GaSb type II superlattice infrared detectors

Infrared Physics & Technology 59 (2013) 53–59

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A kp model of InAs/GaSb type II superlattice infrared detectors P.C. Klipstein a,⇑, Y. Livneh b, O. Klin a, S. Grossman a, N. Snapi a, A. Glozman a, E. Weiss a a b

Semiconductor Devices, P.O. Box 2250, Haifa 31021, Israel Department of Physical Electronics, Tel-Aviv Univ., Tel-Aviv 69978, Israel

a r t i c l e

i n f o

Article history: Available online 29 December 2012 Keywords: Infrared detector pBp detector Type II superlattice InAs/GaSb superlattice Kane model kp theory

a b s t r a c t We present and justify the TVK8 envelope function Hamiltonian for superlattice structures, which is a Kane-like Hamiltonian with a small number of fitting parameters [P.C. Klipstein, Phys. Rev. B 81 (2010) 235314]. In order to predict the bandgaps of type II InAs/GaSb superlattices in which the layer widths are known with a typical uncertainty of 0.2 monolayers (ML), it requires careful fitting of two Luttinger parameters, three interface parameters and the valence band offset. All other parameters, namely standard bandgaps, deformation potentials, electron masses and compliance coefficients, may be found in the literature. We have used the model to calculate the 77 K absorption spectra of representative MWIR and LWIR superlattices in which the layer widths have been determined by in situ beam flux measurements in the MBE reactor. By a comparison with the experimentally measured spectra, a unique set of the six fitting parameters has been determined. Our Luttinger parameters and band offset are close to those deduced by other workers. Using the same set of parameters we have then predicted the bandgaps in more than 30 superlattices with measured bandgap wavelengths in the range 4.3–12 lm. The agreement is very good and is limited by the experimental uncertainty in the layer widths. This is typically 0.2 ML with a maximum value of 0.4 ML. The model also reproduces the main features of the absorption spectrum, including the form and energy of the strong peak from zone boundary HH2 ? E1 transitions, and the value of the bandgap blue shift when the GaSb thickness is increased. We use the modeled absorption spectra to calculate the spectral response of two pBpp barrier detectors with cutoff wavelengths in the MWIR and LWIR, respectively. We find that there is virtually no difference if the experimentally measured absorption spectra are used instead. This shows that the complete detector response can be predicted using only the superlattice period and bandgap as input parameters, together with the optical widths of the layers in the detector structure. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Different sophisticated microscopic approaches and atomistic theories have been developed to compute the free-carrier states in semiconductor heterostructures. These are mostly based on empirical or first-principles pseudopotential and tight-binding methods. However, as in the past development of the physics of bulk semiconductors, approximate non-atomistic theories based on kp perturbation theory (effective mass approximation [1], the envelope-function Kane [2] or Luttinger [3] Hamiltonians, etc.) have been very popular for calculating the electronic properties, due to their simplicity and ease of interpretation. The advantage of the latter theories is a small number of band parameters. However, the elaboration of these theories, in particular the development of a rigorous treatment for the interface region, has lead to a multiplication of the parameters to the point where their attrac-

⇑ Corresponding author. E-mail address: [email protected] (P.C. Klipstein). 1350-4495/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.infrared.2012.12.009

tiveness vis-à-vis other approaches becomes less clear [4]. In this work we show that a kp model can be developed with a small number of parameters for nearly lattice matched materials that is still quite competitive with other approaches (e.g. Ref. [5]) and which has sufficient accuracy to be useful for the interpretation of experimental data. Our model is based on the work of Takhtamirov and Volkov, who showed that in any derivation of the envelope function Hamiltonian, it is important to include all terms up to order l = 2 in  l , where dV is the typical band offset,   is the average dV  ðkaÞ hk momentum modulus of the envelope function, and a is the bulk lattice parameter [6–8]. They derived a general envelope function equation in reciprocal space, but only considered the real space envelope function equation in detail for a single band. One of the present authors used their approach to derive real space 6 band and 8 band Kane-like Hamiltonians, in which all of the important terms up to l = 2 where included [9]. In the following we shall refer to the 8 band form of the Hamiltonian as the Takhtamirov–Volkov– Klipstein-8 Hamiltonian, or TVK8. This model can be further simplified when it is realized that for any realistic comparison with

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P.C. Klipstein et al. / Infrared Physics & Technology 59 (2013) 53–59

experiment, the accuracy in layer width is unlikely to be better than a few tenths of a monolayer and that this uncertainty in thickness will introduce an uncertainty in energy roughly of order l = 2. In this case, it is sufficient to include only those terms in the Hamiltonian of order l = 1, resulting in a considerable simplification, and it is the approach that we have adopted in this work. Our l = 1 TVK8 model is a Kane like Hamiltonian, with bulk piecewise parameters and a diagonal (non-diagonal) three (two) parameter interface matrix for no common atom (common atom) superlattices. We show that four out of the six Luttinger parameters needed to define the Hamiltonian can be eliminated, reducing the parameter set for an InAs/GaSb superlattice to just six: two Luttinger parameters, three interface parameters and the band offset. A seventh parameter is also required if the absorption spectra are calculated, which is related to the strength of the inter-band transitions. All other parameter values can be established with sufficient accuracy from spectroscopic, mechanical or X-ray measurements which may be found in the literature. We show that the model is able to predict the bandgaps of more than 30 midwave-infrared (MWIR) and long-wave-infrared (LWIR) InAs/GaSb superlattices (bandgap wavelengths 4.3–12 lm), with a precision limited only by the accuracy with which the layer widths can be determined. The model is able to reproduce the absorption spectra of MWIR and LWIR superlattices in both form and energy, including a higher energy feature due to zone boundary transitions between the HH2 and E1 bands, and it also predicts the blue shift in the bandgap wavelength quite accurately, when the GaSb thickness is increased [5]. We begin by outlining the main features of the TVK8 Hamiltonian only containing terms of order l = 1. We discuss the microscopic origin of the interface matrix and show that it has different symmetries for common atom (CA), and no common atom (NCA) superlattices. We then describe our method for determining the layer widths of our InAs/GaSb superlattices, which has a typical accuracy of 0.2 monolayers (ML). The superlattices were grown by Molecular Beam Epitaxy (MBE) with ‘‘InSb’’ like interfaces. The TVK8 model is fitted to the experimental absorption spectra of representative superlattices with bandgap wavelengths in the MWIR and the LWIR. From this fit a single set of six fitting parameters is deduced. Using the same set of parameters, the difference between the predicted and measured bandgap wavelengths is then analyzed for about 30 superlattices with bandgap wavelengths from 4.3 to 12 lm. We find a typical difference of less than 0.3 lm in the LWIR and less than 0.1 lm in the MWIR which corresponds well with the typical measured layer width uncertainty of 0.2 ML. We calculate the spectral response of a superlattice detector for both the MWIR and LWIR spectral regions, and show that it is possible to predict the complete detector response using only the superlattice period and bandgap as input parameters, together with the optical widths of the layers in the detector structure. 2. The interface hamiltonian It was shown by one of the authors in Ref. [9] that the l = 1 TVK8 Hamiltonian has a similar form to the standard Kane Hamiltonian given in Eq. (20) of Ref. [2] with B =0 and where Pkz is replaced with (1  f)P kz + f kzP. An expression was given for the ‘‘symmetrization’’ factor, f, in terms of momentum matrix elements with remote bands and their energy separations from the local bands. Its value for InAs/GaSb superlattices is discussed below. This Kane like Hamiltonian is applied in a piecewise fashion using the bulk values for the parameters in each layer. To this is added the Pikus–Bir piecewise strain Hamiltonian (e.g. Eq. (15) of Ref. [10]), the piecewise spin–orbit interaction Hamiltonian (e.g. Eq. (34) of Ref. [9]), and an interface Hamiltonian which has the following form for each spin direction:

2

DS 6 0 X 6 M IF ¼ dðz  zi Þ6 4 0 i

pi b

0 DX

0

pi a

pi a DX 0

0

pi b

3

0 7 7 7 0 5

ð1Þ

DZ

Here, i is the index of the interface at zi, and pi is a sign parameter that takes a value of 1 at a normal interface (e.g. InAs on GaSb) and +1 at an inverted interface (e.g. GaSb on InAs). The interface parameters, a and b, were predicted by Aleı˘ner and Ivchenko in 1992 from considerations of the interface bonding symmetry [11]. Krebs and Voisin [12] explained the anisotropy of  1 0 polarizations in the optical absorption between [1 1 0] and ½1 type I quantum wells, in terms of the interface contribution, a, which has typical values of 1 eV Å in a CA superlattice [13]. In 2010, one of the authors showed that typical values for a and b should be somewhat smaller in NCA superlattices, while the diagonal interface parameters (DS, DX and DZ), which are zero in an ideal CA superlattice are expected to be quite large [9]. It was shown there that the interface parameters are given by matrix elements of the quantity dU  Us0 or dU  Ua0 , for the off-diagonal and diagonal elements, respectively. for example,    Thus,     a ¼ X dU  Us0 Y and DZ ¼ Z dU  Ua0 Z . In these expressions, dU = UGaSb  UInAs where UGaSb, UInAs are the microscopic potentials of bulk GaSb and InAs, respectively. The local potential, is given by V(r)= UInAs + dUG(z) where G(z) is the composition modulation function and InAs is taken to be the reference crystal, whose Bloch functions appear in the interface matrix elements given above. The two functions, Us0 and Ua0 were derived in Ref. [9] and are plotted here in Fig. 1 for both CA and NCA superlattices with perfect, metallurgically abrupt interfaces. For these figures a model potential profile has been assumed, in which the derivative of G is a Gaussian with a Full Width at Half Maximum (FWHM) of 0.8 Å. For a CA superlattice like GaAs/AlAs, the interface is on a plane of As atoms, while in a NCA superlattice like GaSb/InAs with ‘‘InSb’’ like interfaces, the interface is taken to be half way between the planes of In and Sb, and interface strain is not taken into account. While our model is somewhat idealized, Fig. 1 suggests that a and b should be much weaker in a NCA superlattice because the amplitude of Us0 is reduced by a factor of about 5. Moreover, the Dparameters are zero in an ideal CA superlattice, while in an NCA superlattice they are expected to be significant and comparable in strength to the off diagonal interface parameters in a CA superlattice, because Ua0 in a NCA superlattice has a similar amplitude to Us0 in a CA superlattice. The interface functions, Us0 and Ua0 , reduce in amplitude as the width of the derivative of the profile function, G’(z), increases. Since we do not know the exact microscopic potential at the interface we have simply assumed what we think is a reasonable value for its width. Because the interface is always located on a single atomic plane in this model, simply increasing the width of G’(z) cannot be used to model the effect of interface grading where the In-Sb interface bonds are distributed over a few atomic planes. However, Foreman has used an atomistic model to estimate this effect and has found that the interface matrix is actually very insensitive to the grading [4]. Its strength remains essentially unchanged, while the d-function in Eq. (1) must be replaced by a broader ‘‘d-like’’ function whose width is comparable to the width of the grading. This gives us some confidence that the simple model used to derive Eq. (1) should be reasonably robust when comparing nominally similar samples grown at different times or in different laboratories. 3. Fitting parameters As shown in Table 1 of Ref. [9] the main parameters of the Kane Hamiltonian, L’, M and N can be determined from the three Luttin-

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P.C. Klipstein et al. / Infrared Physics & Technology 59 (2013) 53–59

1

1

Φs

0

0.5

(Å)

(Å)

0.5

Φa

-0.5 -1 -1.5

Φaa aΦ Φss sΦ

0 -0.5

-1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

Position (Å)

Position (Å)

(a)

(b)

1

1.5

Fig. 1. Interface functions, Us0 and Ua0 , plotted for (a) a CA superlattice and (b) a NCA superlattice, with metallurgically abrupt interfaces (FWHM of G0 (z) = 0.8 Å).

Table 1 EP values at 77 K for seven binary III–V semiconductors and the parameters that were used to calculate them, according to Eq. (2). All parameters were taken from Refs. [16,17], apart from the electron effective masses for the indirect materials. For these two materials (AlAs and AlSb) the mass values were adjusted until EP was close to the value quoted by Lawaetz [18]. ISO stands for a matching isoelectronic group IV material, whose bandgap values appear in the last three columns of the Table, and which were also taken from Refs. [16,17]. In the case of the group IV alloys, a simple average was taken of the values for the two constituent binaries.

ISO me/m0 E0 D0 E00 D00 EP

GaAs

GaSb

InAs

InP

InSb

AlSb

AlAs

Ge 0.0665 1.519 0.34 4.488 0.171 24.99

GeSn 0.042 0.814 0.76 3.2 0.4 22.75

GeSn 0.022 0.418 0.38 4.55 0.2 22.42

Ge 0.075 1.424 0.108 4.8 0.07 19.58

Sn 0.014 0.237 0.81 3.4 0.4 22.86

Ge 0.14 2.32 0.645 3.7 0.3 18.47

GeSi 0.19 3.13 0.28 4.34 0.14 21.41

Ge

Si

a-Sn

0.29 3.1 0.2

0.04 3.4 0.024

0.8 2.3 0.4

ger parameters, c1, c2 and c3 and the parameter, c4 = EP/E0, where EP = 2m0P2/⁄2 and E0 is the bandgap energy [14]. Using the method of Cardona [15], based on a 5-level kp analysis, the EP values can be determined fairly accurately from spectroscopic measurements of zone centre optical transitions between bands with s- and p-like symmetry in InAs, GaSb, and their group IV isoelectronic counterpart, Ge0.5Sn0.5, together with Faraday rotation or magneto-transport measurements of the InAs and GaSb conduction band effective masses. EP is given by the expression:

confidence in the accuracy of both sets of values. This also means that the parameter, c4, can be determined with reasonable accuracy. It is then possible to use the constraint relations given in Eq. (33) of Ref. [9] to show that c3 for InAs and c1, c2 and c3 for GaSb are not independent but can be determined from c1 and c2 of InAs to an accuracy of l = 2 as follows [19]:

m0 1 i me  0 2   1 2 1 þ E0 þ1D0  PP 3 E0 E00 þ2D00 =3E0  0 2 E0 þ 2D00 =3 þ D0 =3  e00 þ 2d00 =3 þ d0 =3 P  ¼ 00 P E0 þ 2D00 =3 þ D0 =3 þ e00 þ 2d00 =3 þ d0 =3

cGaSb 1

Ep ¼ h

ð2Þ

    where E0, D0(d0), E00 e00 , and D00 d00 , refer to the C6c  C8v, C8v  C7v, C7c  C8v, and C8c  C7c bandgaps respectively for the III– V (IV) material, and me (m0) is the conduction band effective mass (free electron mass). Table 1 shows the values of EP determined in this way for five direct bandgap semiconductors including InAs and GaSb and two indirect bandgap materials, AlAs and AlSb. Also shown are the bandgap data that were used to calculate them, according to Eq. (2). For isoelectronic group IV materials (denoted ISO in the table) which are alloys, we take averages of the bandgaps shown for the group IV elements. All the data in the Table were taken from Refs. [16,17] apart from the conduction band masses of AlSb and AlAs, which cannot be measured with any accuracy in an indirect bandgap material. Since all of our calculated EP values for the direct bandgap materials are within a few percent of those quoted by Lawaetz [18], we have adjusted the mass values for the indirect bandgap materials until their values of EP are also close to those of Lawaetz. The agreement between the EP values for the direct gap materials in Table 1 and those of Lawaetz (which were determined in a different way, by scaling the measured value for Si), gives us some

1 cInAs 2cInAs cInAs þ 1  2 þ 4 12 12 3 4

2v 2v InAs 4v InAs cInAs cGaSb  c1  c2  4 þ 4 þ 1þ 3 3 3 3 3 GaSb v v InAs  v InAs cInAs c    c1 þ 1 þ c2  4 þ 4 6 6 3 6 6





2v þ 1 2v þ 1 InAs v þ 2 InAs cInAs cGaSb þ  c1  c2 þ 4 þ 4 12 12 3 12 6

cInAs  3

cGaSb 2 cGaSb 3

ð3Þ where v ¼ 1  E0G ðGaSbÞ=E0G ðInAsÞ and E0G ¼ E00 þ 2D00 =3 þ D0 =3. Since we only require an accuracy level of l = 1, for the reasons given above, Eq. (3) enables us to eliminate four out of the six Luttinger parameters in our calculation of the superlattice bandstructure, thereby making a significant reduction in the number of fitting parameters. We have chosen to fit the two independent Luttinger parameters, rather than use the values for all six parameters quoted for example by Lawaetz, for several reasons. First, the results of our calculations, particularly for the absorption spectra shown below, are very sensitive to fairly small changes in these values. Second, the Lawaetz values are based on theoretical scaling arguments as well as experimental data, and therefore may be less reliable than our fitted experimental values. Third, other experimental determinations have quite a wide range of values and so cannot be taken as definitive, as may be seen for example in the review by Vurgaftman et al. [20]. These authors have recommended composite averages for each parameter, based on the spread of values that they have

P.C. Klipstein et al. / Infrared Physics & Technology 59 (2013) 53–59 Table 2 The six fitted kp parameters used in the calculations at 77 K. Parameter

Value

DS (eV Å) DX (eV Å) DZ (eV Å) c1 (InAs) c2 (InAs) VBO (eV)

3.0 1.3 1.1 20.0 9.0 0.560

found in the literature. We do not recommend this approach since the values of the Luttinger parameters are not all independent, as shown by Eq. (3), so taking simple averages of each parameter independently may not be the most accurate method. The other important fitting parameters for calculating the superlattice bandstructure are the diagonal interface potentials, DS, DX and DZ, given in Eq. (1), and the valence band offset (VBO), making a total of six fitting parameters in all. We treat the interface parameters, a and b as negligibly small, for the reasons given in Section 2, and because we can obtain a good fit to experimental data in Section 6 with a  b  0. They are therefore not included in our list of fitting parameters, whose final values are given in Table 2. The symmetrization parameter, f (mentioned at the beginning of Section 2), turns out to be unimportant for InAs/GaSb superlattices, since the values of EP for both materials are very similar. We have therefore chosen a convenient value of 0.5. The other parameters used in the calculation, apart from the EP values already given in Table 1, and the layer widths which will be discussed below, can be found by independent spectroscopic, X-ray, and mechanical measurements. These are the bandgaps and electron mass already listed in Table 1, the bulk lattice parameter, the ratio of compliance coefficients (c11/c12) and the Pikus–Bir deformations potentials (ac, av and b). In all cases we used the values in Refs. [16,17]. 4. Layer width determination Our method of layer width determination is based on finding the two rate constants (jIn, jGa) that relate the Indium and Gallium fluxes measured in the MBE machine (/In, /Ga) to the InAs and GaSb layer growth times (tIn, tGa). We can write a set of simultaneous equations for a set of N superlattice samples with InSb-like interfaces as follows:

2

1:1  ðt In /In Þ1

6 1:1  ðt / Þ 6 In In 2 6 6 1:1  ðt In /In Þ3 6 6 .. 6 . 4 1:1  ðtIn /In ÞN

ðtGa /Ga Þ1

3

2

L1

‘‘In’’ at both ends and the GaSb with ‘‘Sb’’ at both ends. In Eq. (4), however, the InAs width, tIn/InjIn is still 10 ML. We have solved Eq. (4) to obtain the rate constants for a set of over 30 samples with InAs layer widths of between about 8 and 16 ML and GaSb widths between about 6 and 16 ML. They span bandgap wavelengths from 4.3 to 12 lm. Next, we determined the complete InAs and GaSb layer widths using the expressions: L0InAs ¼ 1:05  tIn /In jIn and L0GaSb ¼ 0:05  t In /In jIn þ tGa /Ga jGa . In these expressions half of the InSb interface width (i.e. the ‘‘In’’) is assigned to the InAs layer and the other half (i.e. the ‘‘Sb’’) to the GaSb layer. The final expressions for the layer widths (with the   lowest  statistical error) are: LInAs ¼ L  L0InAs þ L0GaSb =2 and  0 0 LGaSb ¼ L þ LInAs  LGaSb =2. When expressed in monolayers, noninteger results for the InAs and GaSb layer widths can be considered to describe layers whose interfaces are not perfectly flat but include monolayer steps, such that their average thicknesses correspond to the non-integer values. By analyzing the distribution of values for the quantity L  L0InAs  L0GaSb we have found a maximum error in any layer width of 0.4 ML and a typical error, based on the standard deviation, of 0.2 ML [19]. We believe that these figures represent the ultimate accuracy with which the individual superlattice layer widths can be determined. 5. Calculation of dispersions and absorption spectra We have solved the TVK8 Hamiltonian using a Fourier Transform method, similar to that described by Gershoni et al. in which we have limited the Fourier components of the envelope functions to the first bulk Brillouin zone [10]. Fig. 2 shows a typical dispersion result for both the in-plane direction (kx or k||) and perpendicular to the layers (q). The superlattice in question has 13.8 ML InAs and 7.8 ML GaSb, which leads to a bandgap wavelength of 9.5 lm, chosen because it lies in a useful range of the LWIR atmospheric window. The in-plane dispersion along [1 0 0] is shown in the left hand panel as a solid line for q = 0 and as a dashed line for q = p/ L. The dispersion in the growth direction is shown in the right hand panel. In the calculation we used values of a = 0 and b = 0.2 eV Å for the off diagonal interface parameters in Eq. (1). Setting b to zero or a to 0.2 eV Å has an effect on the bandgap equivalent to a < 0.2 ML change in InAs width, which is below the level of accuracy for layer width measurement. As already discussed above, these interface parameters are expected to be small and may be treated as essentially negligible without reducing the quality of our fits to experi-

3

6 7 ðtGa /Ga Þ2 7 7 6 L2 7 7 j  6 7 In 6 7 ðtGa /Ga Þ3 7 ¼ 6 L3 7 7 7 jGa 6 . 7 .. 7 6 . 7 . 5 4 . 5 LN ðtGa /Ga ÞN

0.2

ð4Þ

where L1, L2, . . . , LN are the periods of the superlattices measured with high precision by X-ray diffraction. In deriving Eq. (4) we make the usual assumption that the group III atoms have high sticking coefficients and are therefore the species that determine the growth rate. Noting that a superlattice lattice matched to GaSb, must have roughly one additional InSb monolayer (at the interface) for every ten bulk InAs monolayers, we have added this InSb thickness to the total InAs thickness by increasing the latter by 10%. This ignores the small difference in the lattice parameters and assumes similar growth rates for the two materials. For example a superlattice with 10 ML InAs and 10 ML GaSb must contain one additional monolayer of InSb per period. Such a superlattice could be completely described as a 10.5/10.5 superlattice where the InAs and GaSb each continue for an extra 0.5 ML, such that the InAs is terminated with

0.1

Energy (eV)

56

0 -0.1 -0.2 -0.3 -0.4 0.1

0.05

k (× 2 π/a x

00

)

sub

0.5

1

q (× π/L)

Fig. 2. Calculated dispersion relations parallel (left panel), and perpendicular (right panel), to the layers for a 13.8 ML/7.8 ML InAs/GaSb superlattice with InSb like interfaces. The solid (dashed) lines in the left panel are for q = 0 (q = p/L).

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P.C. Klipstein et al. / Infrared Physics & Technology 59 (2013) 53–59

density of states because the in-plane dispersions (dashed lines) are almost parallel to one another.

mental data. In Section 6, we have arbitrarily chosen values of 0.2 eV Å for both parameters. In order to calculate the absorption we divide reciprocal space into cells using cylindrical coordinates. Each cell is wedge shaped with boundaries defined by keeping each of the three coordinates constant in turn. The density of states, q = DN/DE, is calculated from the cell volume which is proportional to the number of transitions, DN, and where DE is taken to be the difference between the maximum and minimum energies at the eight cell corners. The contribution of the cell to the optical absorption is then given by the standard expression [10]:

6. Comparison with experiment Low temperature photoluminescence (PL) measurements at 10 K were used to determine the bandgap wavelength, while photo-absorption was measured at 77 K. Along with other authors, we have found a good correlation between the bandgap wavelength deduced from low temperature PL and that found from the 77 K absorption spectrum [21]. In each case we used a Bruker Equinox 55 Fourier Transform Infrared Spectrometer with a resolution of 4 cm1 to perform the measurements. The absorption coefficient was obtained from the formula: a = ln(T2/T1)/h where T1 and T2 are the transmission coefficients of the superlattice structure (including substrate) before and after etching off a thickness h  1–2 lm of the superlattice. Fig. 3 shows two comparisons between measured and fitted absorption spectra. One of the spectra corresponds to an 8.6/13.5 superlattice with a bandgap wavelength of 4.4 lm and the other to a 14.4/7.2 superlattice with a bandgap wavelength of 10.4 lm. The inhomogeneous broadening of the latter is quite large because a lower quality substrate had to be used, compared with those on which the other samples in this study were grown, in order to obtain a high enough transmission signal for the optical absorption measurement. The sharp feature at energy ED discussed above can be seen quite clearly in both spectra. The quoted layer widths were within a few tenths of a monolayer of the layer widths determined from the MBE flux measurements discussed above, and well within the allowed error of 0.4 ML. On the basis of fits to these two absorption spectra and one other with a bandgap wavelength of 4.6 lm, we found a single set of fitting parameters which is the set quoted in Table 2. Using these fitting parameters we then compared the bandgap wavelengths determined by PL (kPL) and calculated from the layer widths (without adjustment) using our TVK8 model (kkp) for a set of more than 30 samples spanning the wavelength range 4.3– 12 lm, which covers both the MWIR and LWIR atmospheric windows. The difference between the two values: Dk = kPL  kkp is plotted against kPL in Fig. 4a. In the MWIR this difference is extremely small while in the LWIR it is typically less than 0.2 lm with a maximum deviation of 0.6 lm. Note that this maximum deviation is still comparable with the thermal broadening at 10 lm wavelength and 77 K. In Fig. 4b, we plot the difference in the InAs layer width determined from the flux measurements (LInAs) and that predicted by a TVK8 calculation that gives a perfect match to the measured PL bandgap wavelength and X-ray superlattice period (LInAs,

2

daðEÞDE ¼

4p2 afs h

nSL Em20

qSfv ðkÞð1  fc ðkÞÞDE

ð5Þ

in which afs is the fine structure constant and nSL is the real part of the superlattice refractive index, which is taken to be the average of the bulk values, weighted by the respective layer thicknesses. fv and fc are the Fermi–Dirac functions for the valence and conduction bands, and S is the transition oscillator strength, which is calculated from the Fermi Golden Rule. The expression for S contains the momentum matrix element between bulk valence and conduction states. If uS and uX are taken to be the zone center s- and p-like Bloch functions of the bulk reference crystal, then instead of using the standard expression, P = i(⁄/m0)huS|px|uXi, we use instead (gP) where g is an empirical factor introduced to take account of the fact that the real basis states of the 8  8 superlattice Hamiltonian include small (spatially dependent) admixtures from remote states [6,9]. We find that a value of g = 1.45 gives a reasonable match between our calculations of the full absorption spectra and experimental data, and we have used this value throughout. Before calculating the full absorption spectrum, we first replace da (E) DE by a Gaussian of the same area, centered at the same energy, E, and with a standard deviation, r (DEin = 2r is the inhomogeneous broadening energy). The contributions from all the cells and from all transitions are then added up. We use an energy dependent expression for DEin which increases with transition energy, because the dominant mechanism for inhomogeneous broadening in short period superlattices is interface roughness. This expression has the form DEin = DE0 + x(k0  k), where x and DE0 are fitted constants, and k0 is the bandgap wavelength. DE0 can be fitted unambiguously to the shape of the absorption edge, while x can be fitted to the shape of a strong peak in the measured absorption spectrum at energy, ED. This peak corresponds to zone boundary transitions at q = p/L, between the second Heavy Hole band (HH2) and the first Electron band (E1). As can be seen in Fig. 2, these transitions have a very high joint

1 Measured Calculated

1 -1

α ( μm )

-1

α (μ m )

0.8 0.6 0.4

0.5

0.2 0

2

3

4

5

0 2

4

6

8

λ ( μ m)

λ (μm)

(a)

(b)

10

12

Fig. 3. Measured (solid line) and calculated (dashed line) absorption spectrum at 77 K for (a) MWIR superlattice with 8.6 ML InAs and 13.5 ML GaSb and (b) LWIR superlattice with 14.4 ML InAs and 7.2 ML GaSb. Note the strong feature, ED, in (a) at 2.7 eV and in (b) at <2.5 eV due to zone boundary HH2 ? E1 transitions.

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P.C. Klipstein et al. / Infrared Physics & Technology 59 (2013) 53–59

0.6 LW IR MW IR

LW IR

0.5

ΔLInAs (ML)

Δλ ( μ m)

0.4 0.2 0 -0.2

MW IR

0

-0.5

-0.4 -0.6 4

6

8

10

8

12

10

12

λ PL ( μ m)

LInAs (ML)

(a)

(b)

14

16

Fig. 4. Deviation of (a) the predicted bandgap wavelength from the measured PL peak wavelength and (b) the predicted InAs width (that gives a perfect match for the PL peak wavelength) from the nominal InAs width (determined from the in situ flux measurements) for more than 30 superlattices.

p

Bp

p

7. Simulation of p-B-p detector performance

EG

InAs/GaSb

InAs/GaSb

InAs/AlSb

Fig. 5. Schematic band profile of a InAs/GaSb superlattice pBpp structure. The photon absorbing layer is on the left.

In the Figure, this difference, DL = LInAs  LInAs, kp is plotted against LInAs. The typical deviation is 0.2 ML with a maximum value of 0.4 ML. These values correspond very well with the expected layer width uncertainties determined from the analysis of the flux measurements in Section 4. The points for the two MWIR samples plotted in Fig. 4 correspond to 8.6/13.5 and 8.7/15.7 superlattice structures. Their bandgap wavelengths differ by about 0.15 lm. They exhibit the well known blue shift as the GaSb thickness is increased [5]. The extremely low wavelength difference, Dk, for these two samples in Fig. 4a shows that the TVK8 model is able to reproduce the blue shift quite precisely. The results in Fig. 4 also show that the model provides a good prediction of the bandgap wavelength from the MWIR to the LWIR. It also provides a reasonable description of the higher energy feature ED in both MWIR and LWIR samples, as demonstrated in Fig. 3.

100

100

80

80

QE (%)

QE (%)

kp).

Using standard optical transfer matrix techniques [22,23] it is possible to calculate the photoresponse of a realistic detector structure based on absorption spectra, a(k), such as those plotted in Fig. 3. In the calculation the expression n = nSL + i(k/4p)a(k) should be used for the complex refractive index of the photon absorbing layer of the detector. We consider a pBpp barrier detector [24–27] in which the p-type photon absorbing and contact layers are made from the same InAs/GaSb superlattice material and have thicknesses of 4.5 and 0.2 lm respectively. The thin p-type barrier layer (Bp) is an InAs/AlSb superlattice designed to give a smooth conduction band alignment with the photon absorbing layer. A schematic depiction of the typical band alignment in such a pBpp structure is shown in Fig. 5. One of the advantages of this type of heterostructure detector over a standard homostructure photodiode, is that the Generation–Recombination contribution to the dark current can be totally suppressed, leading to a lower overall dark current at a given temperature of operation [24]. Fig. 6 shows the calculated two pass spectral response curves for two detectors, one for the MWIR and one for the LWIR, based on the superlattices whose absorption spectra were shown in Fig. 3. In order to achieve two-pass behavior a highly reflecting metal layer was included on the top of the contact layer (front side). The light enters from the back-side of the detector from which the substrate has been removed and replaced by a dielectric antireflection coating. In both figures we compare the spectral response calculated using the measured absorption spectrum shown in Fig. 3 (solid line) with that calculated from the theoretical absorption spectrum (dashed line) which also appears in Fig. 3. The difference between the two calculations is very small, showing that our model is able to simulate the full detector performance

60 40

Calculation Measurement

20 0

2

3

40 20

4

λ (μm)

60

5

0

5

7

9

11

13

λ (μm)

Fig. 6. Comparison between the detector spectral photoresponse calculated from the calculated (dashed line) and the measured (solid line) absorption spectra, for the two superlattices shown in Fig. 3.

P.C. Klipstein et al. / Infrared Physics & Technology 59 (2013) 53–59

using the X-ray period and InAs layer width (or bandgap PL wavelength) of the superlattice as the only input parameters for the absorption spectrum, which can then be used with the optical widths of the layers in the pBpp (or equivalent diode) structure to complete the calculation. 8. Conclusions We have presented the TVK8 envelope function Hamiltonian and determined the values of its three interface parameters: DS = 3 eV Å, DX = 1.3 eV Å, and DZ = 1.1 eV Å for InAs/GaSb superlattices with ‘‘InSb’’ like interfaces. We find values of c1(InAs) = 20.0 and c2(InAs) = 9.0 for the two independent Luttinger parameters and a value for the overlap between the InAs conduction band and the GaSb valence band of 0.14 eV. Together these form the six fitting parameters required by the model to calculate the superlattice energy dispersion. They were determined by fitting the 77 K absorption spectra of two representative MWIR superlattices and one LWIR superlattice, and then used without change to predict the bandgap wavelengths, measured by photoluminescence spectroscopy at 10 K, of more than 30 superlattices spanning the wavelength range 4.3–12 lm. The superlattice layer widths were determined to a typical uncertainty of 0.2 ML, and a maximum uncertainty of 0.4 ML, using in situ beam flux data from the MBE reactor and high precision X-ray measurements of the superlattice period. The model was able to reproduce the form of the absorption spectra quite well, including a strong high energy feature due to zone boundary transitions between the HH2 and E1 bands, and also the bandgap energies of all the superlattices, to a precision limited by the experimental uncertainty in the layer widths. The blue shift of bandgap energy with increasing GaSb width was also predicted quite accurately. We have presented expressions which can be used to determine the four dependent Luttinger parameters from the two independent ones and found values of c3(InAs) = 9.16 and c1(GaSb) = 11.87, c2(GaSb) = 4.61, and c3(GaSb) = 4.99. Together our six Luttinger parameters deviate by less than 0.65 in any parameter from the widely used values deduced by Lawaetz in 1971 [18]: 19.67, 8.37 and 9.29 for InAs and 11.80, 4.03 and 5.26 for GaSb. In order to fit the absorption spectra we have found an empirical scaling parameter of g = 1.45 for the inter-band matrix element, P, to take account of the admixture of remote states into the eight local states of the 8  8 Hamiltonian. The calculated absorption spectra for both a MWIR and a LWIR superlattice were used to calculate the complex refractive index of the superlattice and hence the spectral response of a complete pBpp bariode detector. The differences between these results and those calculated using the experimentally measured absorption spectra were very small. Our model can therefore be used to predict the complete performance of a superlattice detector, using only the superlattice period and bandgap as input parameters, together with the optical widths of the layers in the detector structure. Acknowledgements We thanks Dr. M. Katz and Mr. O. Westreich of the Soreq Research Centre for assistance with some of the PL measurements.

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