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Modelling of MWIR HgCdTe complementary barrier HOT detector
Solid-State Electronics 80 (2013) 96–104
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Modelling of MWIR HgCdTe complementary barrier HOT detector Piotr Martyniuk ⇑, Antoni Rogalski Institute of Applied Physics, Military University of Technology, 2 Kaliskiego St., 00-908 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 9 July 2012 Received in revised form 11 October 2012 Accepted 26 October 2012 Available online 21 December 2012 The review of this paper was arranged by Prof. E. Calleja Keywords: Complementary barrier infrared detectors Unipolar barrier detectors nBn detectors HgCdTe Type-II InAs/GaSb superlattices
a b s t r a c t The paper reports on the photoelectrical performance of medium wavelength infrared (MWIR) HgCdTe complementary barrier infrared detector (CBIRD) with n-type barriers. CBIRD nB1nB2 HgCdTe/B1,2-n type detector is modelled with commercially available software APSYS by Crosslight Software Inc. The detailed analysis of the detector’s performance such as dark current, photocurrent, responsivity, detectivity versus applied bias, operating temperature, and structural parameters (cap, barriers and absorber doping; and absorber and barriers compositions) are performed pointing out optimal working conditions. Both conduction and valence bands’ alignment of the HgCdTe CBIRD structure are calculated stressing their importance on detectors performance. It is shown that higher operation temperature (HOT) conditions achieved by commonly used thermoelectric (TE) coolers allows to obtain detectivities D⁄ 2 1010 cm Hz1/2/W at T = 200 K and reverse polarisation V = 400 mV, and differential resistance area product RA = 0.9 Xcm2 at T = 230 K for V = 50 mV, respectively. Finally, CBIRD nB1nB2 HgCdTe/B1,2-n type state of the art is compared to unipolar barrier HgCdTe nBn/ B-n type detector, InAs/GaSb/B-Al0.2Ga0.8Sb type-II superlattice (T2SL) nBn detectors, InAs/GaSb T2SLs PIN and the HOT HgCdTe bulk photodiodes’ performance operated at near-room temperature (T = 230 K). It was shown that the RA product of the MWIR CBIRD HgCdTe detector is either comparable or higher (depending on structural parameters) to the state of the art of HgCdTe HOT bulk photodiodes and both AIIIBV 6.1 Å family T2SLs nBn and PIN detectors. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Hitherto, the infrared radiation (IR) industry is conquered by HgCdTe bulk photodiodes [1–3] and GaAs/AlGaAs intersubband quantum well infrared photodetectors (QWIP) [4,5]. The requirement of the infrared detectors’ cryogenic cooling is a major impediment preventing from their extensive application that is why detector’s cooling push boundaries to increase device operating temperature. It is known that the critical condition which must be fulfilled to construct the HOT IR detector is to achieve both low dark current and high values of the quantum efficiency. Among the mechanisms generating the dark current in detector’s structure the following must be enumerated: band-to-band (BTB) tunnelling, trap assisted tunnelling (TAT), Schockley-Read-Hall (SRH) generation-recombination (GR) process, Auger GR process, and leakage currents. It was found that an incorporation of the type II superllatice (T2SLs) e.g. InAs/GaSb 6.1 Å AIIIBV family into detector architecture allows to reduce adverse BTB/TAT currents and GR Auger’s contribution to the total dark current. Therefore T2SLs could be considered as an alternative to the bulk HgCdTe HOT detectors and
⇑ Corresponding author. E-mail address: [email protected] (P. Martyniuk). 0038-1101/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sse.2012.10.021
GaAs/AlGaAs IR material systems [6]. Unfavourable SHR GR and leakage dark current’s components could be limited by the properly selected barriers incorporated into detectors structure. The barrier’s selection plays crucial role due to the lattice constant matching of the detectors’ constituent layers, the barrier’s height in both conduction and valance bands connected directly to the band alignment. It must be stressed that band alignment playing important role in design of the barrier IR structures is often fortuitous and extremely difficult to control from technological perspective [7]. The very first barrier structures were commonly known AIIBVI and AIIIBV heterostructures invented to increase device’s performance by suppression of the diffusion currents from the detector’s active region. The next stage in IR detector’s development was double layer heterojunction (DLHJ) allowing reducing both majority and minority carriers diffusion currents in comparison to the homojunction IR detectors. Another variation of the DLHJ was a graded gap structure incorporated between hole blocking layer and absorber to suppress tunnelling and GR currents [8]. Currently, among the barrier IR detectors (BIRDs) the leading position is occupied by minority carrier devices called unipolar barrier infrared detectors (UBIRD) proposed by Maimon and Wicks [9]. Among electron-blocking UBIRD detectors the most important are these designed with AIIIBV compounds (GaSb, InAs1–zSbz-cap layers, InAs1–ySby-active region, AlSb1–xAsx–barrier), T2SLs nBn
P. Martyniuk, A. Rogalski / Solid-State Electronics 80 (2013) 96–104
InAs/GaSb with AlGaSb/T2SLs barriers and UBIRD nBn HgCdTe detectors, while among hole-blocking UBIRD devices should be considered a four layer architectures: InAs/GaInSb/InAs/AlGaInSb often called ‘‘W’’ structures and GaSb/InAs/GaSb/AlSb referenced as ‘‘M’’ structures [10–12]. The most sophisticated structures containing both electron and hole blocking barriers (AlSb/T2SLs barriers) were proposed by Ting et al. [13] (called either complementary barrier infrared detectors – CBIRD) or PbIbN after Gautam et al. [14] showing possible advantages in suppressing dark currents by blocking majority and minority carriers and circumventing technological problems with making ohmic contact to the widegap layers. Potential interest in InAs/GaSb T2SLs results not only from unique inherited capabilities of the new artificial material with entirely different physical properties in comparison to the constituent layers (InAs and GaSb), but also from the nearly zero band offsets leading to the desirable UBIRD/CBIRD band alignments difficult to attain in HgCdTe [15]. In addition, the 6.1 Å family’s capabilities to tune the position of the conduction and valance band edges in independent way and near lattice matching is extremely helpful in designing of the unipolar and complementary barrier detectors. Although the abovementioned physical properties indicates potential T2SLs’ superiority over bulk materials (including HgCdTe ternary allys), the T2SLs’ quantum efficiency leaves a lot of to be desired (g = 20–30% in MWIR range and g = 8–12% in LWIR range depending on nBn/pBp architecture) which stems from low level of wavefunction overlapping and technological problems connected with growth of uniform and thick enough SLs [16]. What is more, short minority carrier lifetimes (sDIF, sGR < 10 ns in temperature range >200 K) also impedes the development of the T2SLs IR devices [17]. Similarly, theoretical simulations proved quantum dot infrared detectors (QDIPs) to be an alternative to the HgCdTe, but technological problems related to the growth of self-organized QDs led to the suspension of the research on this type of the detector [18,19]. Even though, HgCdTe does not exhibit valance zero band offset, it is commonly known that bulk HgCdTe offers quantum efficiency g = 50–70%, therefore recently research groups have attempted to apply UBIRD architecture to HgCdTe alloy (n type barrier) which offers technological advantages over p–n HgCdTe homojunction (simplifying the fabrication process) [20,21]. Moving forward, it is worth applying CBIRD architecture to HgCdTe alloy incorporating n type barriers to reduce dark current in comparison to UBIRD nBn HgCdTe/B-n type detector. Taking this into consideration, this paper presents the performance estimation of the MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detector with cut-off wavelength of kc = 5.2 lm at temperature T = 200 K. The temperature and bias
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voltage dependences of the dark current and RA product, responsivity, and detectivity of the CBIRD HgCdTe are analysed. Finally, near-room temperature MWIR CBIRD HgCdTe detector state of the art is compared to nBn/B-n type HgCdTe UBIRD, nBn InAs/ GaSb/B-Al0.2Ga0.8Sb T2SLs, PIN InAs/GaSb T2SL and HOT HgCdTe bulk photodiodes’ performance. 2. Simulation procedure Theoretical modelling of the MWIR UBIRD nBn/B-n type HgCdTe detector was performed by Velicu et al. [21] and Martyniuk and Rogalski [22]. For the comparison reasons, the similar structural parameters for MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detector were used. In the same way, for modelling purposes three layers electron barrier (EB–barrier I) was applied in order to mitigate the kinks emerging in energy diagrams between detector’s constituent layers caused by compositional uniformity (see Fig. 1). Interdiffusion was modelled by applying gauss tail doping (dx = 0.05 lm). The modelled structure consists of the 0.16 lm thick n-type HgCdTe cap layer doped to ND = 7 1014 cm–3. After the cap layer, an n-type 0.15 lm thick HgCdTe barrier doped to ND = 2 1015 cm–3 was incorporated. As mentioned, in our model the EB layer was divided on three sub-layers with composition grading fitted to the cap layer and absorber respectively (e.g. x = 0.33–0.6–0.275). The EB thickness was assumed to be thick enough to prevent electron tunnelling between the top contact layer and the absorbing layer, therefore the majority current is blocked by the barrier material under reverse applied bias. Next, n-type HgCdTe absorber with a thickness of 5–10 lm doped to ND = 1014 ? 5 1016 cm–3 and composition x = 0.275 for MWIR range was used. Finally, 0.15–0.5 lm thick hole blocking (HB – barrier II) layer consisted of two n-type sub-layers fitted to the absorber were utilized (e.g. x = 0.275–0.6) and doped to ND = 1014 ? 5 1017 cm–3. Similarly to EB, interdiffusion at absorber–barrier II (HB) interface was modelled by applying gauss tail doping. Numerical calculations were performed utilizing commercial software APSYS by Crosslight Software Inc. Specific equations and relations used in device’s modelling are listed in Table 1 and Appendix. The 50% cut-off wavelength was calculated to be kc = 5.2 lm at T = 200 K. Detector’s area was assumed to be 120 120 lm2. The noise current was calculated using the expression including Johnson–Nyquist noise, optical and electrical shot noises:
in ðVÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4kB T=RA þ 2qIDARK þ 2qIB Þ;
Fig. 1. CBIRD detector with the nB1nB2/B1,2-n type design: (a) schematic of the heterostructure and (b) the device’s structure.
ð1Þ
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Table 1 Parameters taken in modelling of MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detectors. Cap –3
Donor concentration, ND (cm ) Doping concentration’s gauss tail, dx (lm) Composition, x (lm) Geometry, d (lm) Device electrical area, A (lm2) Background temperature, TB, field of view, h Overlap matrix F1F2 Trap energy level, ETrap Trap concentration, NTrap (cm–3) Minority carrier lifetime SHR, sn, sp (ls) Target incident power density, U (W/m2)
Electron barrier (EB – barrier I) 14
7 10 0.05 0.15 ? 0.5 0.16 ? 1 120 120
15
14
2pcq k4
2
sin
10
0.33 ? 0.6 ? 0.275 0.15
0.275 (kc = 5.2 lm at T = 200 K) 2 ? 10
Z kc h ðexpðc=kB T B kÞ 1Þ1 gðkÞ dk; 2 0
R k
gðkÞ ¼ 1:24 i ;
ð2Þ
ð3Þ
The detector’s detectivity is defined by expression:
Ri pffiffiffi A: in ðVÞ
? 5 10
1014 ? 5 1017 0.275 ? 0.6 0.15 ? 0.5 lm
300 K, 20(f# = 2.835) 0.2 Eg/2 1016 0.4, 1 50 104
where TB is a background temperature, h is the detector’s field of view (h = 20°), and TB = 300 K was assumed. The quantum efficiency is a function of the incident radiation wavelength and current responsivity, Ri, according to the relation:
D ¼
Hole barrier (HB – barrier II) 16
2 10
where A is a detector’s area, RA is the dynamic resistance area product, IDARK and IB are the dark current density and background induced current, respectively, and kB is the Boltzmann constant. The background wavelength dependent current was calculated according to the expression:
IB ¼
Absorber
ð4Þ
Fig. 2 depicts calculated dark current versus reverse bias for selected operating temperatures which could be obtained using TE cooling. The electron barrier’s influence is clearly evident in I–V characteristics, where ‘‘turn-on’’ voltage (the voltage required to minimize valance band barrier) was assumed to be V = 0.2 V. For T = 200 K and V = 400 mV dark current reaches 0.1 A/cm2, while for T = 240 K dark current increases to 1 A/cm2. The inset compares simulation results of the structure presented in Fig. 1 with dark current obtained for UBIRD HgCdTe nBn/B-n-type (cap: x = 0.33, d = 0.16 lm, ND = 7 1014 cm–3; barrier: x = 0.33–0.6–0.275,
Fig. 2. IDARK versus voltage for CBIRD nB1nB2 HgCdTe/B1,2-n type and UBIRD nBn HgCdTe/B-n type detectors for selected operating temperatures. Inset: IDARK for CBIRD nB1nB2 HgCdTe/B1,2-n type and UBIRD nBn HgCdTe/B-n type detectors versus temperature for V = 400 mV.
d = 0.15 lm, ND = 2 1015 cm–3; absorber: x = 0.275, d = 10 lm, ND = 1014 cm–3). Within the operating temperature range between 180 and 240 K, the dark current increases from 0.01 to 0.7 A/cm2 while corresponding values for UBIRD structure change from 0.2 to 4 A/cm2 pointing out that dark current for CBIRD architecture decreases more rapidly than for UBIRD counterpart for lower temperatures. The hole barrier incorporation allows dark current reduction more than one order of magnitude depending on the operating temperature. It is clearly seen that for CBIRD nB1nB2 HgCdTe/B1,2-n type, the dark current IDARK = 0.7 A/cm2 is reached at T = 238 K, while the same value of the dark current for UBIRD nBn HgCdTe/B-n type requires extra cooling to T = 192 K. For T = 200 K and the bias voltages within the range V < 0.1 V, IDARK for both types of the detectors keeps the same value which corresponds to the EB II barrier formation with applied voltage. Additionally, it is shown that for voltages V < 200 mV dark current increases sharply (hole concentration increases harshly) while above V > 200 mV a typical photoconductive effect related to the increase of the current versus bias is observed (detector structure consists of four main n doped layers). Fig. 3 depicts D⁄ versus wavelength for selected voltages within turn-on voltage range. The maximum D⁄ was estimated to be 1.6 1010 cm Hz1/2/W for T = 200 K and V = 400 mV (for k = 4.9 lm). Likewise UBIRD nBn HgCdTe/B-n type detector, CBIRD nB1nB2 HgCdTe/B1,2-n type structure operates in minority carrier manner thus dark current is mainly due to the hole transport from absorber’s layer. n-type wide HgCdTe hole blocking layer’s incorporation suppresses dark current, changes turn-on voltage (for UBIRD HgCdTe nBn/B-n detector the turn-on voltage was found to be V = 400 mV) and increases detectivity.
Fig. 3. D⁄ versus wavelength (MWIR range) for CBIRD nB1nB2 HgCdTe/B1,2-n type detector for selected operating voltages (kc = 5.2 lm at T = 200 K).
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3. CBIRD nB1nB2 HgCdTe/B1,2-n type barrier’s band alignment The calculated energy band diagrams for unbiased (V = 0 V) and biased conditions (V = 0.4 V and V = 1 V) are shown in Fig. 4. CBIRD nB1nB2 HgCdTe/B1,2-n type detector is reversely biased, i.e. positive voltage is applied to the hole barrier contact. Unlike 6.1 Å AIIIBV family exhibiting ‘‘staggered’’ zero valance band offset, HgCdTe demonstrates ‘‘nested’’ heterojunction which leads to unintended valance band offset being difficult to control from technological perspective [electron affinity was modelled by Eq. (A.2)]. Comparison of the energy band alignment between unbiased and biased structures directly indicates that likewise UBIRD nBn HgCdTe/B-n type detector, CBIRD nB1nB2 HgCdTe/B1,2-n type one requires a proper level of voltage being applied to the detector (turn-on voltage is V = 0.2 V) to align the valance band (at the cap–barrier I, barrier I–absorber and absorber–barrier II interfaces) to reduce the impediment of desirable minority carrier transport to cap layer. As is mentioned above, the barrier incorporation into detector structure suppresses SHR rate. Magnitude of the SHR GR rate (rSHR) versus position for cap–barrier I (EB) interface for selected compositions is presented in Fig. 5. Barrier’s composition increase within the range x = 0.33–0.6 (sub-layer’s composition is fitted to the cap layer and absorber e.g. x = 0.33–0.4–0.275) and T = 200 K reduces rSHR nearly six orders of magnitude fully confirming the legitimacy of wide gap layers’ incorporation into n type HgCdTe detector’s structure. Ting et al. presented even 10 orders of magnitude suppression of the rSHR for AIIIBV UBIRD nBn structure (InAsSb/B–AlSbAs) and three orders of magnitude for InAs/GaSB T2SLs UBIRD nBn structure at T = 80 K [23]. Fig. 6 presents cap–barrier I (EB), barrier I–absorber and absorber–barrier II (HB) barrier’s heights DEc, DEv versus applied voltage
Fig. 5. The SRH GR rate for cap–barrier I interface of CBIRD nB1nB2 HgCdTe/B1,2 ntype detector for different compositions x at V = 50 mV and at T = 200 K.
respectively. As for as reverse biased CBIRD nB1nB2 HgCdTe/B1,2 ntype detector is concerned the most important is DEc emerging at cap–barrier I interface (desirable majority carrier blocking from cap layer), DEv at barrier I–absorber interface (unfavourable photo generated carriers impediment) and DEc, DEv at absorber–barrier II interface responsible for electron and hole blocking, respectively. The both mentioned DEc and DEv at cap–barrier I directly depend on the applied voltage pointing out that applied voltage is a trade-off between DEc and DEv (e.g. for barrier I–absorber interface DEv 160–50 meV and cap–barrier I interface DEc 350–275 meV for V = 0–1 V, respectively), while DEc at absorber–barrier II
Fig. 4. Calculated energy band structures for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector at: equilibrium (V = 0 V) (a) and under reverse biases: V = 0.4 V (b) and V = 1 V (c).
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Fig. 6. DEc and DEv for cap–barrier I (EB), barrier I–absorber, absorber–barrier II (HB) interfaces versus applied voltage at T = 200 K.
Fig. 7. DEc and DEv for absorber–barrier II (HB) interface versus barrier’s II doping (barrier II: x = 0.275–0.6) and composition (barrier II: ND = 1017 cm–3). V = 0 V, T = 200 K.
interface changes within range DEc 20–70 meV and DEv keeps constant value 450 mV. Above V > 0.2 V, the absorber–barrier II interface, DEc, saturates leading to the dark slight increase (turnon voltage level V = 0.2 V – see Fig. 2), while turn-on voltage for photocurrent is found to be V = 0.4 V (see Fig. 4), where DEv at barrier I–absorber plays crucial role blocking photogenerated carriers. Both barrier’s doping and composition also influence DEc and DEv (see Fig. 7). Once barrier I (EB) composition increases both DEc and DEv at cap–barrier I and barrier I–absorber raises reducing dark and photo currents. The barrier II doping increase reduces absorber–barrier II (HB) DEc (within the range 325–65 mV) and slightly raises DEv (425–450 mV) for ND < 1017 cm–3 while for ND > 1017 cm–3 opposite behaviour is observed. Increase of barrier II composition raises both DEc and DEv at absorber–barrier II interface.
prevent electron tunnelling from cap layer to the absorber. HgCdTe exhibits the potential issues with uniformity of the thin layers due to the interdiffusion at the interfaces. In our model both EB and HB barrier layers were divided on three (electron barrier)/two (hole barrier) sub-layers with composition grading fitted to cap and absorber layers to simulate this unfavourable phenomenon (additionally gauss tail’s doping profile was used, dx = 0.05 lm). Figs. 8 and 9 present IDARK, IPHOTO (for selected voltages) and D⁄ (for selected temperatures) versus HB composition. As it is shown in Fig. 7, both hole barrier (DEc and DEv) increase with HB composition resulting in IDARK decreasing for x < 0.35, while for x > 0.35 IDARK saturates. HB composition influences IPHOTO for 0.35 > x > 0.53 which should be attributed to HB DEc increase (x > 0.53) and IDARK raise for x < 0.35. D⁄ versus HB composition reflects the trend exhibited by IPHOTO dependence on HB x. HB doping influence on detector’s performance is depicted in Figs. 10 and 11. For hole barrier’s doping below ND < 1016 cm–3, both dark and photocurrent decrease two orders of magnitude in comparison to values obtained for ND = 1017 cm–3 for all analyzed voltages. This behaviour is directly connected with HB DEc decrease versus HB doping. The results presented in Fig. 11 points out that the highest detectivity is reached only for highly doped HB (ND > 1016 cm–3) for all simulated voltages. Increase of the HB doping within the range 1014–5 1017 cm–3 raises D⁄ from 108 to 1010 cm Hz1/2/W for V = 400 mV. Above ND > 1016 cm–3, increase of HB doping saturates to D⁄ 1.6 1010 cm Hz1/2/W. Similar considerations were conducted for EB barrier’s composition. Direct dependence of the EB DEc and DEv on composition and voltage is responsible for the both dark and photocurrent characteristics. Both EB DEc and DEv raise with composition resulting in IDARK and IPHOTO decrease (see Fig. 12). Below x < 0.4 both currents keep nearly constant value for simulated voltages. To attain the highest detectivity, the optimal composition increases with applied voltage (see Fig. 13). The simulated structure reaches D⁄ 1.6 1010 cm Hz/1/2/W for V = 400 mV and T = 200 K, being one order of magnitude higher than UBIRD nBn HgCdTe/B-n type single barrier structure, whereas detectivity was estimated to be D⁄ = 3.5 109 cm Hz1/2/W for the same EB parameters, working conditions and structural parameters (absorber’s doping and width). 5. Optimization of the CBIRD nB1nB2 HgCdTe/B1,2-n type absorber Doping of absorbers also plays critical role and must be optimized for assumed voltages and operating temperatures. Figs. 14
4. Optimization of the CBIRD nB1nB2 HgCdTe/B1,2-n type barriers The choice of the both electron and hole barriers’ doping and composition plays crucial role in designing CBIRD nB1nB2 HgCdTe/B1,2 n-type structures. HB width was checked thoroughly (within the range d = 0.15–5 lm) not revealing strict influence on D⁄. The EB width was chosen to be thick enough (d = 0.15 lm) to
Fig. 8. IDARK and IPHOTO for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus hole barrier composition for selected voltages. T = 200 K. HB doping ND = 1017 cm3.
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Fig. 9. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus hole barrier composition for selected temperatures. V = 400 mV. k = 4.95 lm.
Fig. 12. IDARK and IPHOTO for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus electron barrier’s composition for selected voltages.
Fig. 10. IDARK and IPHOTO for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector versus hole barrier doping concentration for selected voltages. T = 200 K.
Fig. 13. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus applied voltage for selected electron barrier’s composition. T = 200 K, k = 4.95 lm.
photocurrents keep nearly constant value for simulated voltages. Above ND > 1015 cm–3 photocurrent decreases sharply which could be attributed to Burstein–Moss effect (minority carrier density drops). Once absorber’s doping raises, the dark current increases sharply (mainly due to Auger 1 process), while the photocurrent decreases (free minority carrier concentration decreases). Clearly
Fig. 11. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2 n-type structure versus barrier doping concentration for selected voltages. T = 200 K, k = 4.95 lm.
and 15 present IDARK and IPHOTO versus applied voltage (for selected absorber’s doping) and absorber’s doping, respectively. Once absorber’s doping increases, the photocurrent exhibits decreasing trend lowering responsivity Ri = 2.3–1.3 A/W and quantum efficiency g = 59–32% (within the range 1014 < ND < 5 1016 cm–3). In doping range 1014 < ND < 1015 cm–3 both dark current and
Fig. 14. IDARK and IPHOTO for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector versus voltage for different absorbers doping. T = 200 K.
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Fig. 15. IDARK and IPHOTO for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector versus absorber’s doping for selected voltages. T = 200 K.
Fig. 17. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2-n type structure versus absorber’s width for selected voltages. T = 200 K, k = 4.95 lm.
6. Comparison of the InAs/GaSb T2SLs and HgCdTe technologies
Fig. 16. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2-n type structure versus applied voltage for selected absorber’s doping concentrations. T = 200 K.
seen are turn-on voltages for both dark and photocurrents. The barriers’ parameters were assumed to be as follows: EB x = 0.33– 0.6–0.275, d = 0.15 lm, ND = 2 1015 cm3; HB x = 0.275–0.6, d = 0.4 lm, ND = 1017 cm3. Since CBIRD nB1nB2 HgCdTe/B1,2 n-type detector is a minority carrier device, absorber’s doping decreases concentration of the free holes lowering IDARK. Further absorber’s doping increase contributes to the IDARK current raise due to EB DEc lowering and Auger 1 effect. The lowest value of IDARK = 0.5 A/cm2 for V = 400 mV could be obtained for ND = 5 1015 cm–3 for V = 100 mV but for this absorber doping IPHOTO assumes the lowest value which indicates that optimal working conditions appears for ND < 1015 cm–3. Fig. 16 presents D⁄ versus voltage for selected absorber’s doping. Presented results indicate that the maximum value of the D⁄ = 1.6 1010 cmHz1/2/W for given structure could be obtained for absorbers doping ND = 1014 cm–3 and V = 400 mV while at doping level above 1014 cm–3, detectivity decreases rapidly due to dark current increase and lowering of the photocurrent. Once absorber’s doping increases, the optimal voltage to be applied to the structure to attain the highest detectivity decreases. Absorber width influences the IR absorption what is shown in Fig. 17. For voltage V = 400 mV the optimal thickness is estimated to be d = 5 lm which allows reaching detectivity D⁄ = 2 1010 cm Hz1/2/W. Once bias decreases optimal thickness decreases and for V = 300 mV equals d = 4 lm.
The very last figure (Fig. 18) shows the RoA and RA products versus temperature for MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detector, UBIRD nBn HgCdTe/B-n type (kc = 5.2 lm) detector, InAs/GaSb/B–AlGaSb T2SL nBn (kc = 5.4 lm) detector, InAs/GaSb PIN photodiode (kc = 6.2 lm), and finally HOT HgCdTe bulk photodiodes (kc = 5.4 lm) fabricated at the joint laboratory run by Institute of Applied Physics, Military University of Technology/Vigo System SA. Theoretical estimation for the MWIR UBIRD nBn HgCdTe/B-n type was conducted by Martyniuk and Rogalski in Ref. [22], whereas the performance of T2SLs nBn InAs/GaSb/B-AlGaSb detector’s performance was presented in Ref. [24] where an analytical approach was used to model the detectors’s performance. PIN T2SLs photodiodes were analysed by Wrobel et al. [25]. It is clearly seen that the performance of CBIRD nB1nB2 HgCdTe/B1,2-n type structure has reached a comparable level determined by the state of the art of HgCdTe bulk photodiodes and put itself in superior position with reference to UBIRD nBn HgCdTe/B-n type, T2SLs nBn, and PIN detectors. The particular significance of the incorporation of the extra barrier for minority carriers in CBIRD structures versus single barrier (majority carriers’ blocking) in UBIRD detectors is clearly evident by RA product increase from 0.1 to 0.5 Xcm2 for T = 230 K (for the same absorber’s
Fig. 18. Temperature dependence of the RA and RoA products for MWIR nB1nB2 HgCdTe/B1,2-n type detector, nBn HgCdTe/B-n type detector, nBn InAs/GaSb/BAl0.2Ga0.8Sb T2SL detector, HgCdTe HOT bulk diodes and PIN InAs/GaSb T2SL diodes operating at near-room temperature (T = 230 K).
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doping ND = 1014 cm–3 and thickness d = 10 lm) resulting in raising of detectivity e.g. from 3.5 109 to 1.6 1010 cm Hz1/2/W for V = 400 mV and T = 200 K. Once absorber’ thickness is reduced to the optimal size to attain the highest detectivity, the RA product increases to 0.9 Xcm2 for T = 230 K and detectivity reaches 2 1010 cm Hz1/2/W at V = 400 mV and T = 200 K. In fact, the RA products of MWIR 5.2 lm CBIRD nB1nB2 HgCdTe/B1,2-n type is slightly higher in comparison to bulk HgCdTe photodiodes, but it was calculated for V = 50 mV reverse bias.
Band-gap energy:
Eg ðx; TÞ ¼ 0:302 þ 1:93x 0:81x2 þ 0:832x3 þ 5035 104 Tð1 2xÞ:
ðA:1Þ
Electron affinity:
c ¼ 4023 0:813½Eg ðx; TÞ 0:083:
ðA:2Þ
Carriers’ effective masses:
me ¼ 8:035 102 Eg ðx; TÞm0 ;
ðA:3Þ
mh ¼ 0:55m0 :
ðA:4Þ
7. Conclusions In the paper we theoretically estimated the performance of the CBIRD nB1nB2 HgCdTe/B1,2-n type detector versus operating conditions and structural parameters. The unfavourable compositional uniformity and interdiffusion at the interfaces were modelled by proper barrier grading matched to the cap and absorber’s composition respectively. The maximum RA product of the detector with 5.2 lm cut-off wavelength is higher than 0.9 Xcm2 at 230 K while maximum detectivity was estimated to be 2 1010 cm Hz1/2/W at T = 200 K assuming an absorber thickness d = 5 lm and V = 400 mV. Inherited barriers in both conduction and valance bands were analysed in detail pointing the optimal operating conditions as for as bias and doping are concerned. Turn-on voltage above which dark current increases slowly was estimated to be V = 0.2 V for dark current and 0.4 V for photocurrent respectively. Although, it is unfeasible to attain wanted band alignment in valance band, the theoretically predicted CBIRD nB1nB2 HgCdTe/ B1,2-n type structure demonstrate performance which underline significance and full legitimacy of the barriers’ incorporation to the detector’s architectures. Similarly to UBIRD HgCdTe/B-n type, the analysed structure allows circumventing requirements for ptype doping reducing number of the processing steps. Barrier’s doping and composition should be perceived as the most important parameters in CBIRD nB1nB2 HgCdTe/B1,2-n structure optimization. The proper doping and composition choice leads to either building up or lowering the barriers in both conduction and valance bands. It was shown that the raise of the operating temperature by nearly 50 K could be reached by additional incorporation of the minority carrier barrier into UBIRD structure. The extra barrier leads to suppressing of the dark current by nearly one order of magnitude. As mentioned in introduction, T2SL InAs/GaSb 6.1 Å compound family is the only one infrared material system theoretically predicted to achieve higher performance than HgCdTe bulk photodiodes. However, so far the HOT HgCdTe photodiode performance has not been overcome by T2SL PIN and UBIRD nBn structures because of the low quantum efficiency and presence of the SRH recombination characterized by a relatively short carrier lifetime. Mentioned limitations could be circumvented by simplified CBIRD nB1nB2 HgCdTe/B1,2-n type structures and UBIRD nBn HgCdTe/B-n type detectors. Finally, unlike T2SLs InAs/GaSb/B–AlGaSb nBn detector, CBIRD nB1nB2 HgCdTe/B1,2-n type and UBIRD nBn HgCdTe/B-n type detector do not require highly doped layers which should be perceived as a technological advantage. Acknowledgement This paper has been done under financial support of the Polish National Science Centre, Project: DEC-2011/01/B/ST5/06283. Appendix A The CBIRD nB1nB2 HgCdTe/B1,2-n type detector was simulated using the following material parameters [1,26]:
Dielectric constant:
e ¼ 20:5 15:5x þ 5:7x2 :
ðA:5Þ
The radiative recombination rate:
B ¼ 5:9052
1018 n2 i
eT
3=2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þx Eg exp ðE2g ð81:9 þ TÞ kB T
2
þ 3kB TEg þ 3:75kB T 2 Þ:
ðA:6Þ
The Auger recombination coefficients Cn and Cp:
C n ¼ 5 1012 jF 1 F 2 j
"
" n2 3:8 1018 e2 i
!#1=2 3 mg Eg Eg exp 1 þ 2 kB T mh kB Tðmg =mh Þ ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #1 mg mg 1 1 þ 2 1þ ; ðA:7Þ mg mh mh
C p ¼ 0:1C n :
ðA:8Þ
The absorption coefficient: for: k > kc
n¼
ag n ; a0
ðA:9Þ
1:24=kc e0 ; eg e0
ðA:10Þ
a ¼ a0
for: k 6 kc
a ¼ ag exp
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B½1:24=kc eg ;
ðA:11Þ
Eg ¼ 0:295 þ 1:87x 0:28x2 þ ð6 14x þ 3x2 Þ 104 T þ 0:35x4 ;
ðA:12Þ
ag ¼ 65 þ 1:883T þ ð8694 10:314TÞx;
ðA:13Þ
a0 ¼ expð18:5 þ 45:68xÞ;
ðA:14Þ
e0 ¼ 0:355 þ 1:77x;
ðA:15Þ
B¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þ 0:083T þ ð21 0:13TÞx:
ðA:16Þ
References [1] Rogalski A. Infrared detectors. 2nd ed. Boca Raton: CRC Press; 2011. [2] Rogalski A. HgCdTe infrared detector material: history, status and outlook. Rep Prog Phys 2005;68:2267–336. [3] Norton P. HgCdTe infrared detectors. Opto-Electron Rev 2002;10:159–74. [4] Gunapala SD, Bandara SV. GaAs/AlGaAs based quantum well infrared photodetector focal plane arrays. In: Henini M, Razeghi M, editors. Handbook of infrared detection technologies. Oxford: Elsevier; 2002. p. 83–119. [5] Rogalski A. Quantum well photoconductors in infrared detectors technology. J Appl Phys 2003;93:4355–91.
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[6] Rogalski A, Martyniuk P. InAs/GaInSb superlattices as a promising material system for third generation infrared detectors. Infrared Phys Technol 2006;48:39–52. [7] Ting DZ-Y, Soibel A, Höglund L, Nguyen J, Hill CJ, Khoshakhlagh A, et al. Type-II superlattice infrared detectors. In: Gunapala SD, Rhiger DR, Jagadish C, editors. Semiconductors and semimetals, vol. 84. Amsterdampp: Elsevier; 2011. p. 1– 57. [8] Vurgaftman I, Aifer EH, Canedy CL, Tischler JG, Meyer JR, Warner JH. Graded band gap for dark-current suppression in long-wave infrared W-structured type-II superlattice photodiodes. Appl Phys Lett 2006;89:121114. [9] Maimon S, Wicks G. nBn detector, an infrared detector with reduced dark current and higher operating temperature. Appl Phys Lett 2006;89. 151109-1– 3. [10] Klipstein P. XBn’ barrier photodetectors for high sensitivity and high operating temperature infrared sensors. Proc SPIE 2008;6940. 69402U-01–11. [11] Aifer EH, Tischler JG, Warner JH, Vurgaftman I, Kim JC, Meyer JR, et al. Wstructured type-II superlattice based long and very-long wavelength infrared photodiodes. Proc SPIE 2005;5732:259–72. [12] Nguyen BM, Hoffman D, Delaunay PY, Razeghi M. Dark current suppression in type II InAs/GaSb superlattice longwavelength infrared photodiodes with Mstructure barrier. Appl Phys Lett 2007;91:163511. [13] Ting DZ-Y, Hill CJ, Soibel A, Keo SA, Mumolo JM, Nguyen J, et al. A highperformance long wavelength superlattice complementary barrier infrared detector. Appl Phys Lett 2009;95:023508. [14] Gautam N, Kim HS, Kutty MN, Plis E, Dawson LR, Krishna S. Performance improvement of longwave infrared photodetector based on type-II InAs/GaSb superlattices using unipolar current blocking layers. Appl Phys Lett 2010;96:231107.
[15] Rodriguez JB, Plis E, Bishop G, Sharma YD, Kim H, Dawson LR, et al. nBn structure based on InAs/GaSb type-II strained layer superlattices. Appl Phys Lett 2007;91. 043514-1-2. [16] Myers S, Plis E, Morath C, Cowan V, Gautam N, Klein B, et al. Comparison of superlattice based dual color nBn and pBp infrared detectors. Proc SPIE 2011;8155:815507–11. [17] Wróbel J, Martyniuk P, Plis E, Madejczyk P, Gawron W, Krishna S, et al. Dark current modeling of MWIR type-II superlattice detectors. Proc SPIE 2012;8353:835316. [18] Martyniuk P, Rogalski A. Comparison of performance of quantum dot and other types infrared photodetectors. Proc SPIE 2008;6940:694004. [19] Martyniuk P, Rogalski A. Insight into performance of quantum dot infrared photodetectors. Bull Polish Acad Sci: Tech Sci 2009;57:103–16. [20] Itsuno AM, Philips JD, Velicu S. Design and modeling of HgCdTe nBn detectors. J Elect Mater 2011;40:1624–9. [21] Velicu S, Zhao J, Morley M, Itsuno AM, Philips JD. Theoretical investigation of MWIR HgCdTe nBn detectors. Proc SPIE 2012;8268:82682X. [22] Martyniuk P, Rogalski A. Theoretical modeling of MWIR thermoelectrically cooled nBn HgCdTe detector, Bull Polish Acad Sci: Tech Sci, in press. [23] Ting DZ, Hill CJ, Soibel A, Nguyen J, Keo S, Lee MC, et al. Antimonide-based barrier infrared detectors. Proc SPIE 2010;7660:76601R. [24] Martyniuk P, Wrobel J, Plis E, Madejczyk P, Kowalewski A, Gawron W, et al. Performance modeling of MWIR InAs/GaSb/B–Al0.2Ga0.8Sb type-II superlattice nBn detector. Semicond Sci Technol 2012;27:055002. [25] Wrobel J, Plis E, Madejczyk P, Gawron W, Martyniuk P, Krishna S, et al. Analysis of temperature dependence of dark current mechanisms for mid wavelength infrared InAs/GaSb superlattice photodiodes, in press. [26] APSYS macro/user’s manual ver. 2011. Crosslight Software, Inc.; 2011.
Contents lists available at SciVerse ScienceDirect
Solid-State Electronics journal homepage: www.elsevier.com/locate/sse
Modelling of MWIR HgCdTe complementary barrier HOT detector Piotr Martyniuk ⇑, Antoni Rogalski Institute of Applied Physics, Military University of Technology, 2 Kaliskiego St., 00-908 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 9 July 2012 Received in revised form 11 October 2012 Accepted 26 October 2012 Available online 21 December 2012 The review of this paper was arranged by Prof. E. Calleja Keywords: Complementary barrier infrared detectors Unipolar barrier detectors nBn detectors HgCdTe Type-II InAs/GaSb superlattices
a b s t r a c t The paper reports on the photoelectrical performance of medium wavelength infrared (MWIR) HgCdTe complementary barrier infrared detector (CBIRD) with n-type barriers. CBIRD nB1nB2 HgCdTe/B1,2-n type detector is modelled with commercially available software APSYS by Crosslight Software Inc. The detailed analysis of the detector’s performance such as dark current, photocurrent, responsivity, detectivity versus applied bias, operating temperature, and structural parameters (cap, barriers and absorber doping; and absorber and barriers compositions) are performed pointing out optimal working conditions. Both conduction and valence bands’ alignment of the HgCdTe CBIRD structure are calculated stressing their importance on detectors performance. It is shown that higher operation temperature (HOT) conditions achieved by commonly used thermoelectric (TE) coolers allows to obtain detectivities D⁄ 2 1010 cm Hz1/2/W at T = 200 K and reverse polarisation V = 400 mV, and differential resistance area product RA = 0.9 Xcm2 at T = 230 K for V = 50 mV, respectively. Finally, CBIRD nB1nB2 HgCdTe/B1,2-n type state of the art is compared to unipolar barrier HgCdTe nBn/ B-n type detector, InAs/GaSb/B-Al0.2Ga0.8Sb type-II superlattice (T2SL) nBn detectors, InAs/GaSb T2SLs PIN and the HOT HgCdTe bulk photodiodes’ performance operated at near-room temperature (T = 230 K). It was shown that the RA product of the MWIR CBIRD HgCdTe detector is either comparable or higher (depending on structural parameters) to the state of the art of HgCdTe HOT bulk photodiodes and both AIIIBV 6.1 Å family T2SLs nBn and PIN detectors. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Hitherto, the infrared radiation (IR) industry is conquered by HgCdTe bulk photodiodes [1–3] and GaAs/AlGaAs intersubband quantum well infrared photodetectors (QWIP) [4,5]. The requirement of the infrared detectors’ cryogenic cooling is a major impediment preventing from their extensive application that is why detector’s cooling push boundaries to increase device operating temperature. It is known that the critical condition which must be fulfilled to construct the HOT IR detector is to achieve both low dark current and high values of the quantum efficiency. Among the mechanisms generating the dark current in detector’s structure the following must be enumerated: band-to-band (BTB) tunnelling, trap assisted tunnelling (TAT), Schockley-Read-Hall (SRH) generation-recombination (GR) process, Auger GR process, and leakage currents. It was found that an incorporation of the type II superllatice (T2SLs) e.g. InAs/GaSb 6.1 Å AIIIBV family into detector architecture allows to reduce adverse BTB/TAT currents and GR Auger’s contribution to the total dark current. Therefore T2SLs could be considered as an alternative to the bulk HgCdTe HOT detectors and
⇑ Corresponding author. E-mail address: [email protected] (P. Martyniuk). 0038-1101/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sse.2012.10.021
GaAs/AlGaAs IR material systems [6]. Unfavourable SHR GR and leakage dark current’s components could be limited by the properly selected barriers incorporated into detectors structure. The barrier’s selection plays crucial role due to the lattice constant matching of the detectors’ constituent layers, the barrier’s height in both conduction and valance bands connected directly to the band alignment. It must be stressed that band alignment playing important role in design of the barrier IR structures is often fortuitous and extremely difficult to control from technological perspective [7]. The very first barrier structures were commonly known AIIBVI and AIIIBV heterostructures invented to increase device’s performance by suppression of the diffusion currents from the detector’s active region. The next stage in IR detector’s development was double layer heterojunction (DLHJ) allowing reducing both majority and minority carriers diffusion currents in comparison to the homojunction IR detectors. Another variation of the DLHJ was a graded gap structure incorporated between hole blocking layer and absorber to suppress tunnelling and GR currents [8]. Currently, among the barrier IR detectors (BIRDs) the leading position is occupied by minority carrier devices called unipolar barrier infrared detectors (UBIRD) proposed by Maimon and Wicks [9]. Among electron-blocking UBIRD detectors the most important are these designed with AIIIBV compounds (GaSb, InAs1–zSbz-cap layers, InAs1–ySby-active region, AlSb1–xAsx–barrier), T2SLs nBn
P. Martyniuk, A. Rogalski / Solid-State Electronics 80 (2013) 96–104
InAs/GaSb with AlGaSb/T2SLs barriers and UBIRD nBn HgCdTe detectors, while among hole-blocking UBIRD devices should be considered a four layer architectures: InAs/GaInSb/InAs/AlGaInSb often called ‘‘W’’ structures and GaSb/InAs/GaSb/AlSb referenced as ‘‘M’’ structures [10–12]. The most sophisticated structures containing both electron and hole blocking barriers (AlSb/T2SLs barriers) were proposed by Ting et al. [13] (called either complementary barrier infrared detectors – CBIRD) or PbIbN after Gautam et al. [14] showing possible advantages in suppressing dark currents by blocking majority and minority carriers and circumventing technological problems with making ohmic contact to the widegap layers. Potential interest in InAs/GaSb T2SLs results not only from unique inherited capabilities of the new artificial material with entirely different physical properties in comparison to the constituent layers (InAs and GaSb), but also from the nearly zero band offsets leading to the desirable UBIRD/CBIRD band alignments difficult to attain in HgCdTe [15]. In addition, the 6.1 Å family’s capabilities to tune the position of the conduction and valance band edges in independent way and near lattice matching is extremely helpful in designing of the unipolar and complementary barrier detectors. Although the abovementioned physical properties indicates potential T2SLs’ superiority over bulk materials (including HgCdTe ternary allys), the T2SLs’ quantum efficiency leaves a lot of to be desired (g = 20–30% in MWIR range and g = 8–12% in LWIR range depending on nBn/pBp architecture) which stems from low level of wavefunction overlapping and technological problems connected with growth of uniform and thick enough SLs [16]. What is more, short minority carrier lifetimes (sDIF, sGR < 10 ns in temperature range >200 K) also impedes the development of the T2SLs IR devices [17]. Similarly, theoretical simulations proved quantum dot infrared detectors (QDIPs) to be an alternative to the HgCdTe, but technological problems related to the growth of self-organized QDs led to the suspension of the research on this type of the detector [18,19]. Even though, HgCdTe does not exhibit valance zero band offset, it is commonly known that bulk HgCdTe offers quantum efficiency g = 50–70%, therefore recently research groups have attempted to apply UBIRD architecture to HgCdTe alloy (n type barrier) which offers technological advantages over p–n HgCdTe homojunction (simplifying the fabrication process) [20,21]. Moving forward, it is worth applying CBIRD architecture to HgCdTe alloy incorporating n type barriers to reduce dark current in comparison to UBIRD nBn HgCdTe/B-n type detector. Taking this into consideration, this paper presents the performance estimation of the MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detector with cut-off wavelength of kc = 5.2 lm at temperature T = 200 K. The temperature and bias
97
voltage dependences of the dark current and RA product, responsivity, and detectivity of the CBIRD HgCdTe are analysed. Finally, near-room temperature MWIR CBIRD HgCdTe detector state of the art is compared to nBn/B-n type HgCdTe UBIRD, nBn InAs/ GaSb/B-Al0.2Ga0.8Sb T2SLs, PIN InAs/GaSb T2SL and HOT HgCdTe bulk photodiodes’ performance. 2. Simulation procedure Theoretical modelling of the MWIR UBIRD nBn/B-n type HgCdTe detector was performed by Velicu et al. [21] and Martyniuk and Rogalski [22]. For the comparison reasons, the similar structural parameters for MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detector were used. In the same way, for modelling purposes three layers electron barrier (EB–barrier I) was applied in order to mitigate the kinks emerging in energy diagrams between detector’s constituent layers caused by compositional uniformity (see Fig. 1). Interdiffusion was modelled by applying gauss tail doping (dx = 0.05 lm). The modelled structure consists of the 0.16 lm thick n-type HgCdTe cap layer doped to ND = 7 1014 cm–3. After the cap layer, an n-type 0.15 lm thick HgCdTe barrier doped to ND = 2 1015 cm–3 was incorporated. As mentioned, in our model the EB layer was divided on three sub-layers with composition grading fitted to the cap layer and absorber respectively (e.g. x = 0.33–0.6–0.275). The EB thickness was assumed to be thick enough to prevent electron tunnelling between the top contact layer and the absorbing layer, therefore the majority current is blocked by the barrier material under reverse applied bias. Next, n-type HgCdTe absorber with a thickness of 5–10 lm doped to ND = 1014 ? 5 1016 cm–3 and composition x = 0.275 for MWIR range was used. Finally, 0.15–0.5 lm thick hole blocking (HB – barrier II) layer consisted of two n-type sub-layers fitted to the absorber were utilized (e.g. x = 0.275–0.6) and doped to ND = 1014 ? 5 1017 cm–3. Similarly to EB, interdiffusion at absorber–barrier II (HB) interface was modelled by applying gauss tail doping. Numerical calculations were performed utilizing commercial software APSYS by Crosslight Software Inc. Specific equations and relations used in device’s modelling are listed in Table 1 and Appendix. The 50% cut-off wavelength was calculated to be kc = 5.2 lm at T = 200 K. Detector’s area was assumed to be 120 120 lm2. The noise current was calculated using the expression including Johnson–Nyquist noise, optical and electrical shot noises:
in ðVÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4kB T=RA þ 2qIDARK þ 2qIB Þ;
Fig. 1. CBIRD detector with the nB1nB2/B1,2-n type design: (a) schematic of the heterostructure and (b) the device’s structure.
ð1Þ
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P. Martyniuk, A. Rogalski / Solid-State Electronics 80 (2013) 96–104
Table 1 Parameters taken in modelling of MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detectors. Cap –3
Donor concentration, ND (cm ) Doping concentration’s gauss tail, dx (lm) Composition, x (lm) Geometry, d (lm) Device electrical area, A (lm2) Background temperature, TB, field of view, h Overlap matrix F1F2 Trap energy level, ETrap Trap concentration, NTrap (cm–3) Minority carrier lifetime SHR, sn, sp (ls) Target incident power density, U (W/m2)
Electron barrier (EB – barrier I) 14
7 10 0.05 0.15 ? 0.5 0.16 ? 1 120 120
15
14
2pcq k4
2
sin
10
0.33 ? 0.6 ? 0.275 0.15
0.275 (kc = 5.2 lm at T = 200 K) 2 ? 10
Z kc h ðexpðc=kB T B kÞ 1Þ1 gðkÞ dk; 2 0
R k
gðkÞ ¼ 1:24 i ;
ð2Þ
ð3Þ
The detector’s detectivity is defined by expression:
Ri pffiffiffi A: in ðVÞ
? 5 10
1014 ? 5 1017 0.275 ? 0.6 0.15 ? 0.5 lm
300 K, 20(f# = 2.835) 0.2 Eg/2 1016 0.4, 1 50 104
where TB is a background temperature, h is the detector’s field of view (h = 20°), and TB = 300 K was assumed. The quantum efficiency is a function of the incident radiation wavelength and current responsivity, Ri, according to the relation:
D ¼
Hole barrier (HB – barrier II) 16
2 10
where A is a detector’s area, RA is the dynamic resistance area product, IDARK and IB are the dark current density and background induced current, respectively, and kB is the Boltzmann constant. The background wavelength dependent current was calculated according to the expression:
IB ¼
Absorber
ð4Þ
Fig. 2 depicts calculated dark current versus reverse bias for selected operating temperatures which could be obtained using TE cooling. The electron barrier’s influence is clearly evident in I–V characteristics, where ‘‘turn-on’’ voltage (the voltage required to minimize valance band barrier) was assumed to be V = 0.2 V. For T = 200 K and V = 400 mV dark current reaches 0.1 A/cm2, while for T = 240 K dark current increases to 1 A/cm2. The inset compares simulation results of the structure presented in Fig. 1 with dark current obtained for UBIRD HgCdTe nBn/B-n-type (cap: x = 0.33, d = 0.16 lm, ND = 7 1014 cm–3; barrier: x = 0.33–0.6–0.275,
Fig. 2. IDARK versus voltage for CBIRD nB1nB2 HgCdTe/B1,2-n type and UBIRD nBn HgCdTe/B-n type detectors for selected operating temperatures. Inset: IDARK for CBIRD nB1nB2 HgCdTe/B1,2-n type and UBIRD nBn HgCdTe/B-n type detectors versus temperature for V = 400 mV.
d = 0.15 lm, ND = 2 1015 cm–3; absorber: x = 0.275, d = 10 lm, ND = 1014 cm–3). Within the operating temperature range between 180 and 240 K, the dark current increases from 0.01 to 0.7 A/cm2 while corresponding values for UBIRD structure change from 0.2 to 4 A/cm2 pointing out that dark current for CBIRD architecture decreases more rapidly than for UBIRD counterpart for lower temperatures. The hole barrier incorporation allows dark current reduction more than one order of magnitude depending on the operating temperature. It is clearly seen that for CBIRD nB1nB2 HgCdTe/B1,2-n type, the dark current IDARK = 0.7 A/cm2 is reached at T = 238 K, while the same value of the dark current for UBIRD nBn HgCdTe/B-n type requires extra cooling to T = 192 K. For T = 200 K and the bias voltages within the range V < 0.1 V, IDARK for both types of the detectors keeps the same value which corresponds to the EB II barrier formation with applied voltage. Additionally, it is shown that for voltages V < 200 mV dark current increases sharply (hole concentration increases harshly) while above V > 200 mV a typical photoconductive effect related to the increase of the current versus bias is observed (detector structure consists of four main n doped layers). Fig. 3 depicts D⁄ versus wavelength for selected voltages within turn-on voltage range. The maximum D⁄ was estimated to be 1.6 1010 cm Hz1/2/W for T = 200 K and V = 400 mV (for k = 4.9 lm). Likewise UBIRD nBn HgCdTe/B-n type detector, CBIRD nB1nB2 HgCdTe/B1,2-n type structure operates in minority carrier manner thus dark current is mainly due to the hole transport from absorber’s layer. n-type wide HgCdTe hole blocking layer’s incorporation suppresses dark current, changes turn-on voltage (for UBIRD HgCdTe nBn/B-n detector the turn-on voltage was found to be V = 400 mV) and increases detectivity.
Fig. 3. D⁄ versus wavelength (MWIR range) for CBIRD nB1nB2 HgCdTe/B1,2-n type detector for selected operating voltages (kc = 5.2 lm at T = 200 K).
P. Martyniuk, A. Rogalski / Solid-State Electronics 80 (2013) 96–104
99
3. CBIRD nB1nB2 HgCdTe/B1,2-n type barrier’s band alignment The calculated energy band diagrams for unbiased (V = 0 V) and biased conditions (V = 0.4 V and V = 1 V) are shown in Fig. 4. CBIRD nB1nB2 HgCdTe/B1,2-n type detector is reversely biased, i.e. positive voltage is applied to the hole barrier contact. Unlike 6.1 Å AIIIBV family exhibiting ‘‘staggered’’ zero valance band offset, HgCdTe demonstrates ‘‘nested’’ heterojunction which leads to unintended valance band offset being difficult to control from technological perspective [electron affinity was modelled by Eq. (A.2)]. Comparison of the energy band alignment between unbiased and biased structures directly indicates that likewise UBIRD nBn HgCdTe/B-n type detector, CBIRD nB1nB2 HgCdTe/B1,2-n type one requires a proper level of voltage being applied to the detector (turn-on voltage is V = 0.2 V) to align the valance band (at the cap–barrier I, barrier I–absorber and absorber–barrier II interfaces) to reduce the impediment of desirable minority carrier transport to cap layer. As is mentioned above, the barrier incorporation into detector structure suppresses SHR rate. Magnitude of the SHR GR rate (rSHR) versus position for cap–barrier I (EB) interface for selected compositions is presented in Fig. 5. Barrier’s composition increase within the range x = 0.33–0.6 (sub-layer’s composition is fitted to the cap layer and absorber e.g. x = 0.33–0.4–0.275) and T = 200 K reduces rSHR nearly six orders of magnitude fully confirming the legitimacy of wide gap layers’ incorporation into n type HgCdTe detector’s structure. Ting et al. presented even 10 orders of magnitude suppression of the rSHR for AIIIBV UBIRD nBn structure (InAsSb/B–AlSbAs) and three orders of magnitude for InAs/GaSB T2SLs UBIRD nBn structure at T = 80 K [23]. Fig. 6 presents cap–barrier I (EB), barrier I–absorber and absorber–barrier II (HB) barrier’s heights DEc, DEv versus applied voltage
Fig. 5. The SRH GR rate for cap–barrier I interface of CBIRD nB1nB2 HgCdTe/B1,2 ntype detector for different compositions x at V = 50 mV and at T = 200 K.
respectively. As for as reverse biased CBIRD nB1nB2 HgCdTe/B1,2 ntype detector is concerned the most important is DEc emerging at cap–barrier I interface (desirable majority carrier blocking from cap layer), DEv at barrier I–absorber interface (unfavourable photo generated carriers impediment) and DEc, DEv at absorber–barrier II interface responsible for electron and hole blocking, respectively. The both mentioned DEc and DEv at cap–barrier I directly depend on the applied voltage pointing out that applied voltage is a trade-off between DEc and DEv (e.g. for barrier I–absorber interface DEv 160–50 meV and cap–barrier I interface DEc 350–275 meV for V = 0–1 V, respectively), while DEc at absorber–barrier II
Fig. 4. Calculated energy band structures for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector at: equilibrium (V = 0 V) (a) and under reverse biases: V = 0.4 V (b) and V = 1 V (c).
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Fig. 6. DEc and DEv for cap–barrier I (EB), barrier I–absorber, absorber–barrier II (HB) interfaces versus applied voltage at T = 200 K.
Fig. 7. DEc and DEv for absorber–barrier II (HB) interface versus barrier’s II doping (barrier II: x = 0.275–0.6) and composition (barrier II: ND = 1017 cm–3). V = 0 V, T = 200 K.
interface changes within range DEc 20–70 meV and DEv keeps constant value 450 mV. Above V > 0.2 V, the absorber–barrier II interface, DEc, saturates leading to the dark slight increase (turnon voltage level V = 0.2 V – see Fig. 2), while turn-on voltage for photocurrent is found to be V = 0.4 V (see Fig. 4), where DEv at barrier I–absorber plays crucial role blocking photogenerated carriers. Both barrier’s doping and composition also influence DEc and DEv (see Fig. 7). Once barrier I (EB) composition increases both DEc and DEv at cap–barrier I and barrier I–absorber raises reducing dark and photo currents. The barrier II doping increase reduces absorber–barrier II (HB) DEc (within the range 325–65 mV) and slightly raises DEv (425–450 mV) for ND < 1017 cm–3 while for ND > 1017 cm–3 opposite behaviour is observed. Increase of barrier II composition raises both DEc and DEv at absorber–barrier II interface.
prevent electron tunnelling from cap layer to the absorber. HgCdTe exhibits the potential issues with uniformity of the thin layers due to the interdiffusion at the interfaces. In our model both EB and HB barrier layers were divided on three (electron barrier)/two (hole barrier) sub-layers with composition grading fitted to cap and absorber layers to simulate this unfavourable phenomenon (additionally gauss tail’s doping profile was used, dx = 0.05 lm). Figs. 8 and 9 present IDARK, IPHOTO (for selected voltages) and D⁄ (for selected temperatures) versus HB composition. As it is shown in Fig. 7, both hole barrier (DEc and DEv) increase with HB composition resulting in IDARK decreasing for x < 0.35, while for x > 0.35 IDARK saturates. HB composition influences IPHOTO for 0.35 > x > 0.53 which should be attributed to HB DEc increase (x > 0.53) and IDARK raise for x < 0.35. D⁄ versus HB composition reflects the trend exhibited by IPHOTO dependence on HB x. HB doping influence on detector’s performance is depicted in Figs. 10 and 11. For hole barrier’s doping below ND < 1016 cm–3, both dark and photocurrent decrease two orders of magnitude in comparison to values obtained for ND = 1017 cm–3 for all analyzed voltages. This behaviour is directly connected with HB DEc decrease versus HB doping. The results presented in Fig. 11 points out that the highest detectivity is reached only for highly doped HB (ND > 1016 cm–3) for all simulated voltages. Increase of the HB doping within the range 1014–5 1017 cm–3 raises D⁄ from 108 to 1010 cm Hz1/2/W for V = 400 mV. Above ND > 1016 cm–3, increase of HB doping saturates to D⁄ 1.6 1010 cm Hz1/2/W. Similar considerations were conducted for EB barrier’s composition. Direct dependence of the EB DEc and DEv on composition and voltage is responsible for the both dark and photocurrent characteristics. Both EB DEc and DEv raise with composition resulting in IDARK and IPHOTO decrease (see Fig. 12). Below x < 0.4 both currents keep nearly constant value for simulated voltages. To attain the highest detectivity, the optimal composition increases with applied voltage (see Fig. 13). The simulated structure reaches D⁄ 1.6 1010 cm Hz/1/2/W for V = 400 mV and T = 200 K, being one order of magnitude higher than UBIRD nBn HgCdTe/B-n type single barrier structure, whereas detectivity was estimated to be D⁄ = 3.5 109 cm Hz1/2/W for the same EB parameters, working conditions and structural parameters (absorber’s doping and width). 5. Optimization of the CBIRD nB1nB2 HgCdTe/B1,2-n type absorber Doping of absorbers also plays critical role and must be optimized for assumed voltages and operating temperatures. Figs. 14
4. Optimization of the CBIRD nB1nB2 HgCdTe/B1,2-n type barriers The choice of the both electron and hole barriers’ doping and composition plays crucial role in designing CBIRD nB1nB2 HgCdTe/B1,2 n-type structures. HB width was checked thoroughly (within the range d = 0.15–5 lm) not revealing strict influence on D⁄. The EB width was chosen to be thick enough (d = 0.15 lm) to
Fig. 8. IDARK and IPHOTO for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus hole barrier composition for selected voltages. T = 200 K. HB doping ND = 1017 cm3.
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101
Fig. 9. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus hole barrier composition for selected temperatures. V = 400 mV. k = 4.95 lm.
Fig. 12. IDARK and IPHOTO for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus electron barrier’s composition for selected voltages.
Fig. 10. IDARK and IPHOTO for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector versus hole barrier doping concentration for selected voltages. T = 200 K.
Fig. 13. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2 n-type versus applied voltage for selected electron barrier’s composition. T = 200 K, k = 4.95 lm.
photocurrents keep nearly constant value for simulated voltages. Above ND > 1015 cm–3 photocurrent decreases sharply which could be attributed to Burstein–Moss effect (minority carrier density drops). Once absorber’s doping raises, the dark current increases sharply (mainly due to Auger 1 process), while the photocurrent decreases (free minority carrier concentration decreases). Clearly
Fig. 11. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2 n-type structure versus barrier doping concentration for selected voltages. T = 200 K, k = 4.95 lm.
and 15 present IDARK and IPHOTO versus applied voltage (for selected absorber’s doping) and absorber’s doping, respectively. Once absorber’s doping increases, the photocurrent exhibits decreasing trend lowering responsivity Ri = 2.3–1.3 A/W and quantum efficiency g = 59–32% (within the range 1014 < ND < 5 1016 cm–3). In doping range 1014 < ND < 1015 cm–3 both dark current and
Fig. 14. IDARK and IPHOTO for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector versus voltage for different absorbers doping. T = 200 K.
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Fig. 15. IDARK and IPHOTO for the CBIRD nB1nB2 HgCdTe/B1,2 n-type detector versus absorber’s doping for selected voltages. T = 200 K.
Fig. 17. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2-n type structure versus absorber’s width for selected voltages. T = 200 K, k = 4.95 lm.
6. Comparison of the InAs/GaSb T2SLs and HgCdTe technologies
Fig. 16. D⁄ for CBIRD nB1nB2 HgCdTe/B1,2-n type structure versus applied voltage for selected absorber’s doping concentrations. T = 200 K.
seen are turn-on voltages for both dark and photocurrents. The barriers’ parameters were assumed to be as follows: EB x = 0.33– 0.6–0.275, d = 0.15 lm, ND = 2 1015 cm3; HB x = 0.275–0.6, d = 0.4 lm, ND = 1017 cm3. Since CBIRD nB1nB2 HgCdTe/B1,2 n-type detector is a minority carrier device, absorber’s doping decreases concentration of the free holes lowering IDARK. Further absorber’s doping increase contributes to the IDARK current raise due to EB DEc lowering and Auger 1 effect. The lowest value of IDARK = 0.5 A/cm2 for V = 400 mV could be obtained for ND = 5 1015 cm–3 for V = 100 mV but for this absorber doping IPHOTO assumes the lowest value which indicates that optimal working conditions appears for ND < 1015 cm–3. Fig. 16 presents D⁄ versus voltage for selected absorber’s doping. Presented results indicate that the maximum value of the D⁄ = 1.6 1010 cmHz1/2/W for given structure could be obtained for absorbers doping ND = 1014 cm–3 and V = 400 mV while at doping level above 1014 cm–3, detectivity decreases rapidly due to dark current increase and lowering of the photocurrent. Once absorber’s doping increases, the optimal voltage to be applied to the structure to attain the highest detectivity decreases. Absorber width influences the IR absorption what is shown in Fig. 17. For voltage V = 400 mV the optimal thickness is estimated to be d = 5 lm which allows reaching detectivity D⁄ = 2 1010 cm Hz1/2/W. Once bias decreases optimal thickness decreases and for V = 300 mV equals d = 4 lm.
The very last figure (Fig. 18) shows the RoA and RA products versus temperature for MWIR CBIRD nB1nB2 HgCdTe/B1,2-n type detector, UBIRD nBn HgCdTe/B-n type (kc = 5.2 lm) detector, InAs/GaSb/B–AlGaSb T2SL nBn (kc = 5.4 lm) detector, InAs/GaSb PIN photodiode (kc = 6.2 lm), and finally HOT HgCdTe bulk photodiodes (kc = 5.4 lm) fabricated at the joint laboratory run by Institute of Applied Physics, Military University of Technology/Vigo System SA. Theoretical estimation for the MWIR UBIRD nBn HgCdTe/B-n type was conducted by Martyniuk and Rogalski in Ref. [22], whereas the performance of T2SLs nBn InAs/GaSb/B-AlGaSb detector’s performance was presented in Ref. [24] where an analytical approach was used to model the detectors’s performance. PIN T2SLs photodiodes were analysed by Wrobel et al. [25]. It is clearly seen that the performance of CBIRD nB1nB2 HgCdTe/B1,2-n type structure has reached a comparable level determined by the state of the art of HgCdTe bulk photodiodes and put itself in superior position with reference to UBIRD nBn HgCdTe/B-n type, T2SLs nBn, and PIN detectors. The particular significance of the incorporation of the extra barrier for minority carriers in CBIRD structures versus single barrier (majority carriers’ blocking) in UBIRD detectors is clearly evident by RA product increase from 0.1 to 0.5 Xcm2 for T = 230 K (for the same absorber’s
Fig. 18. Temperature dependence of the RA and RoA products for MWIR nB1nB2 HgCdTe/B1,2-n type detector, nBn HgCdTe/B-n type detector, nBn InAs/GaSb/BAl0.2Ga0.8Sb T2SL detector, HgCdTe HOT bulk diodes and PIN InAs/GaSb T2SL diodes operating at near-room temperature (T = 230 K).
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doping ND = 1014 cm–3 and thickness d = 10 lm) resulting in raising of detectivity e.g. from 3.5 109 to 1.6 1010 cm Hz1/2/W for V = 400 mV and T = 200 K. Once absorber’ thickness is reduced to the optimal size to attain the highest detectivity, the RA product increases to 0.9 Xcm2 for T = 230 K and detectivity reaches 2 1010 cm Hz1/2/W at V = 400 mV and T = 200 K. In fact, the RA products of MWIR 5.2 lm CBIRD nB1nB2 HgCdTe/B1,2-n type is slightly higher in comparison to bulk HgCdTe photodiodes, but it was calculated for V = 50 mV reverse bias.
Band-gap energy:
Eg ðx; TÞ ¼ 0:302 þ 1:93x 0:81x2 þ 0:832x3 þ 5035 104 Tð1 2xÞ:
ðA:1Þ
Electron affinity:
c ¼ 4023 0:813½Eg ðx; TÞ 0:083:
ðA:2Þ
Carriers’ effective masses:
me ¼ 8:035 102 Eg ðx; TÞm0 ;
ðA:3Þ
mh ¼ 0:55m0 :
ðA:4Þ
7. Conclusions In the paper we theoretically estimated the performance of the CBIRD nB1nB2 HgCdTe/B1,2-n type detector versus operating conditions and structural parameters. The unfavourable compositional uniformity and interdiffusion at the interfaces were modelled by proper barrier grading matched to the cap and absorber’s composition respectively. The maximum RA product of the detector with 5.2 lm cut-off wavelength is higher than 0.9 Xcm2 at 230 K while maximum detectivity was estimated to be 2 1010 cm Hz1/2/W at T = 200 K assuming an absorber thickness d = 5 lm and V = 400 mV. Inherited barriers in both conduction and valance bands were analysed in detail pointing the optimal operating conditions as for as bias and doping are concerned. Turn-on voltage above which dark current increases slowly was estimated to be V = 0.2 V for dark current and 0.4 V for photocurrent respectively. Although, it is unfeasible to attain wanted band alignment in valance band, the theoretically predicted CBIRD nB1nB2 HgCdTe/ B1,2-n type structure demonstrate performance which underline significance and full legitimacy of the barriers’ incorporation to the detector’s architectures. Similarly to UBIRD HgCdTe/B-n type, the analysed structure allows circumventing requirements for ptype doping reducing number of the processing steps. Barrier’s doping and composition should be perceived as the most important parameters in CBIRD nB1nB2 HgCdTe/B1,2-n structure optimization. The proper doping and composition choice leads to either building up or lowering the barriers in both conduction and valance bands. It was shown that the raise of the operating temperature by nearly 50 K could be reached by additional incorporation of the minority carrier barrier into UBIRD structure. The extra barrier leads to suppressing of the dark current by nearly one order of magnitude. As mentioned in introduction, T2SL InAs/GaSb 6.1 Å compound family is the only one infrared material system theoretically predicted to achieve higher performance than HgCdTe bulk photodiodes. However, so far the HOT HgCdTe photodiode performance has not been overcome by T2SL PIN and UBIRD nBn structures because of the low quantum efficiency and presence of the SRH recombination characterized by a relatively short carrier lifetime. Mentioned limitations could be circumvented by simplified CBIRD nB1nB2 HgCdTe/B1,2-n type structures and UBIRD nBn HgCdTe/B-n type detectors. Finally, unlike T2SLs InAs/GaSb/B–AlGaSb nBn detector, CBIRD nB1nB2 HgCdTe/B1,2-n type and UBIRD nBn HgCdTe/B-n type detector do not require highly doped layers which should be perceived as a technological advantage. Acknowledgement This paper has been done under financial support of the Polish National Science Centre, Project: DEC-2011/01/B/ST5/06283. Appendix A The CBIRD nB1nB2 HgCdTe/B1,2-n type detector was simulated using the following material parameters [1,26]:
Dielectric constant:
e ¼ 20:5 15:5x þ 5:7x2 :
ðA:5Þ
The radiative recombination rate:
B ¼ 5:9052
1018 n2 i
eT
3=2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þx Eg exp ðE2g ð81:9 þ TÞ kB T
2
þ 3kB TEg þ 3:75kB T 2 Þ:
ðA:6Þ
The Auger recombination coefficients Cn and Cp:
C n ¼ 5 1012 jF 1 F 2 j
"
" n2 3:8 1018 e2 i
!#1=2 3 mg Eg Eg exp 1 þ 2 kB T mh kB Tðmg =mh Þ ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #1 mg mg 1 1 þ 2 1þ ; ðA:7Þ mg mh mh
C p ¼ 0:1C n :
ðA:8Þ
The absorption coefficient: for: k > kc
n¼
ag n ; a0
ðA:9Þ
1:24=kc e0 ; eg e0
ðA:10Þ
a ¼ a0
for: k 6 kc
a ¼ ag exp
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B½1:24=kc eg ;
ðA:11Þ
Eg ¼ 0:295 þ 1:87x 0:28x2 þ ð6 14x þ 3x2 Þ 104 T þ 0:35x4 ;
ðA:12Þ
ag ¼ 65 þ 1:883T þ ð8694 10:314TÞx;
ðA:13Þ
a0 ¼ expð18:5 þ 45:68xÞ;
ðA:14Þ
e0 ¼ 0:355 þ 1:77x;
ðA:15Þ
B¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þ 0:083T þ ð21 0:13TÞx:
ðA:16Þ
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