iπ ( h + k + l )/2 ]


Iobs – Icalc RI = 0.017 30

+ exp ( – iπh ) + exp ( – iπk ) + exp ( – iπl ) }

+ exp [ – iπ ( 3h + k + 3l )/2 ]

20000 10000

Fig. 5. Positions of the lead atoms (closed circles) and sul fur atoms (open circles) in the PbSfilm structure (space group Fm3m ). The occupancies of positions 4(b) and 8(c) with the sulfur atoms are ~0.84 and ~0.08, respectively. The 8(c) positions are connected inside the unit cell with the dotdashed lines.

+ exp [ – iπ ( 3h + 3k + l )/2 ]

30000

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Figure 6 shows, as an example, the experimental Xray diffraction patterns of the PbS film measured in situ at temperatures of 293 and 393 K and the Xray diffraction patterns calculated in the approximation of the new cubic structure of the film, in which sulfur atoms are placed not only in the octahedral positions 4(b), but also in the tetrahedral positions 8(c). The 2009

2382

SADOVNIKOV, REMPEL

sulfur atom, at least one of the neighboring octahedral interstitial positions is unoccupied.

Counts

900 K

asprepared 20

40

60

80 100 2θ, deg

120

140

160

Fig. 7. Xray diffraction patterns of the synthesized nanoc rystalline PbS powder and the same powder after annealing at 900 K for 2 h. The powders contain only the phase with the B1 structure. Cu K α radiation. 1, 2

minimization of the experimental Xray diffraction patterns of the PbS films shows that, over entire tem perature range from 293 to 423 K, the occupancies of the 4(b) and 8(c) positions with S atoms are ~0.84 and ~0.08, respectively (Table 2). For all the Xray diffrac tion patterns, the convergence factor RI does not exceed 0.017. In the cubic (space group Fm3m ) structure of lead sulfide PbS, the radii of the octahedral and tetrahedral 2+ interstitial positions are rocta = a/2 – r Pb and rtetra = 2+

a 3/4 – r Pb , respectively. The lattice parameter of the PbS film studied is 0.5940 nm, the radii of the Pb2+ and S2– ions are 0.121 and 0.184 nm, respectively [26]. With inclusion of these results, the radii of the octahe dral and tetrahedral interstitial positions are ~0.176 2– and 0.136 nm, respectively. As r S > rtetra, the location of the S2– ion in the tetrahedral interstitial position significantly displaces four neighboring Pb atoms, which is observed experimentally: the microstrains ε in the film is from 0.20 to 0.30%. It is clear as well that, as the tetrahedral interstitial position is occupied with

Moreover, we studied the phase composition and crystal structure of the lead sulfide powder prepared by chemical deposition. The Xray diffraction patterns were measured for the initial (assynthesized) nanoc rystalline PbS powder and for the same powder after annealing at a temperature of 900 K for 2 h (Fig. 7). The annealing was performed in an evacuated quartz ampoule with a residual pressure of 10–4 Pa. As seen from Fig. 7, the Xray diffraction reflections of the ini tial powder are strongly broadened because of a small particle size, while, in the Xray diffraction pattern of the annealed powder, all the reflections are narrow, and its broadening is practically absent. The refine ment of the structure using the X’Pert Plus program shows that the initial nanocrystalline and annealed coarsegrained PbS powders contain the only cubic sulfide phase with the B1type structure, which is con firmed by the small factor RI (0.019 for the initial nanocrystalline powder and 0.024 fro the powder annealed at 900 K). Thus, the new cubic (space group Fm3m ) phase of lead sulfide exists only in the thin nanostructured films. 5. CONCLUSIONS The studies performed in this work show that the real structure of the nanostructured films of lead sul fide studied differs from the known B1type structure. The new structure revealed of the PbS films is cubic, belongs to space group Fm3m , and remains stable during the longterm annealing over the temperature range from 293 to 423 K. The specific feature of the revealed structure is the location of the sulfur atoms in the octahedral (positions 4(b)) and tetrahedral (posi tions 8(c)) interstitial positions of the facecentered cubic sublattice formed by the lead atoms. The occu pancies of the positions 4(b) and 8(c) with the sulfur atoms are ~0.84 and 0.08, respectively. The absence of the superstructure reflections demonstrates that the placement of the sulfur atoms in the positions of each type is statistical. ACKNOWLEDGMENTS

Table 2. Cubic (space group Fm3 m) structure of the nano structured PbS film at a temperature of 293 K (a = 0.59395 nm) Position Atom and mul tiplicity Pb S S

4(a) 4(b) 8(c)

x/a

y/b

z/c

Occu pancy

We are grateful to N.S. Kozhevnikova for her assis tance in synthesizing the films, A. Magerl for provid ing an opportunity to perform the diffraction experi ment, and A.I. Gusev for his participation in helpful discussions of the results obtained in this work.

0 0.5 0.25

0 0.5 0.25

0 0.5 0.25

1 0.84 0.08

This study was supported by the Russian Founda tion for Basic Research (project no. 090300039a) and the Presidium of the Russian Academy of Sci ences (program no. 27).

Atomic coordinates


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16. A. I. Gusev Nanomaterials, Nanostructures, and Nano technologies, 2nd ed. (Fizmatlit, Moscow, 2007) [in Russian]. 17. W. H. Hall, Proc. Phys. Soc., London, Sect. A 62, 741 (1949). 18. W. H. Hall and G. K. Williamson, Proc. Phys. Soc., London, Sect. B 64, 937 (1951). 19. J. B. Nelson and D. P. Riley, Proc. Phys. Soc., London 57, 160 (1945). 20. S. I. Novikova and N. Kh. Abrikosov, Fiz. Tverd. Tela (Leningrad) 5 (7), 1913 (1963) [Sov. Phys. Solid State 5 (7), 1397 (1963)]. 21. R. W. James, The Optical Principles of the Diffraction of XRays (Bell, London, 1954). 22. A. I. Gusev, A. A. Rempel, and A. A. Magerl, Disorder and Order in Strongly NonStoichiometric Compounds: Transition Metal Carbides, Nitrides, and Oxides (Springer, Berlin, 2001). 23. A. I. Gusev, NonStoichiometry, Disorder, ShortRange and LongRange Order in the Solid State (Fizmatlit, Moscow, 2007) [in Russian]. 24. A. A. Rempel’, Physics of the Solid State (Ural State Technical University–Ural Polytechnic Institute, Yekat erinburg, 2007) [in Russian]. 25. International Tables for XRay Crystallography, Vol. C: Mathematical, Physical, and Chemical Tables, Ed. by A. J. C. Wilson (Kluwer, Dordrecht, 1992). 26. H. M. Rietveld, J. Appl. Crystallogr. 2, 65 (1969). 27. L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, 3rd ed. (Cornell Univer sity Press, Ithaca, NY, United States, 1960).

Translated by Yu. Ryzhkov

2009