The band offsets of PbSe/Pb1xEuxSe multi-quantum wells grown by molecular beam epitaxy are determined as a function of temperature and europium content using temperature-modulated differential transmission spectroscopy. The confined quantum well states in the valence and conduction bands are analyzed using a k·p model with envelope function approximation. From the fit of the experimental data, the normalized conduction band offset is determined as 0.45±0.15 of the band gap difference, independently of Eu content up to 14% and temperature from 20 to 300 K.

Lead salt heterostructures have been extensively used for fabrication of coherent mid-infrared light sources such as buried stripe lasers,1,2 vertical cavity surface emitting lasers,3,4 microdisk lasers,5,6 photonic crystal lasers,7 as well as external cavity disk lasers with tunable emission and high output powers.8–11 In these devices, ternary lead salt alloys with Sr or Eu are usually employed as barriers for the quantum wells (QWs), providing a large tunability of the bands gaps Eg as required for band gap engineering. As a result, room temperature mid-infrared laser operation in cw-mode has been achieved at wavelengths up to 4.3μm.6 

For the design of these lasers, the band alignment between the wells and barrier materials is of crucial importance. It determines not only the energy levels but also the overlap of the electron and hole wave functions, and thus, the oscillator strengths of the optical transitions. While for the PbSe/PbSrSe12,13 and PbTe/PbEuTe14 systems, a type I alignment has been established at temperatures up to 300 K, controversial results have been reported for the PbSe/PbEuSe system, where some experiments were interpreted as giving evidence for a staggered type II band alignment,15,16 while others have indicated an almost symmetric type I configuration.17,18 Moreover, a pronounced temperature dependence, changing from type I at low temperatures to staggered type II above 180 K has been proposed.15,16 This has been used to explain the lower high-temperature performance of PbSe/PbEuSe lasers compared to those using PbSe/PbSrSe.11 Nevertheless, efficient above room temperature photoluminescence has been reported for both material systems19,20 and operation temperatures as high as 250 K have been achieved for PbSe/PbEuSe QW lasers.21,22

To resolve this issue, in this letter, we employ temperature modulated differential transmission spectroscopy to determine the band alignments of PbSe/Pb1xEuxSe multi-quantum wells (MQWs). From comparison with detailed envelope function calculations, we demonstrate a type I band alignment with a relative conduction band offset of ΔEc/ΔEg=0.45±0.15, independently of temperature and europium content up to xEu=14%. Thus, the reduction of the efficiency of PbSe/PbEuSe lasers at higher operating temperatures cannot be caused by a change in band alignment but must be of a different origin.

The samples were grown by molecular beam epitaxy onto 34μm thick, fully relaxed Pb1xEuxSe buffer layers predeposited onto (111) BaF2 substrates. The MQW structure consisted of 20–80 periods of PbSe quantum wells alternating with Pb1xEuxSe barrier layers with Eu content xEu equal to that of the buffer layers. A large series of samples was grown in which the QW thickness dQW varied over a wide range from 34 to 155 Å, while the barrier thickness was held constant at around 250-350 Å. The Eu content was varied from xEu=7.5% to 13.8%, which yields barrier band gap in the range of 390 to 575 meV at 77 K compared to 176 meV for PbSe. For the given Eu contents, the lattice mismatch between the wells and barrier layers is rather small (below 0.25%). Thus, pseudomorphic structures were obtained in all cases. All relevant sample parameters are listed in Table I.

Table I.

Structural parameters of the investigated PbSe/Pb1xEuxSe multi-quantum well samples derived from x-ray diffraction and FTIR measurements. dQW and dbarr represent the PbSe QW and PbEuSe barrier thicknesses, xEu the Eu content, ε||QW the in-plane QW strain, and N the number of MQW periods.

SampleVA378VA372VA399M1644VA592VA594
dQW (Å) 34 73 100 116 155 142 
dbar (Å) 253 272 310 345 291 265 
xEu(%) 9.3 8.8 9.0 7.0 13.3 11.6 
ε||QW(%) 0.16 0.14 0.26 0.13 0.21 0.22 
N periods 80 50 50 50 20 20 
SampleVA378VA372VA399M1644VA592VA594
dQW (Å) 34 73 100 116 155 142 
dbar (Å) 253 272 310 345 291 265 
xEu(%) 9.3 8.8 9.0 7.0 13.3 11.6 
ε||QW(%) 0.16 0.14 0.26 0.13 0.21 0.22 
N periods 80 50 50 50 20 20 

The structural properties of the MQWs were determined precisely by high-resolution x-ray diffraction. This is an essential prerequisite for exact theoretical modelling of the electronic transitions as required for band offset determination. The corresponding diffraction spectra of three representative samples with QW thicknesses of 34, 73, and 116 Å are shown in Figs. 1(a)–1(c), respectively. All spectra show sharp superlattice satellite peaks centered around the zero order “SL0” peak. This peak essentially coincides with the buffer peak “B” due to the close PbSe and PbEuSe lattice matching. By reciprocal space mapping around the asymmetric (264) reflection, all MQWs are found to be fully pseudomorphic to the buffer layers. From the superlattice period calculated from the satellite peak spacing and the Eu content xEu derived from infrared transmission measurements (see Fig. 2 below), precise values for the QW and barrier thicknesses were obtained as listed in Table I. Also listed are the measured in-plane strain values ε||QW in the QWs determined from reciprocal space mapping.

FIG. 1.

High-resolution (222) x-ray diffraction spectra of three PbSe/Pb1xEuxSe MQW samples, with different QW thicknesses of dQW=116, 73, and 34 Å and xEu = 7.0%, 8.8%, and 9.3% from (a) to (c), respectively. The peaks from buffer layer, substrate, and the superlattice stack are labeled by “B”, “S,” and “SL0,” respectively. The sample parameters derived from these measurements are listed in Table I.

FIG. 1.

High-resolution (222) x-ray diffraction spectra of three PbSe/Pb1xEuxSe MQW samples, with different QW thicknesses of dQW=116, 73, and 34 Å and xEu = 7.0%, 8.8%, and 9.3% from (a) to (c), respectively. The peaks from buffer layer, substrate, and the superlattice stack are labeled by “B”, “S,” and “SL0,” respectively. The sample parameters derived from these measurements are listed in Table I.

Close modal
FIG. 2.

Normalized differential transmission spectra ΔT/T of PbSe/PbEuSe MQWs with different QW thicknesses of 34, 73, 100, and 142 Å (from top to bottom) measured at T = 80 K (left) and 250 K (right). The peaks indicated by arrows and labeled by (ii)l,o correspond to the QW interband transitions in the longitudinal (l) and oblique (o) valleys. The dashed lines represent the theoretical spectra calculated using the matrix transfer method.

FIG. 2.

Normalized differential transmission spectra ΔT/T of PbSe/PbEuSe MQWs with different QW thicknesses of 34, 73, 100, and 142 Å (from top to bottom) measured at T = 80 K (left) and 250 K (right). The peaks indicated by arrows and labeled by (ii)l,o correspond to the QW interband transitions in the longitudinal (l) and oblique (o) valleys. The dashed lines represent the theoretical spectra calculated using the matrix transfer method.

Close modal

Fourier transform (FTIR) spectroscopy was used to measure the transmission of the samples as a function of temperature between 20 and 300 K. To reduce the strong Fabry Perot interference fringes caused by the large refractive index contrast between the lead salt layers and the BaF2 substrate,14 an impedance matching 470 Å thick NiCr anti-reflection coating was deposited on the top of the samples as described in Refs. 23 and 24. The fundamental band gap Eg(x,T) of the PbEuSe barriers was derived from the absorption edge in the FTIR spectra and its temperature as well as xEu-dependence was found to be in good agreement with the expression given by Maurice et al.25 

The transition energies between the confined quantum well states were determined from temperature modulated differential transmission spectra (DTS) ΔT/T. These were obtained by subtracting the transmission measured at two slightly different temperatures, where the temperature difference of 3 K was chosen such that the corresponding change in the PbSe band gap was about 1 meV. The resulting spectra are presented in Fig. 2 for four MQWs with dQW increasing from 34, 73, 100 to 143 Å from (a)–(d), respectively. In these spectra, sharp peaks corresponding to the QW interband transitions appear as indicated by the arrows. The peaks coincide exactly with the onset of the 2D QW absorption edges seen in the transmission spectra. For the thinnest QWs, only two peaks from the ground state (1–1) transitions are observed. With increasing QW thickness, a higher number of energy levels become confined and therefore, also the higher order (2–2) and even (3–3) transitions appear. This is shown by the lowest panel of Fig. 2 for the sample with dQW=155Å.

Due to the (111) growth direction and the anisotropic many valley band structure of PbSe with the band extrema at the L-points of the Brillouin zone,26–28 the energy levels in the quantum wells split up into two independent subband systems, one for the longitudinal valley (l) oriented along the [111] growth direction and one for the three oblique valleys (o) with the valley axes inclined by 70.5° to the growth direction. Due to the three-fold valley degeneracy, the peaks for the oblique valleys are three times higher than those of the longitudinal valley. In addition, they are blue-shifted to higher energies due to the smaller effective electron and hole masses in the quantization direction compared to those for the longitudinal valley. As can be seen in Fig. 2, the valley splitting ΔEol strongly depends on the QW thickness and quantum number i. For the ground state (1–1) transition, the splitting decreases from 44 to 15 meV at 80 K when the QW thickness increases from 34 to 155 Å, respectively. Likewise, the valley splitting strongly increases for the higher order transitions as shown e.g. in Figs. 2(e) and 2(g).

Comparing the spectra at 80 and 250 K (left and right hand sides of Fig. 2, respectively), we see that with increasing temperature the peaks shift to higher energies due to the concomitant increase of the band gaps Eg(x,T) as described by Eq. (1) below. In addition, the longitudinal and oblique ground state transitions increasingly overlap and at 250 K are no longer resolved for the thicker QWs. This results, on the one hand, from the reduced valley splitting at higher temperatures due to the increasing effective masses and, on the other hand, from the strong increase in the line widths due to thermal broadening. However, for the higher order transitions, the valley splitting is still easily resolved (cf. Fig. 2) due to the strong increase of the splitting with quantum number. Using the spectral dependence of the absorption coefficients and refractive indices of lead salt QWs and barriers as described in Ref. 14, the transmission spectra can be modelled by the transfer matrix method using the transition energies from the DTS peaks as input parameters.14,25 The resulting theoretical differential transmission spectra are shown as dashed red lines in Fig. 2, demonstrating that the experimental data can be nicely reproduced without any further fit parameters.

The measured interband transition energies of the samples with xEu10% are plotted in Fig. 3 for T = 80 K as a function of QW thickness. The longitudinal and oblique transitions are represented by the dots and diamonds, respectively, and the solid and dashed lines are the theoretical calculations based on the envelope function model described below and a band offset of 0.45. Evidently, for all samples, a good agreement between the experimental data and the calculation is obtained. As the transition energies rapidly decrease with increasing QW thickness, at dQW>60Å, the higher second order transitions appear in the calculations and the third level enters at dQW>100Å, in nice agreement with the experiments.

FIG. 3.

Transition energies of PbSe/PbEuSe MQWs as function of quantum well width at T = 80 K. Dots and diamonds: Experimental data for the longitudinal and oblique valleys, respectively, for the MQWs with 34, 73, 100, 116, and 142 Å well thicknesses and Eu content 10%. Solid and dashed lines: Calculations based on the envelope function model described in the text. Due to the effective mass anisotropy mt/ml=0.54 of PbSe, the transitions of the longitudinal valleys are always lower in energy than those of the oblique valleys.

FIG. 3.

Transition energies of PbSe/PbEuSe MQWs as function of quantum well width at T = 80 K. Dots and diamonds: Experimental data for the longitudinal and oblique valleys, respectively, for the MQWs with 34, 73, 100, 116, and 142 Å well thicknesses and Eu content 10%. Solid and dashed lines: Calculations based on the envelope function model described in the text. Due to the effective mass anisotropy mt/ml=0.54 of PbSe, the transitions of the longitudinal valleys are always lower in energy than those of the oblique valleys.

Close modal

To determine the PbSe/PbEuSe band alignment, an envelope function model was derived to calculate the QW energy levels as a function of band offsets, QW thickness, xEu content, and sample temperature. The confinement potentials V(z) for the electrons and holes are set by the conduction and valence band offsets, i.e., V(z) is set to zero within the QWs and equal to ΔEc=δ·ΔEg and ΔEv=(1δ)·ΔEg in the barriers for the conduction and valence bands, respectively, where δ=ΔEc/ΔEg is the normalized conduction band offset. Because of the temperature and xEu-dependence of the PbSe and PbEuSe band gaps25 given by

Eg(x,T)=146+0.475T2(13x)/[T+40.7]+3000x(meV),
(1)

the absolute values of the confining potentials V(z) depend on xEu as well as temperature T. In addition, the relative partitioning δ of the PbSe/PbEuSe band gap difference between the conduction and valence band can also vary in our model with temperature and Eu content.

For the resulting potential V(z) for electrons and holes, the energy levels in the QWs were calculated by solving the Schrödinger equation using the Hamilton operator29 

H(εn)=22ddz1mz(z,εn)ddz+h2kx22mx(z,εn)+h2ky22my(z,εn)+V(z).
(2)

To account for the strong band non-parabolicity of the lead salt compounds, energy dependent effective masses m(z,εn) derived from magneto-optical data and a six-band k·p EFA model25,30,31 were used. The corresponding longitudinal () and transversal () effective masses parallel and perpendicular, respectively, to the 111 directions are thus represented by

1m,c,v(z,|εnl,o|)=2m02P,2(z,x)(|εnl,o|±Eg,QW)+moml,t±(z).
(3)

In this equation, P, denotes the momentum matrix element parallel, respectively, normal to the 111 directions and ml,t±(z) denotes the far band contributions in the conduction (−) and valence (+) bands. Moreover, the temperature dependence of the effective masses is encoded in the temperature dependence of the PbSe band gap Eg,QW(T) as given by Eq. (1). Therefore, the Schrödinger equation must be solved self-consistently. It is noted that for the longitudinal valley, the confinement mass mzc,v in the growth direction is equal to mc,v, whereas for the oblique valleys mzc,v=9mc,vmc,v/(mc,v+8mc,v). All used band parameters are listed in Table II. For PbEuSe, these have been determined only for low europium contents xEu<0.8%.31 For higher Eu contents, we use the expression of

2P2/m0=1.963.98xEu(eV),
(4)

based on linear extrapolation of the data derived in Ref. 31, where P was found to be essentially independent of xEu.

Table II.

Band parameters of PbSe and PbEuSe at 4 K used for the envelope function calculations.

MaterialEg2P2m0PP||mtm0mlm0mt+m0ml+m0Refs.
PbSe 146.3 3.6 1.96 0.27 0.95 −0.29 −0.37 30 
PbEuSe Eq. (1) 3.6 Eq. (4) 0.27 0.95 −0.29 −0.37 25 and 31 
MaterialEg2P2m0PP||mtm0mlm0mt+m0ml+m0Refs.
PbSe 146.3 3.6 1.96 0.27 0.95 −0.29 −0.37 30 
PbEuSe Eq. (1) 3.6 Eq. (4) 0.27 0.95 −0.29 −0.37 25 and 31 

It is also noted that even though the strain in our QWs is rather small, it still has a noticeable effect on the confinement energies due to the deformation potentials.32 For the longitudinal valleys, the tensile QW strain ε||QW0.2% (see Table I) leads to a slight increase of the longitudinal band gap, and thus, the (ii)l QW transition energies increase by about 5 meV. We take this influence into account in our calculations using the expressions derived in Refs. 24 and 32, as will be described in detail elsewhere. The strain effect on the oblique valleys is negligible for the given strain values in our samples.

Using the parameters of Tables I and II, the transition energies of all samples were calculated as a function of the band offset parameter δ=ΔEc/ΔEg. The results are presented in Fig. 4 for the samples with dQW=100 and 155 Å and a temperature of T = 20 K (left) and 300 K, respectively, 250 K (right). Evidently, the energies of the strongly confined ground state transitions do not depend much on the band offset value. Thus, these do not yield much information on the actual band alignment. On the contrary, the higher order transitions show a pronounced maximum at δ=0.5 where the band alignment is nearly symmetric. Thus, the transitions near the barrier band gap are most sensitive to changes in band alignment. Comparing the theoretical calculations with the measured experimental values of Fig. 2, represented as solid horizontal lines in Fig. 4, clearly shows that the best agreement is found for ΔEc/ΔEg=0.45±0.15, independent of QW thickness and Eu content. Most importantly, the same band alignment of δ=0.45 is found for all samples and the whole temperature range from 20 to 300 K, demonstrating that no band alignment change occurs as a function of temperature in this material system.

FIG. 4.

Calculated transition energies () for the (ii)l,o QW transitions obtained by EFA as a function of the relative band offset ΔEc/ΔEg for PbSe/PbEuSe MQWs with dQW=100Å ((a) and (b)) and 155 Å ((c) and (d)) for T = 20 K ((a) and (c)), 300 K (b) and 250 K (d). The indices i denote the ith state in the QWs and l and o the longitudinal and oblique valleys, respectively. The solid horizontal lines show the measured transition energies for the corresponding samples determined by differential transmission as shown in Fig. 2.

FIG. 4.

Calculated transition energies () for the (ii)l,o QW transitions obtained by EFA as a function of the relative band offset ΔEc/ΔEg for PbSe/PbEuSe MQWs with dQW=100Å ((a) and (b)) and 155 Å ((c) and (d)) for T = 20 K ((a) and (c)), 300 K (b) and 250 K (d). The indices i denote the ith state in the QWs and l and o the longitudinal and oblique valleys, respectively. The solid horizontal lines show the measured transition energies for the corresponding samples determined by differential transmission as shown in Fig. 2.

Close modal

As a further proof, we present in Fig. 5 the theoretical (solid lines) and measured (full symbols) transition energies as a function of temperature from 20 to 300 K for samples with QW thickness of 34, 73, 100, and 142 Å. Evidently, the complete set of experimental data is in excellent agreement with the theoretical calculations for a constant band offset of δ=0.45. The same result is also found for samples with different Eu contents. Therefore, we can also conclude that the band alignment does not depend on Eu content up to values of at least xEu14%.

FIG. 5.

Measured (symbols) and calculated (solid lines) temperature dependent transition energies of PbSe/PbEuSe MQWs with different well thicknesses of 34, 73, 100, and 142 Å, from (a) to (d), respectively, for a constant conduction band offset of ΔEC/ΔEG=0.45. Evidently, an excellent agreement between theory and experiments at all temperatures is found. The shaded regions indicate the PbSe and PbEuSe band gaps for xEu = 9.3%, 8.8%, 9%, and 11.6% in the samples (see Table I).

FIG. 5.

Measured (symbols) and calculated (solid lines) temperature dependent transition energies of PbSe/PbEuSe MQWs with different well thicknesses of 34, 73, 100, and 142 Å, from (a) to (d), respectively, for a constant conduction band offset of ΔEC/ΔEG=0.45. Evidently, an excellent agreement between theory and experiments at all temperatures is found. The shaded regions indicate the PbSe and PbEuSe band gaps for xEu = 9.3%, 8.8%, 9%, and 11.6% in the samples (see Table I).

Close modal

In summary, a quasi symmetric type I band alignment of ΔEc/ΔEg=0.45±0.15 was derived for the PbSe/PbEuSe heteroepitaxial system, independent of temperature and Eu content. The resulting strong overlap of the electron and hole wave functions provides good conditions for efficient room temperature mid-infrared emission and absorption. Thus, the finding that room temperature emission of PbSe/PbEuSe quantum wells tends to be less efficient than for PbSe/PbSrSe quantum wells must be caused by other effects such as a faster non-radiative carrier recombination and/or stronger alloy scattering, which could be also affected by the epitaxial growth conditions. The excellent agreement between the experimental data and theoretical calculations over a wide range of sample parameters and temperatures demonstrates that our presented envelope function approach allows a highly reliable and predictive modelling of the electronic structure of this quantum well system as required for device applications.

This work was supported by the Austrian Science Funds, Project No. SFB-025 “IR-ON.”

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