We calculate the energy dispersion relations in Si quantum wells (QW), E(k2D), and quantum wires (QWR), E(k1D), focusing on the regions with negative effective mass (NEM) in the valence band. The existence of such NEM regions is a necessary condition for the current oscillations in ballistic quasineutral plasma in semiconductor structures.  The frequency range of such oscillations can be extended to the terahertz region by scaling down the length of structures.  Our analysis shows that silicon is a promising material for prospective NEM-based terahertz wave generators. We also found that comparing to Si QWRs, Si QWs are preferable structures for NEM-based generation in the terahertz range

Klimeck, Gerhard

Luisier, Mathieu

Mitin, Vladimir

Vagidov, Nizami

English

Purdue E-Pubs

Purdue UniversityPurdue e-PubsOther Nanotechnology Publications Birck Nanotechnology Center6-1-2007Energy dispersion relations for holes inn siliconquantum wells and quantum wiresVladimir MitinNizami VagidovMathieu LuisierGerhard KlimeckPurdue University - Main Campus, gekco@purdue.eduFollow this and additional works at: http://docs.lib.purdue.edu/nanodocsThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Mitin, Vladimir; Vagidov, Nizami; Luisier, Mathieu; and Klimeck, Gerhard, "Energy dispersion relations for holes inn silicon quantumwells and quantum wires" (2007). Other Nanotechnology Publications. Paper 97.http://docs.lib.purdue.edu/nanodocs/97J Comput Electron (2007) 6:227–230DOI 10.1007/s10825-006-0103-9Energy dispersion relations for holes in silicon quantum wellsand quantum wiresVladimir Mitin · Nizami Vagidov · Mathieu Luisier ·Gerhard KlimeckPublished online: 9 December 2006C© Springer Science + Business Media, LLC 2007Abstract We calculate the energy dispersion relations in Siquantum wells (QW), E(k2D), and quantum wires (QWR),E(k1D), focusing on the regions with negative effective mass(NEM) in the valence band. The existence of such NEMregions is a necessary condition for the current oscillationsin ballistic quasineutral plasma in semiconductor structures.The frequency range of such oscillations can be extendedto the terahertz region by scaling down the length of struc-tures. Our analysis shows that silicon is a promising materialfor prospective NEM-based terahertz wave generators. Wealso found that comparing to Si QWRs, Si QWs are prefer-able structures for NEM-based generation in the terahertzrange.Keywords Energy dispersion relations . Tight-bindingmodel . Negative effective mass . Quantum well . QuantumwireV. Mitin (!) · N. VagidovUniversity at Buffalo, SUNY, 332D Bonner Hall, Buffalo,NY 14260, USAe-mail: vmitin@buffalo.eduN. Vagidove-mail: nizami@eng.buffalo.eduM. LuisierIntegrated Systems Laboratory, ETH Zurich, 8092 Zurich,Switzerlande-mail: mluisier@iis.ee.ethz.chG. KlimeckNetwork for Computational Nanotechnology, Purdue University,West Lafayette, IN 47907, USAe-mail: gekco@purdue.edu1 IntroductionEnergy dispersion curves with negative-effective-mass(NEM) regions can be found in bulk semiconductors, quan-tum wells, quantum wires, as well as in artificially band-engineered structures such as heterostructure obtained by thecleaved edge overgrowth technique (see [1] and referencestherein). The existence of such NEM regions may lead tocurrent instabilities in ballistic diodes and transistors [2]. Forsubmicrometer base lengths of these devices the frequencyof current oscillations falls into the terahertz range. Tight-binding simulations of the energy band structure is an effec-tive tool for identification and engineering of structures withwell-pronounced NEM regions in energy dispersions.In this paper we report the results of band structure calcula-tions of Si quantum wells (QW) and quantum wires (QWR).We calculated the energy dispersion relations, E (k2D) and E(k1D), for different orientations of two-dimensional wavevec-tor, k2D, in QWs and one-dimensional wavevector, k1D, inQWRs. We used an empirical sp3d5s∗ tight-binding modelwhich takes into account 10 atomic orbitals: s- and exciteds∗-orbitals, three p-, and five d-orbitals. The inclusion in themodel of higher energy d-orbitals and spin-orbital couplinghas dramatically improved the precision of the calculatedelectron and hole energy dispersion relations. In our calcu-lations we used tight-binding parameters of Si from Ref.[3]. All other details of the exploited model can be found inRef. [4].The simplest model of energy dispersion relations withNEM regions is based on anti-crossing of two bands withlight and heavy carriers. It can be described by the followingequation:E(k) = 12[ε1(k) + ε2(k)−√[ε1(k)− ε2(k)]2 + 4δ2](1)Springer228 J Comput Electron (2007) 6:227–230NEM regionkE(arb. units)ε1 (k)e2 (k)ε0ki1 ki201/m*(arb. units)+-1234Fig. 1 The graph of E(k) defined by Eq. (1) is shown by solid line1, inverse effective mass, 1/m∗, for E(k) is shown by dash-dotted line2, and two parabolic energy dispersion relations, ε1(k) and ε2(k), areshown by dashed lines 3 and 4. The region of negative effective massis located between two inflection points, ki1 and ki2where ε1(k) = h¯2k2/2m, ε2(k) = h¯2k2/2M + ε0, and δ <<ε0 (see Fig. 1) [5]. Quasineutral semiconductor plasma withcurrent carriers having dispersion E(k) of Eq. (1), may haveself-organized oscillatory regimes only at certain ratio of twoeffective masses, m and M, - light and heavy effective masses,respectively. This ratio must satisfy the following inequality[5]:M/m > 2. (2)Large ratios of M/m are preferable for NEM-based currentoscillations to be established.While this model ignores many important details, it iswidely used for qualitative estimates in nonlinear carriertransport. For this reason, in the next sections we will fitour data to the above model.2 Energy dispersion relations in Si quantum wellsFirst, we will present results for Si QWs grown on (100)wafers. Analysis of the peculiarities of the energy dispersionrelations associated with the NEM is our main interest here.We calculated energy dispersion relations E(k2D) for holesin Si QWs with thicknesses from 0.8 nm to 7 nm for k2D|| [110] and k2D || [100]. As an example, E(k2D) of QWwith thickness 2.72 nm and k2D || [110] is shown in Fig. 2.Figure 2 demonstrates well-defined NEM regions, suitablefor occurrence of NEM-based current instability, and wideenergy interval (eVc, eVK ) or interval of applied voltage (Vc,VK ), where the stationary current in ballistic semiconductordevices with NEM carriers is unstable. The range of this in-terval is defined by kc and kK , where kc is the tangency pointand kK is the intersection of tangent and E(k2D), as shownin Fig. 2. Calculated dependences of these two parameterson the well thickness are shown in Fig. 3. The increase ofkc by 50% and increase of kK by almost factor of 5 with the0.0 0.2 0.4 0.6 0.8 1.0-0.13-0.12-0.11-0.10-0.09kc kKk2DaE(eV)eVceVK1 ∆2345Fig. 2 Dispersion relations E(k2D) for holes in a Si QW for k2D || [110].The thickness of QW is 2.72 nm.# is the energy gap between the lowestand next subband, kc is a tangency point around which the NEM regionis located; kc and kK define the beginning and the end of the intervalwhere instability takes place, and a is the Si lattice constant equal to5.43 A˚. Solid lines (1, 3) and dash-dotted lines (2, 4) relate to two energydispersion relations with opposite spins. Tangent to curve 1 is shownby dashed line 50 1 2 3 4 5 6 70.100.120.140.160.180.200.00.30.60.91.21.51.8k caQW width (nm)k Ka12Fig. 3 Dependences of critical values of kc and kK on QW width areshown by solid line 1 and dashed line 2, respectively; k2D || [110]decrease of thickness of QW leads to a substantial increaseof energy interval where the NEM instability exists.Although, the obtained energy dispersion relations are notdescribed by Eq. (1), qualitatively we can introduce two ef-fective masses, m and M, that characterize calculated E(k2D).As we can see from Fig. 4 the inequality (2) for Si QW withk2D || [110] is fully satisfied for widths ranging from 0.8 to7 nm. Another important characteristic of the energy disper-sion relations is the energy distance between the lowest andthe next subband, #. For QWs with thicknesses less than3 nm this value is about 25 meV. As we can see from Figs. 3and 4 there is a good control of kc, kK , #, and M/m over awide range of QW thicknesses.In Si QWs with k2D || [100] the energy dispersion relationsdo not satisfy the inequality (2) for any QW thicknesses andthe NEM regions are weakly pronounced.In addition, the carried out calculations showed that theKramers degeneracy [6] of holes at k2D || [110] is lifted fork2D $= 0 and there are two different branches of dispersionSpringerJ Comput Electron (2007) 6:227–230 229051015(meV∆)QW width (nm)M/m120 1 2 3 4 5 6 710152025Fig. 4 Dependences of M/m ratio and energy gap # on Si QW widthfor E(k2D) with k2D || [110] are shown by solid line 1 and dashed line 2,respectively. Dependence on M/m was calculated for the valence band’slowest energy subbandE(eV)-0.52-0.48-0.44-0.40(b)0.0 0.2 0.4 0.6 0.8 1k1Da12E(eV)-0.5-0.4-0.3-0.2(a)123456Fig. 5 Dispersion relations E(k1D) for holes in a Si QWR: (a) withcross-section 1.63 × 1.72 nm2 (in directions [001] and [−110], re-spectively) and with k1D || [110]; (b) with cross-section 1.63 nm2 (indirections [001] and [010]) and with k1D || [100]. Solid (1, 2, 3) anddashed (4, 5, 6) lines correspond to holes with opposite spinsrelations corresponding to two different spin states as it isseen in Fig. 2 [7]. In contrast to that, for the hole states withwavevector k2D || [100] the subbands stay spin degenerate forany value of k2D.3 Energy dispersion relations in Si quantum wiresThe calculated Si QWR energy dispersion relations, E(k1D),for k1D || [110] demonstrate well enough pronounced NEMregions, whereas dispersion curves for k1D || [100] are almostflat (see Figs. 5(a) and (b)).The model shows that as in the case of Si QWs, the in-equality (2) for energy dispersion relations in QWRs for k1D|| [100] is never satisfied.Analysis of dependences of inverse effective mass, 1/m∗,versus k1D shows that only for QWRs with small cross-0.0 0.2 0.4 0.6 0.8 1.0-8-4048k1Da1/m*NEMkcaki1 ki2Fig. 6 Dependence of inverse effective mass, 1/m∗, on k1Da for holesof the lowest energy state in a Si QWR with the cross-section 1.63 ×1.73 nm2 (in directions [001] and [−110], respectively) and with k1D|| [110]. The NEM region is located between two inflection points, ki1and ki2; the ratio M/m ≈ 3sections the inequality (2) may be satisfied. An exampleshown in Fig. 6 for QWR with cross-section 1.63 × 1.73nm2 (in directions [001] and [−110], respectively) has theratio M/m ≈ 3. If we compare the energy dispersion rela-tions of QWs and QWRs we observe that the energy intervalof NEM-induced instability is shifted to higher energies (orhigher values of k) in the case of QWRs. The shift of NEMregion to higher energies makes it more difficult to maintainballistic regime of the device.As in the case of QWs, the spin degeneracy of states fork1D $= 0 in QWRs with k1D || [110] is lifted whereas in thecase of k1D || [100] the spin degeneracy is conserved as it isshown in Figs. 5(a) and (b).4 ConclusionWe calculated energy dispersion relations in Si QWs, E(k2D),and QWRs, E(k1D). The results were obtained for differ-ent directions of two-dimensional wavevector, k2D, and one-dimensional wavevector, k1D. The analysis of the obtaineddependences shows that the occurrence of NEM-based os-cillations are most likely to be found in Si samples in [110]crystallographic direction. Dispersion curves in [100] direc-tion for both QWs and QWRs are flat and their NEM regionsdo not satisfy the necessary condition for instability occur-rence. Another interesting detail is that the energy bands arespin degenerate in [100] direction but the spin degeneracy islifted in [110] direction for both QWs and QWRs.The ratio of the effective masses after and before the NEMregion, M/m, behaves differently in QWs and QWRs. ForQWs in [110] direction this ratio for wide range of wellthicknesses is significantly greater than in QWRs. The en-ergy interval of NEM region in QWRs is shifted to higherenergies (or higher momenta). This may be an obstacle formaintaining of ballistic oscillatory regime of a device. InSpringer230 J Comput Electron (2007) 6:227–230addition, the energy interval of NEM region is smaller in thecase of QWRs in comparison to QWs. The fourth parameterthat characterizes NEM oscillatory regime,#, is the distancebetween the lowest and the next subbands. The larger is thisparameter, the better are the conditions to achieve larger am-plitudes of oscillations. And this is the only parameter thatis superior in QWRs in comparison with the case of QWs.In summary, we demonstrate that Si QWs are promisingcandidates for NEM-based terahertz generation. The dataobtained in the framework of empirical tight-binding modelcan be fitted by the two-band anti-crossing model with theratio, M/m, changing in the range from 3 to 13 (see Fig. 4).Large values of M/m result in effective current oscillationsin wide energy intervals. Using the anti-crossing model, onecan estimate the frequencies of generation band [2]. For QWstructures with the base lengths 0.1–0.3 µm we obtain theestimates of generation band of 0.5–2 THz.Acknowledgment This research was supported by Petroleum Re-search Fund Grant No. 41317-AC10, NSF Grant No. EEC-0228390,and the Indiana 21st Century Fund.References1. Gribnikov, Z.S., et al.: Quantum real-space transfer in a heterostuc-ture overgrown on the cleaved edge of a superlattice. J. Appl. Phys.93(1), 330 (2003)2. Gribnikov, Z.S., et al.: Negative-effective-mass ballistic field-effecttransistor: Theory and modeling. J. Appl. Phys. 87(10), 7466(2000)3. Boykin, T.B., et al.: Valence band effective-mass expressions inthe sp3d5s∗ empirical tight-binding model applied to a Si and Geparametrization. Phys. Rev. B 69(11), 115201 (2004)4. Rahman, A., et al.: Atomistic approach for nanoscale devices at thescaling limit and beyond – valley splitting in Si. Jpn. J. Appl. Phys.44(4B), 2187 (2005)5. Gribnikov, Z.S., et al.: Terahertz ballistic current oscillations forcarriers with negative effective mass. J. Appl. Phys. 80(10), 5799(1996)6. Schmid, U., et al.: Relativistic band structure of Si, Ge, andGeSi: Inversion-asymmetry effects. Phys. Rev. B 41(9), 5919(1990)7. The sp3d5s∗ model includes spin and spin-orbital coupling explicitly.These simulations do not include any external magnetic field thatwould provide spin selection. The splitting of the states is purely dueto symmetry breaking and the two split states are a mixture of up anddown spinsSpringer

Energy dispersion relations for holes inn silicon quantum wells and quantum wires

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Energy dispersion relations for holes inn silicon quantum wells and quantum wires

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