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Semiconductor-nanoheterointerface eigenstate photonic modification
Journal of Non-Crystalline Solids 354 (2008) 4233–4237
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Semiconductor-nanoheterointerface eigenstate photonic modification E.A. Anagnostakis Solid State Physics Section, Department of Physics, University of Athens, Greece
a r t i c l e
i n f o
Article history: Available online 8 August 2008 PACS: 73.20.–r 73.63.–b 78.67.–n 73.40.Kp Keyword: Devices
a b s t r a c t As a crucial issue permeating the optoelectronic functionality of a technological semiconductor nanodevice, modification of its two-dimensional electron gas (2DEG) eigenstate by absorption of regulated successive photon-doses is studied for a generic case of a conventional nanoheterodiode in terms of the 2DEG fundamental-sublevel eigenenergy correlation with respective 2DEG areal density, versus instantaneous cumulative photonic intake. Application of this treatment to the experimental photoresponse of a typical AlxGa1xAs/GaAs modulation – doped heterodiode leads to a realistic tracing of the pertinent 2DEG-eigenstate photonic modification. Indirect justification of the scheme appears to be provided by the measured 2DEG mobility photonic evolution being compatible with the evolution deduced for the nanoheterointerface fundamental-wavefunction penetration-length into the energetic-barrier region adjacent to the 2DEG quantum well. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Carrier transport within semiconductor nanodevices digresses exotically from a Boltzmann-like scenery owing to the mesoscopic regime engineered, fruitfully rendering characteristic structural dimensions actually smaller than the appropriate mean free-paths concerning both mobility and wavefunction (WF) inelastic phasebreaking. The advanced epitaxial-growth techniques applied for composing functional nanoscale-heterostructures permit incorporation of the desired depth-profile quantum wells (QW) into nanoheterointerfaces (NHI) established as successive heterogeneous layers are compiled. For the total energy lower than the NHI QW potential-energy top, the carrier WF gets confined, oscillating between the QW boundaries – thus conditioning the carrier to acquire a quasi-two-dimensionality perpendicular to the QW spatial bottom (whence two-dimensional electron gas (2DEG) designates the thermodynamic ensemble of such electrons) – and exhibiting momentum and eigenenergy quantisation [1]. The major nanodevice-performance enhancement achieved through the introduction of modulation doping by Dingle et al. [2] has ever since been providing NHIs hosting separately a dense distribution of dopants on their wider-bandgap part and a highmobility 2DEG along the QW extension, belonging to the narrower-bandgap side. The investigation of such NHIs has often been undertaken due to their crucial importance for the tunability of numerous, established and yet-to-be-invented, optoelectronic nanodevices such as the Bloch oscillator [3,4], the resonant-tunnelling double heterodiode [5], the hot-electron tunnelling-transistor [6], the revolutionary quantum-cascade LASER [7,8]. E-mail address: [email protected] 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.06.029
For the last two decades, a major part of our research activity has been evolving within the field of designing, characterising, and analyzing photonic epitaxial semiconductor nanodevices [9– 24]. In particular, frequent monitoring of the photoconductive response of NHI structures has ultimately inspired us to assess the 2DEG-eigenstate modification being effected within a generic modulation-doped nanoheterodiode by absorption of regulated successive photon-doses. In this work, this photonic modification is studied in terms of the NHI 2DEG fundamental-sublevel eigenenergy correlation with the respective 2DEG areal density, versus instantaneous cumulative photonic intake. The scheme is being plausibly applied to the experimental photoresponse of a typical AlxGa1-xAs / GaAs Si – modulation-doped nanoheterodiode (NHD) and it is also indirectly justified through a juxtaposition of its measured 2DEG mobility photonic evolution and the evolution deduced for the NHI fundamental-wavefunction penetrationlength. 2. Predictive scheme Within the extension of a typical NHI, the energy-band bendings of the energetic-barrier portion and the conductive channel are determined by its neighboring (ionised-impurity) electriccharge density and overall field (with any effective non-built-in field incorporated). Thus, on the one hand, the 2DEG confined sublevels are quantum-mechanically calculable and, on the other hand, the thermodynamic-equilibrium requirement mirrors an ultimate uniform Fermi-energy level throughout the nanoheterojunction to a realistic sheet-concentration of QW two-dimensional electrons.
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At each instance of such dynamic equilibrium, the energetic top of actually filled NHI 2DEG states aligns with the uniform Fermi level operative:
E0 þ
f
q
¼ EF
ð1Þ
with EF being exactly the Fermi-energy level, E0 being the energetic bottom of the 2DEG fundamental subband (considered as the only one occupied near the electric quantum limit, approached by conventional NHIs primarily functioning at ambient temperatures in the vicinity of absolute zero), f being the instantaneous 2DEG sheet-concentration within the NHI QW, and q being the theoretical two-dimensional per-unit-area (normal to the QW spatial extension) density of states accessible to conductivity electrons having their normal wavevector-component quantised, given [22,24] in the parabolic approximation by
q¼
m
ph2
;
ð2Þ
where m* denotes the conductivity electron effective mass at the NHI QW fundamental subband, and h Planck’s action constant divided by 2p. For the fruitfully common case, modulation doping of an epitaxially composed nanodevice embeds a heavily dense donor-distribution within the wider-bandgap-semiconductor part of the established NHI, with the donor energy level getting located sufficiently deep – with reference to the bandgap profile – for the Fermi level to be plausibly regarded as pinned to it and remaining there (with an effectively negligible rate of change) for the entire regime of quantum-limit-like nanodevice functioning [28,30,18]. Indeed, around absolute zero, the highest occupied electron-states in the doped energetic-barrier nanodevice-side would be those at the donor level, pinning the thermodynamic-equilibrium (equihigh for both NHI sides) Fermy energy there and keeping it there regardless of alterations in the ionised-donor percentage, insofar as the donor-population is dense enough for accommodating (recaptured) electrons and the ambient temperature is adequately cryogenic for the donor level to continue providing a trustworthy demarcation between (partially) filled and empty energy states within the wider-bandgap nanodevice-part. Such a Fermi level immunity against evolving cumulative photonic intake (tantamount to increasing the 2DEG population hosted by the NHI QW) by the nanoheterostructure would be sustainable through a mechanism successively lowering the 2DEG fundamental sublevel E0 (as well as the first excited sublevel, which nevertheless would not be participating in the hosting of QW conductivity electrons as always lying over the Fermi-energy demarcation) within the QW being commensurately broadened whilst keeping its energetic depth dictated by the invariable NHI conduction-band discontinuity. Believably, such a mechanism would be based on the formation of ever new (after each further photonic intaking) additional effective electron-attractive centers [12,24] accountable for evolving photowidening of the enclosingpotential profile. Thus, for each photoaugmentation of the QW 2DEG number, effected in parallel to the respective hole photoenhancement within the neutral portion of the lower-bandgap nanoheterostructure side, the above QW progressive modification, indexed by fundamental-sublevel eigenenergy photolowering occurring each time, would make it possible to accommodate the increasing 2DEG ensemble by ever more energy states appearing between the sinking subband bottom and the plausibly still Fermi level. Considering, therefore, the dynamics of energy level positions against regulated successive photon-doses being absorbed by the probed NHI, we obtain through partial differentiation of (1) with respect to the instantaneous cumulative photonic dose d:
oE0 1 of oEF ¼ þ ffi0 od q od od
ð3Þ
or
oE0 1 of ¼ od q od
ð4Þ
providing the photonic dose-rate of evolution of the 2DEG fundamental-eigenstate-sublevel E0 in terms of the experimentally traceable photon-dose-rate of augmentation of the 2DEG areal density f within the QW of the monitored NHI. In this sense, Eq. (4) furnishes the prime quest of our simplistic scheme: namely, following the dynamics of the evolving photonic modification of the NHI fundamental-eigenstate, during the procedure of successive intakings of appropriate photon-transmissions. Whenever an experimental measurement Df(d) for the augmentation of the 2DEG sheet-concentration f (with respect to its pre-illumination initial value fd) consequent upon some instantaneous total photon-dose d is obtained, the respective eigenstate-sublevel shift E0(f) (away from its dark locus E0d) drawn by this cumulative dose d is predicted, according to (4), to be
1 DE0 ðfÞ ¼ DfðdÞ:
q
ð5Þ
3. Application The NHI probed is formulated within a typical molecular beam epitaxy (MBE) – grown functional semiconductor structure, consisting of an upper, wide-bandgap, n-type (Si donor – doped at a concentration of the order of 1 1017 cm3) AlxGa1xAs (x 0.3) epitaxial layer with a thickness of the order of 2 lm developed upon a lower, narrower-bandgap, relatively p-type-like, non-intentionally doped (NID) GaAs epitaxial layer of a thickness of the order of 100 nm mounted on top of a semi-insulating (SI) GaAs substrate. The NHI of the device comprises the ionised-donor depletion zone on the upper-epilayer side and the 2DEG QW hosted within the lower epilayer. The NHI device, subjected to persistent-photoconductivity (PP) experimental conditions [9,11] by the provision of successive regulated photon-doses evolving within six orders of magnitude is illuminated from the upper epitaxial-layer side with normally incident light filtred to photon energies greater than the bandgap of each semiconductor and, thus, capable of producing interband transitions. The device is kept at a constant temperature of around 4 K and a magnetic field of about 0.5 T is applied within an automated Hall apparatus for the pertinent van der Pauw technique measuring the persistent photoenhancement (PPE) of both the NHI 2DEG population and the NHI 2DEG mobility. Figs. 1 and 2 present [24] the PPE Df in the NHI 2DEG sheetdensity f and the reduced (with respect to the unperturbed darkvalue ld) PPE Dll in the NHI 2DEG mobility l, respectively, versus d relative values of the incoming cumulative photon-dose d, prior to any saturation or negative differential feature of either evolution. For interpreting this regime of the NHI 2DEG PPE experimental curves, we come to the conclusion that, as the cumulative photon dose d absorbed by the device increases, the conduction electrons confined within the NHI QW (and expected to be occupying solely the QW fundamental conduction-subband in the electric quantum limit of operation of the device at 4 K) evolve to higher numbers, due to the augmentation of their initial dark-concentration by the rising persistent-photoelectron density. Photoaugmentation of 2DEG areal density f, now, is considered tantamount to its mobility PPE, especially in view of a previous theoretical finding of ours [13] correlating the the NHI 2DEG mobility l(f*) with
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7.5
7.0
ΔE0 (meV)
-2
Δζ (x10 cm )
-20
11
6.5
6.0
5.5
5.0 -30 1
2
3
4
5
1
6
2
3
4
5
6
log δrel
logδ rel Fig. 1. NHI 2DEG sheet-density PPE Df versus decimal logarithm of relative values drel for the instantaneous cumulative photon-dose d intaken.
Fig. 3. Predicted red photoshift DE of the NHI fundamental-eigenstate sublevel E0 against decimal algorithm of relative values drel for the instantaneous total photondose d intaken.
0.8
concerning the electron de Boglie time-independent wavefunction w(x) and taking into account the spatial variation m*(x) of the carrier effective mass, reads
0.7
0.9
Δμ/μd
2
d h dwðxÞ ½ þ UðxÞwðxÞ ¼ EwðxÞ; dx 2m ðxÞ dx
ð7Þ
with E being the allowed energy-eignevalue conjugate to each physically meaningful wavefunction w(x), solving (7) and vanishing asymptotically at infinities, i.e.
0.6 0.5
wð1Þ ¼ 0
0.4
ð8Þ
and h being Planck’s action constant divided by 2p. Performing, now, an independent variable transformation, namely,
0.3 1
2
3
4
5
6
x ax Arctanh ðnÞ $ /ðnÞ w½xðnÞ;
log δrel Fig. 2. Reduced (with respect to the unperturbed dark-value ld) PPE Dl/ld in the NHI 2DEG mobility l versus decimal logarithm of relative instantaneous total photonic dose values drel.
respective dimensionless reduced instantaneous 2DEG sheet-density f* through
1 pffiffiffiffi lðf Þ ¼ lr 1 f ef
ð6Þ
ð9Þ
We obtain in place of (7) the Sturm–Liouville differential equation
ðd=dnÞfMðnÞ½d/ðnÞ=dng uðnÞ/ðnÞ þ krðnÞ/ðnÞ ¼ 0
ð10Þ
under the boundary conditions
/ð1Þ ¼ 0 and /ðþ1Þ ¼ 0
ð11Þ
with functions M(n), u(n), and r(n) in the new dimensionless variable n (belonging to the universal interval [1, +1]) defined as
with lr being an appropriate, device-wise meaningful, reference mobility-value. Employing, furthermore, our predictive scheme we arrive, through Eq.(5), to Fig. 3 presenting the awaited photoshif DE0 evolution of the NHI fundamental-eigenstate sublevel against instantaneous total photon dose d intaken: This predicted gradual photolowering of NHI eigenstate-subband bottom is interpretable as quantum-mechanically compatible with our previous studies [13,17] approximating the modification of a generic nanodevice’s conductive-channel extension as positively linearly proportional to the respective 2DEG areal-concentration alteration.
MðnÞ ¼ ð1=aÞð1—n2 Þm0 =m ½xðnÞ;
ð12Þ
uðnÞ ¼ ½2a=ð1—n2 ÞU½xðnÞ=E ;
ð13Þ
rðnÞ ¼ 2a=ð1 n2 Þ
ð14Þ
4. NHI WF penetration-length photonics
rendering the characteristic confinement-length x* entering the independent variable transformation (9) after the dimensionless scale factor a equal to 2.76043 Å, mo giving the electron rest mass. For converting the Sturm–Liouville differential equation concerning the NHI 2DEG transformed wavefunction u(n) via a
With respect to the generic situation of a conductivity electron being hosted by the QW of potential-energy profile U(x) against the growth-axis coordinate x within a conventional semiconductor nanodevice heterointerface, the pertinent Schrödinger equation,
and dimensionless new, ‘reduced-energy’, eigenvalue k defined as
k¼
E ; E
ð15Þ
where T* denotes a convenient energy-scale 2
E
h 1eV; mo x2
ð16Þ
E.A. Anagnostakis / Journal of Non-Crystalline Solids 354 (2008) 4233–4237
linearized system of difference equations into a tridiagonal-matrix eigenvalue-problem, we have previously [23,24] composed a finite-difference-method iterative algorithm. Therewith, the opposite of the eigenvalues of this matrix lead to the approximations to the NHI WF exact reduced-energy eigenvalues k, thus computing (Eq. (15)) the allowed QW 2DEG subband-energies E = kE*. On the other hand, the determined Strum–Liouville eigenvectors |u(n)i conjugate to these numerical eigenvalues unveil through transformation (9) the quantum-mechanically allowed wavefunctions w(x) for the 2DEG dwelling within the nanodevice heterointerface QW and underlying the crucial optoelectronic effects exhibited by the generic semiconductor-nanostructure. In particular, such determination of the NHI 2DEG wavefunction may lead to the computation of its entailed penetration-length into the nanodevice neighbouring energy-barrier layer, thus, facilitating the prediction of 2DEG mobility behavior [25–27] and, furthermore, the consideration of quantum-mechanical-tunnelling transmission probability [28,29] for conductivity electrons escaping the heterointerface and traveling through the nanodevice – by virtue of a normal transport mechanism advantageously exploitable, especially at nanoelectronic cryogenic ambient temperatures. The above outlined finite-difference-method algorithm is incorporated in the following procedure, repeatedly performed for each successive experimental cumulative photon-dose d: Step 1. Extraction from predictive scheme Eq.(5) of the awaited 2DEG fundamental-sublevel photolowering DE0(d) conjugate to experimentally measured 2DEG sheet-density PPE Df(d) driven by current total photonic intake d. Step 2. Deduction of current 2DEG fundamental sublevel E0(d) = E0,d |DE0(d)| by subtraction of the absolute value |DE0(d)| of current fundamental sublevel photolowering DE0(d) from sublevel dark (prior to exposure to photonic doses) locus E0,d. Step 3. Iterative applications of the algorithm with respect to the sought 2DEG QW spatial width w(d) compatible with the respective fundamental-eigenstate sublevel E0(d) predicted for the current cumulative photonic dose d. The entailed potential-energy profile U(x; d) is simplifyingly simulated by a rectangular one, of energetic depth always expressible by the NHI conduction-band discontinuity DEC and resting on the boundaries of the photowidened QW spatial extension w(d). Thus, the parametrisation of the QW potentialenergy profile by total photonic intake d is effected through letting a simulative fixed-energetic-depth rectangular model-QW expand spatially at a rate induced by each current intaken cumulative photon-dose d. Step 4. Once the iterative algorithm has converged for the optimum QW spatial width w(d) conjugate to the current total photonic intake d, the adjoint 2DEG fundamental-eigenstate wavefunction w0(x; d) is obtained, thus exhibiting its traceable penetration-length l(d) into the NHI energeticbarrier part. Therefore, tracing the fundamental-eigenstate penetrationlength l(d) in the optimum wavefunction w0(x; d) obtained by convergence (with respect to the optimum QW width w(d) compatible with predicted photoredefined eigenstate locus E0(d)) of the iterative algorithm for each total photon-dose d intaken, we manage to map its photonic evolution (Fig. 4). This, juxtaposed with the experimentally measured 2DEG mobility PPE Dl(d) (Fig. 2), appears adequately justified in the common trend of growing photoinduced facilitation of 2DEG intra-QW transport at shortening mobility-limitting influence by external scatterers.
0.95 0.90 0.85
l/ld
4236
0.80 0.75 0.70 0.65 1
2
3
4
5
6
log δ re l Fig. 4. Reduced (with respect to the unperturbed dark-value ld) NHI 2DEG fundamental-wavefunction penetration-length.
5. Conclusion The modification of the NHI 2DEG-eigenstate by absorption of regulated successive photon-doses is studied for the generic case of a conventional nanoheterodiode, in terms of the 2DEG fundamental-sublevel eigenenergy correlation with the respective 2DEG areal density, versus instantaneous cumulative photonic intake. The scheme is plausibly applied to the experimental photoresponse of a typical AlxGa1-xAs/GaAsSi-modulation-doped NHD and is, also, indirectly justified through juxtaposition of its measured 2DEG mobility photonic evolution and the evolution deduced for the NHI fundamental-wavefunction penetration-length, as computed through an iterative algorithm converting the Sturm–Liouville differential equation concerning the NHI 2DEG transformed wavefunction into a tridiagonal-matrix eigenvalue-problem. Thus, the predicted trend of evolving red photoshift for the NHI eigenstate sublevel is obtained compatible with a proceeding NHI QW-extension photowidening, on the one hand, and a NHI wavefunction penetration-length shrinking being tantamount to the 2DEG mobility continuing PPE, on the other hand. Acknowledgements Valuable support and enlightening consultation offered by University of Athens Professors G.J. Papadopoulos and N. Guskos are gratefully acknowledged. References [1] Ch.P. Poole, F.J. Owens, Introduction to Nanotechnology, Wiley-Interscience, New Jersey, 1997 (Chapter 9). [2] R. Dingle, H.L. Stoermer, A.C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33 (1978) 665. [3] R.O. Grondin, W. Porod, J. Ho, D.K. Ferry, G.J. Iafratte, Superlattices Microstruct. 1 (1985) 183. [4] J.N. Churchill, F.E. Holmstrom, Phys. Lett. A 85 (1981) 453. [5] T.C.L. Sollner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker, D.D. Peck, Appl. Phys. Lett. 43 (1983) 588. [6] N. Yokoyama, K. Imamura, T. Oshima, H. Nishi, S. Mutto, K. Kondo, S. Hiyamizu, Jpn. J. Appl. Phys. 23 (1984) L311. [7] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, S.N.G. Chu, A.Y. Cho, Science 264 (1994) 553. [8] J. Faist, F. Capasso, C. Sirtori, D.L. Sivco, A.L. Hutchinson, A.Y. Cho, Electron. Lett. 32 (1996) 560. [9] E.A. Anagnostakis, Phys. Stat. Sol. A 126 (1991) 397. [10] E.A. Anagnostakis, Phys. Stat. Sol. A 127 (1991) 153. [11] D.E. Theodorou, E.A. Anagnostakis, Phys. Rev. B 44 (1991) 3352. [12] E.A. Anagnostakis, Appl. Phys. A 54 (1992) 68. [13] E.A. Anagnostakis, Phys. Status Soilidi B 171 (1992) K75. [14] E.A. Anagnostakis, Phys. Stat. Sol. B 172 (1992) K61. [15] E.A. Anagnostakis, Phys. Rev. B 46 (1992) 7593. [16] E.A. Anagnostakis, Phys. Stat. Sol. A 136 (1993) 247. [17] E.A. Anagnostakis, Phys. Stat. Sol. B 177 (1993) 533. [18] E.A. Anagnostakis, D.E. Theodorou, J. Appl. Phys. 73 (1993) 4550.
E.A. Anagnostakis / Journal of Non-Crystalline Solids 354 (2008) 4233–4237 [19] [20] [21] [22] [23] [24] [25] [26]
E.A. Anagnostakis, Phys. Stat. Sol. B 181 (1994) K15. E.A. Anagnostakis, Phys. Stat. Sol. A 141 (1994) 373. E.A. Anagnostakis, Phys. Stat. Sol. A 146 (1994) K9. E.A. Anagnostakis, D.E. Theodorou, Phys. Stat. Sol. B 188 (1995) 689. E.A. Anagnostakis, Task Quart. 8 (3) (2004) 385. E.A. Anagnostakis, Rev. Adv. Mater. Sci. 12 (2006) 182. S. Mori, T. Ando, J. Phys. Soc. Jpn. 48 (1980) 865. H.L. Stormer, A.C. Gossard, W. Weigmann, Solid State Commun. 41 (1982) 707.
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[27] M. Jaros, Physics and Applications of Semiconductor Microstructures, Oxford University Press, 1989 (Chapters 6 and 7). [28] C. Weisbuch, Semiconductors and Semimetals, vol. 24, Academic Press, New York, 1987 (Chapter 1). [29] D.V. Ferry, S.M. Goodnick, Transport in Nanostructures, Cambridge University Press, 1997 (Chapter 3). [30] G. Bastard, Wave mechanics applied to semiconductor heterostructures (Les Editions), de Physique, Les Ulis Cedex, France, 1988 (Chapter V).
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Semiconductor-nanoheterointerface eigenstate photonic modification E.A. Anagnostakis Solid State Physics Section, Department of Physics, University of Athens, Greece
a r t i c l e
i n f o
Article history: Available online 8 August 2008 PACS: 73.20.–r 73.63.–b 78.67.–n 73.40.Kp Keyword: Devices
a b s t r a c t As a crucial issue permeating the optoelectronic functionality of a technological semiconductor nanodevice, modification of its two-dimensional electron gas (2DEG) eigenstate by absorption of regulated successive photon-doses is studied for a generic case of a conventional nanoheterodiode in terms of the 2DEG fundamental-sublevel eigenenergy correlation with respective 2DEG areal density, versus instantaneous cumulative photonic intake. Application of this treatment to the experimental photoresponse of a typical AlxGa1xAs/GaAs modulation – doped heterodiode leads to a realistic tracing of the pertinent 2DEG-eigenstate photonic modification. Indirect justification of the scheme appears to be provided by the measured 2DEG mobility photonic evolution being compatible with the evolution deduced for the nanoheterointerface fundamental-wavefunction penetration-length into the energetic-barrier region adjacent to the 2DEG quantum well. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Carrier transport within semiconductor nanodevices digresses exotically from a Boltzmann-like scenery owing to the mesoscopic regime engineered, fruitfully rendering characteristic structural dimensions actually smaller than the appropriate mean free-paths concerning both mobility and wavefunction (WF) inelastic phasebreaking. The advanced epitaxial-growth techniques applied for composing functional nanoscale-heterostructures permit incorporation of the desired depth-profile quantum wells (QW) into nanoheterointerfaces (NHI) established as successive heterogeneous layers are compiled. For the total energy lower than the NHI QW potential-energy top, the carrier WF gets confined, oscillating between the QW boundaries – thus conditioning the carrier to acquire a quasi-two-dimensionality perpendicular to the QW spatial bottom (whence two-dimensional electron gas (2DEG) designates the thermodynamic ensemble of such electrons) – and exhibiting momentum and eigenenergy quantisation [1]. The major nanodevice-performance enhancement achieved through the introduction of modulation doping by Dingle et al. [2] has ever since been providing NHIs hosting separately a dense distribution of dopants on their wider-bandgap part and a highmobility 2DEG along the QW extension, belonging to the narrower-bandgap side. The investigation of such NHIs has often been undertaken due to their crucial importance for the tunability of numerous, established and yet-to-be-invented, optoelectronic nanodevices such as the Bloch oscillator [3,4], the resonant-tunnelling double heterodiode [5], the hot-electron tunnelling-transistor [6], the revolutionary quantum-cascade LASER [7,8]. E-mail address: [email protected] 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.06.029
For the last two decades, a major part of our research activity has been evolving within the field of designing, characterising, and analyzing photonic epitaxial semiconductor nanodevices [9– 24]. In particular, frequent monitoring of the photoconductive response of NHI structures has ultimately inspired us to assess the 2DEG-eigenstate modification being effected within a generic modulation-doped nanoheterodiode by absorption of regulated successive photon-doses. In this work, this photonic modification is studied in terms of the NHI 2DEG fundamental-sublevel eigenenergy correlation with the respective 2DEG areal density, versus instantaneous cumulative photonic intake. The scheme is being plausibly applied to the experimental photoresponse of a typical AlxGa1-xAs / GaAs Si – modulation-doped nanoheterodiode (NHD) and it is also indirectly justified through a juxtaposition of its measured 2DEG mobility photonic evolution and the evolution deduced for the NHI fundamental-wavefunction penetrationlength. 2. Predictive scheme Within the extension of a typical NHI, the energy-band bendings of the energetic-barrier portion and the conductive channel are determined by its neighboring (ionised-impurity) electriccharge density and overall field (with any effective non-built-in field incorporated). Thus, on the one hand, the 2DEG confined sublevels are quantum-mechanically calculable and, on the other hand, the thermodynamic-equilibrium requirement mirrors an ultimate uniform Fermi-energy level throughout the nanoheterojunction to a realistic sheet-concentration of QW two-dimensional electrons.
4234
E.A. Anagnostakis / Journal of Non-Crystalline Solids 354 (2008) 4233–4237
At each instance of such dynamic equilibrium, the energetic top of actually filled NHI 2DEG states aligns with the uniform Fermi level operative:
E0 þ
f
q
¼ EF
ð1Þ
with EF being exactly the Fermi-energy level, E0 being the energetic bottom of the 2DEG fundamental subband (considered as the only one occupied near the electric quantum limit, approached by conventional NHIs primarily functioning at ambient temperatures in the vicinity of absolute zero), f being the instantaneous 2DEG sheet-concentration within the NHI QW, and q being the theoretical two-dimensional per-unit-area (normal to the QW spatial extension) density of states accessible to conductivity electrons having their normal wavevector-component quantised, given [22,24] in the parabolic approximation by
q¼
m
ph2
;
ð2Þ
where m* denotes the conductivity electron effective mass at the NHI QW fundamental subband, and h Planck’s action constant divided by 2p. For the fruitfully common case, modulation doping of an epitaxially composed nanodevice embeds a heavily dense donor-distribution within the wider-bandgap-semiconductor part of the established NHI, with the donor energy level getting located sufficiently deep – with reference to the bandgap profile – for the Fermi level to be plausibly regarded as pinned to it and remaining there (with an effectively negligible rate of change) for the entire regime of quantum-limit-like nanodevice functioning [28,30,18]. Indeed, around absolute zero, the highest occupied electron-states in the doped energetic-barrier nanodevice-side would be those at the donor level, pinning the thermodynamic-equilibrium (equihigh for both NHI sides) Fermy energy there and keeping it there regardless of alterations in the ionised-donor percentage, insofar as the donor-population is dense enough for accommodating (recaptured) electrons and the ambient temperature is adequately cryogenic for the donor level to continue providing a trustworthy demarcation between (partially) filled and empty energy states within the wider-bandgap nanodevice-part. Such a Fermi level immunity against evolving cumulative photonic intake (tantamount to increasing the 2DEG population hosted by the NHI QW) by the nanoheterostructure would be sustainable through a mechanism successively lowering the 2DEG fundamental sublevel E0 (as well as the first excited sublevel, which nevertheless would not be participating in the hosting of QW conductivity electrons as always lying over the Fermi-energy demarcation) within the QW being commensurately broadened whilst keeping its energetic depth dictated by the invariable NHI conduction-band discontinuity. Believably, such a mechanism would be based on the formation of ever new (after each further photonic intaking) additional effective electron-attractive centers [12,24] accountable for evolving photowidening of the enclosingpotential profile. Thus, for each photoaugmentation of the QW 2DEG number, effected in parallel to the respective hole photoenhancement within the neutral portion of the lower-bandgap nanoheterostructure side, the above QW progressive modification, indexed by fundamental-sublevel eigenenergy photolowering occurring each time, would make it possible to accommodate the increasing 2DEG ensemble by ever more energy states appearing between the sinking subband bottom and the plausibly still Fermi level. Considering, therefore, the dynamics of energy level positions against regulated successive photon-doses being absorbed by the probed NHI, we obtain through partial differentiation of (1) with respect to the instantaneous cumulative photonic dose d:
oE0 1 of oEF ¼ þ ffi0 od q od od
ð3Þ
or
oE0 1 of ¼ od q od
ð4Þ
providing the photonic dose-rate of evolution of the 2DEG fundamental-eigenstate-sublevel E0 in terms of the experimentally traceable photon-dose-rate of augmentation of the 2DEG areal density f within the QW of the monitored NHI. In this sense, Eq. (4) furnishes the prime quest of our simplistic scheme: namely, following the dynamics of the evolving photonic modification of the NHI fundamental-eigenstate, during the procedure of successive intakings of appropriate photon-transmissions. Whenever an experimental measurement Df(d) for the augmentation of the 2DEG sheet-concentration f (with respect to its pre-illumination initial value fd) consequent upon some instantaneous total photon-dose d is obtained, the respective eigenstate-sublevel shift E0(f) (away from its dark locus E0d) drawn by this cumulative dose d is predicted, according to (4), to be
1 DE0 ðfÞ ¼ DfðdÞ:
q
ð5Þ
3. Application The NHI probed is formulated within a typical molecular beam epitaxy (MBE) – grown functional semiconductor structure, consisting of an upper, wide-bandgap, n-type (Si donor – doped at a concentration of the order of 1 1017 cm3) AlxGa1xAs (x 0.3) epitaxial layer with a thickness of the order of 2 lm developed upon a lower, narrower-bandgap, relatively p-type-like, non-intentionally doped (NID) GaAs epitaxial layer of a thickness of the order of 100 nm mounted on top of a semi-insulating (SI) GaAs substrate. The NHI of the device comprises the ionised-donor depletion zone on the upper-epilayer side and the 2DEG QW hosted within the lower epilayer. The NHI device, subjected to persistent-photoconductivity (PP) experimental conditions [9,11] by the provision of successive regulated photon-doses evolving within six orders of magnitude is illuminated from the upper epitaxial-layer side with normally incident light filtred to photon energies greater than the bandgap of each semiconductor and, thus, capable of producing interband transitions. The device is kept at a constant temperature of around 4 K and a magnetic field of about 0.5 T is applied within an automated Hall apparatus for the pertinent van der Pauw technique measuring the persistent photoenhancement (PPE) of both the NHI 2DEG population and the NHI 2DEG mobility. Figs. 1 and 2 present [24] the PPE Df in the NHI 2DEG sheetdensity f and the reduced (with respect to the unperturbed darkvalue ld) PPE Dll in the NHI 2DEG mobility l, respectively, versus d relative values of the incoming cumulative photon-dose d, prior to any saturation or negative differential feature of either evolution. For interpreting this regime of the NHI 2DEG PPE experimental curves, we come to the conclusion that, as the cumulative photon dose d absorbed by the device increases, the conduction electrons confined within the NHI QW (and expected to be occupying solely the QW fundamental conduction-subband in the electric quantum limit of operation of the device at 4 K) evolve to higher numbers, due to the augmentation of their initial dark-concentration by the rising persistent-photoelectron density. Photoaugmentation of 2DEG areal density f, now, is considered tantamount to its mobility PPE, especially in view of a previous theoretical finding of ours [13] correlating the the NHI 2DEG mobility l(f*) with
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7.5
7.0
ΔE0 (meV)
-2
Δζ (x10 cm )
-20
11
6.5
6.0
5.5
5.0 -30 1
2
3
4
5
1
6
2
3
4
5
6
log δrel
logδ rel Fig. 1. NHI 2DEG sheet-density PPE Df versus decimal logarithm of relative values drel for the instantaneous cumulative photon-dose d intaken.
Fig. 3. Predicted red photoshift DE of the NHI fundamental-eigenstate sublevel E0 against decimal algorithm of relative values drel for the instantaneous total photondose d intaken.
0.8
concerning the electron de Boglie time-independent wavefunction w(x) and taking into account the spatial variation m*(x) of the carrier effective mass, reads
0.7
0.9
Δμ/μd
2
d h dwðxÞ ½ þ UðxÞwðxÞ ¼ EwðxÞ; dx 2m ðxÞ dx
ð7Þ
with E being the allowed energy-eignevalue conjugate to each physically meaningful wavefunction w(x), solving (7) and vanishing asymptotically at infinities, i.e.
0.6 0.5
wð1Þ ¼ 0
0.4
ð8Þ
and h being Planck’s action constant divided by 2p. Performing, now, an independent variable transformation, namely,
0.3 1
2
3
4
5
6
x ax Arctanh ðnÞ $ /ðnÞ w½xðnÞ;
log δrel Fig. 2. Reduced (with respect to the unperturbed dark-value ld) PPE Dl/ld in the NHI 2DEG mobility l versus decimal logarithm of relative instantaneous total photonic dose values drel.
respective dimensionless reduced instantaneous 2DEG sheet-density f* through
1 pffiffiffiffi lðf Þ ¼ lr 1 f ef
ð6Þ
ð9Þ
We obtain in place of (7) the Sturm–Liouville differential equation
ðd=dnÞfMðnÞ½d/ðnÞ=dng uðnÞ/ðnÞ þ krðnÞ/ðnÞ ¼ 0
ð10Þ
under the boundary conditions
/ð1Þ ¼ 0 and /ðþ1Þ ¼ 0
ð11Þ
with functions M(n), u(n), and r(n) in the new dimensionless variable n (belonging to the universal interval [1, +1]) defined as
with lr being an appropriate, device-wise meaningful, reference mobility-value. Employing, furthermore, our predictive scheme we arrive, through Eq.(5), to Fig. 3 presenting the awaited photoshif DE0 evolution of the NHI fundamental-eigenstate sublevel against instantaneous total photon dose d intaken: This predicted gradual photolowering of NHI eigenstate-subband bottom is interpretable as quantum-mechanically compatible with our previous studies [13,17] approximating the modification of a generic nanodevice’s conductive-channel extension as positively linearly proportional to the respective 2DEG areal-concentration alteration.
MðnÞ ¼ ð1=aÞð1—n2 Þm0 =m ½xðnÞ;
ð12Þ
uðnÞ ¼ ½2a=ð1—n2 ÞU½xðnÞ=E ;
ð13Þ
rðnÞ ¼ 2a=ð1 n2 Þ
ð14Þ
4. NHI WF penetration-length photonics
rendering the characteristic confinement-length x* entering the independent variable transformation (9) after the dimensionless scale factor a equal to 2.76043 Å, mo giving the electron rest mass. For converting the Sturm–Liouville differential equation concerning the NHI 2DEG transformed wavefunction u(n) via a
With respect to the generic situation of a conductivity electron being hosted by the QW of potential-energy profile U(x) against the growth-axis coordinate x within a conventional semiconductor nanodevice heterointerface, the pertinent Schrödinger equation,
and dimensionless new, ‘reduced-energy’, eigenvalue k defined as
k¼
E ; E
ð15Þ
where T* denotes a convenient energy-scale 2
E
h 1eV; mo x2
ð16Þ
E.A. Anagnostakis / Journal of Non-Crystalline Solids 354 (2008) 4233–4237
linearized system of difference equations into a tridiagonal-matrix eigenvalue-problem, we have previously [23,24] composed a finite-difference-method iterative algorithm. Therewith, the opposite of the eigenvalues of this matrix lead to the approximations to the NHI WF exact reduced-energy eigenvalues k, thus computing (Eq. (15)) the allowed QW 2DEG subband-energies E = kE*. On the other hand, the determined Strum–Liouville eigenvectors |u(n)i conjugate to these numerical eigenvalues unveil through transformation (9) the quantum-mechanically allowed wavefunctions w(x) for the 2DEG dwelling within the nanodevice heterointerface QW and underlying the crucial optoelectronic effects exhibited by the generic semiconductor-nanostructure. In particular, such determination of the NHI 2DEG wavefunction may lead to the computation of its entailed penetration-length into the nanodevice neighbouring energy-barrier layer, thus, facilitating the prediction of 2DEG mobility behavior [25–27] and, furthermore, the consideration of quantum-mechanical-tunnelling transmission probability [28,29] for conductivity electrons escaping the heterointerface and traveling through the nanodevice – by virtue of a normal transport mechanism advantageously exploitable, especially at nanoelectronic cryogenic ambient temperatures. The above outlined finite-difference-method algorithm is incorporated in the following procedure, repeatedly performed for each successive experimental cumulative photon-dose d: Step 1. Extraction from predictive scheme Eq.(5) of the awaited 2DEG fundamental-sublevel photolowering DE0(d) conjugate to experimentally measured 2DEG sheet-density PPE Df(d) driven by current total photonic intake d. Step 2. Deduction of current 2DEG fundamental sublevel E0(d) = E0,d |DE0(d)| by subtraction of the absolute value |DE0(d)| of current fundamental sublevel photolowering DE0(d) from sublevel dark (prior to exposure to photonic doses) locus E0,d. Step 3. Iterative applications of the algorithm with respect to the sought 2DEG QW spatial width w(d) compatible with the respective fundamental-eigenstate sublevel E0(d) predicted for the current cumulative photonic dose d. The entailed potential-energy profile U(x; d) is simplifyingly simulated by a rectangular one, of energetic depth always expressible by the NHI conduction-band discontinuity DEC and resting on the boundaries of the photowidened QW spatial extension w(d). Thus, the parametrisation of the QW potentialenergy profile by total photonic intake d is effected through letting a simulative fixed-energetic-depth rectangular model-QW expand spatially at a rate induced by each current intaken cumulative photon-dose d. Step 4. Once the iterative algorithm has converged for the optimum QW spatial width w(d) conjugate to the current total photonic intake d, the adjoint 2DEG fundamental-eigenstate wavefunction w0(x; d) is obtained, thus exhibiting its traceable penetration-length l(d) into the NHI energeticbarrier part. Therefore, tracing the fundamental-eigenstate penetrationlength l(d) in the optimum wavefunction w0(x; d) obtained by convergence (with respect to the optimum QW width w(d) compatible with predicted photoredefined eigenstate locus E0(d)) of the iterative algorithm for each total photon-dose d intaken, we manage to map its photonic evolution (Fig. 4). This, juxtaposed with the experimentally measured 2DEG mobility PPE Dl(d) (Fig. 2), appears adequately justified in the common trend of growing photoinduced facilitation of 2DEG intra-QW transport at shortening mobility-limitting influence by external scatterers.
0.95 0.90 0.85
l/ld
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0.80 0.75 0.70 0.65 1
2
3
4
5
6
log δ re l Fig. 4. Reduced (with respect to the unperturbed dark-value ld) NHI 2DEG fundamental-wavefunction penetration-length.
5. Conclusion The modification of the NHI 2DEG-eigenstate by absorption of regulated successive photon-doses is studied for the generic case of a conventional nanoheterodiode, in terms of the 2DEG fundamental-sublevel eigenenergy correlation with the respective 2DEG areal density, versus instantaneous cumulative photonic intake. The scheme is plausibly applied to the experimental photoresponse of a typical AlxGa1-xAs/GaAsSi-modulation-doped NHD and is, also, indirectly justified through juxtaposition of its measured 2DEG mobility photonic evolution and the evolution deduced for the NHI fundamental-wavefunction penetration-length, as computed through an iterative algorithm converting the Sturm–Liouville differential equation concerning the NHI 2DEG transformed wavefunction into a tridiagonal-matrix eigenvalue-problem. Thus, the predicted trend of evolving red photoshift for the NHI eigenstate sublevel is obtained compatible with a proceeding NHI QW-extension photowidening, on the one hand, and a NHI wavefunction penetration-length shrinking being tantamount to the 2DEG mobility continuing PPE, on the other hand. Acknowledgements Valuable support and enlightening consultation offered by University of Athens Professors G.J. Papadopoulos and N. Guskos are gratefully acknowledged. References [1] Ch.P. Poole, F.J. Owens, Introduction to Nanotechnology, Wiley-Interscience, New Jersey, 1997 (Chapter 9). [2] R. Dingle, H.L. Stoermer, A.C. Gossard, W. Wiegmann, Appl. Phys. Lett. 33 (1978) 665. [3] R.O. Grondin, W. Porod, J. Ho, D.K. Ferry, G.J. Iafratte, Superlattices Microstruct. 1 (1985) 183. [4] J.N. Churchill, F.E. Holmstrom, Phys. Lett. A 85 (1981) 453. [5] T.C.L. Sollner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker, D.D. Peck, Appl. Phys. Lett. 43 (1983) 588. [6] N. Yokoyama, K. Imamura, T. Oshima, H. Nishi, S. Mutto, K. Kondo, S. Hiyamizu, Jpn. J. Appl. Phys. 23 (1984) L311. [7] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, S.N.G. Chu, A.Y. Cho, Science 264 (1994) 553. [8] J. Faist, F. Capasso, C. Sirtori, D.L. Sivco, A.L. Hutchinson, A.Y. Cho, Electron. Lett. 32 (1996) 560. [9] E.A. Anagnostakis, Phys. Stat. Sol. A 126 (1991) 397. [10] E.A. Anagnostakis, Phys. Stat. Sol. A 127 (1991) 153. [11] D.E. Theodorou, E.A. Anagnostakis, Phys. Rev. B 44 (1991) 3352. [12] E.A. Anagnostakis, Appl. Phys. A 54 (1992) 68. [13] E.A. Anagnostakis, Phys. Status Soilidi B 171 (1992) K75. [14] E.A. Anagnostakis, Phys. Stat. Sol. B 172 (1992) K61. [15] E.A. Anagnostakis, Phys. Rev. B 46 (1992) 7593. [16] E.A. Anagnostakis, Phys. Stat. Sol. A 136 (1993) 247. [17] E.A. Anagnostakis, Phys. Stat. Sol. B 177 (1993) 533. [18] E.A. Anagnostakis, D.E. Theodorou, J. Appl. Phys. 73 (1993) 4550.
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