Quantum Kinetic Transport High Electric...
6
VLSI DESIGN 1998, Vol. 6, Nos. (1-4), pp. 3-7 Reprints available directly from the publisher Photocopying permitted by license only 1998 OPA (Ov+rseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint. Printed in India. Quantum Kinetic Transport under High Electric Fields NOBUYUKI SANO *’* and AKIRA YOSHII NTT LSI Laboratories, 3-1 Morinosato Wakamiya, Atsugi-shi, Kanagawa 243-01, Japan Quantum kinetic transport under high electric fields is investigated with emphasis on the intracollisional field effect (ICFE) in low-dimensional structures. It is shown that the ICFE in GaAs one-dimensional quantum wires is already significant under moderate electric field strengths (> a few hundreds V/cm). This is a marked contrast to the cases in bulk, where the ICFE is expected to be significant under extremely strong electric fields (> MV/cm). Employ- ing the Monte Carlo method including the ICFE, the electron drift velocity in quantum wires is shown to be much smaller than that expected from earlier investigations. Keywords: quantum transport, hot electron, monte carlo, quantum wire, intracollisional field effect, col- lisional broadening INTRODUCTION The fast development of semiconductor devices has brought about renewed interest in high-field carrier transport in semiconductors. Among others, the breakdown of the semiclassical Boltzmann transport equation under high electric fields, on which most conventional analyses of carrier transport in semicon- ductor devices are based, has been widely pre- dicted.J1] However, the predicted electric field strengths under which the semiclassical treatment of carrier transport is-supposed to break down vary from several tens kV/cm to MV/cm. Therefore, our under- standing of quantum carrier transport under high elec- tric fields is still immature. In the present paper, we investigate the quantum effects in electron transport from a rather different point of view; we apply the quantum kinetic transport equation (the Barker-Ferry equation [2]) to low-dimensional structures so that the quantum effects are, in some cases, more pronounced than in the three-dimensional (3-D) bulk cases and their physical meanings become more transparent. We consider ideal 1-D GaAs quantum wires and investigate the intracollisional field effect (ICFE) on electron transport via the Monte Carlo method.[3] To the best of our knowledge, this is the first time the ICFE on high-temperature electron transport in low- dimensional structures has been taken into account. Since the ICFE is mainly ascribed to the energy change of an electron by the electric field during the collision duration with phonons,[2] it is expected that the ICFE could be most significant when the electron motion is restricted to the dimension parallel or anti- parallel to the electric field. As we shall show, this is indeed the case in 1-D quantum wire structures; the ICFE becomes significant even under moderate elec- tric field strengths and greatly affects transport char- acteristics such as the drift velocity. * Corresponding author, e-mail: [email protected], phone: +81-462-40-2616, FAX: +81-462-40-4306 t Present address: Institute of Applied Physics, University of Tsukuba, Tsukuba, lbaraki 305, Japan
Transcript of Quantum Kinetic Transport High Electric...
VLSIDESIGN1998, Vol. 6, Nos. (1-4), pp. 3-7Reprints available directly from the publisherPhotocopying permitted by license only
1998 OPA (Ov+rseas Publishers Association) N.V.Published by license under
the Gordon and Breach SciencePublishers imprint.
Printed in India.
Quantum Kinetic Transport under High Electric FieldsNOBUYUKI SANO*’* and AKIRA YOSHII
NTTLSI Laboratories, 3-1 Morinosato Wakamiya, Atsugi-shi, Kanagawa 243-01, Japan
Quantum kinetic transport under high electric fields is investigated with emphasis on theintracollisional field effect (ICFE) in low-dimensional structures. It is shown that the ICFE inGaAs one-dimensional quantum wires is already significant under moderate electric fieldstrengths (> a few hundreds V/cm). This is a marked contrast to the cases in bulk, where theICFE is expected to be significant under extremely strong electric fields (> MV/cm). Employ-ing the Monte Carlo method including the ICFE, the electron drift velocity in quantum wiresis shown to be much smaller than that expected from earlier investigations.
Keywords: quantum transport, hot electron, monte carlo, quantum wire, intracollisional field effect, col-lisional broadening
INTRODUCTION
The fast development of semiconductor devices hasbrought about renewed interest in high-field carriertransport in semiconductors. Among others, thebreakdown of the semiclassical Boltzmann transportequation under high electric fields, on which most
conventional analyses of carrier transport in semicon-ductor devices are based, has been widely pre-dicted.J1] However, the predicted electric fieldstrengths under which the semiclassical treatment ofcarrier transport is-supposed to break down vary fromseveral tens kV/cm to MV/cm. Therefore, our under-standing of quantum carrier transport under high elec-tric fields is still immature. In the present paper, weinvestigate the quantum effects in electron transportfrom a rather different point of view; we apply thequantum kinetic transport equation (the Barker-Ferryequation [2]) to low-dimensional structures so that the
quantum effects are, in some cases, more pronouncedthan in the three-dimensional (3-D) bulk cases andtheir physical meanings become more transparent.
We consider ideal 1-D GaAs quantum wires andinvestigate the intracollisional field effect (ICFE) onelectron transport via the Monte Carlo method.[3] Tothe best of our knowledge, this is the first time theICFE on high-temperature electron transport in low-dimensional structures has been taken into account.
Since the ICFE is mainly ascribed to the energychange of an electron by the electric field during thecollision duration with phonons,[2] it is expected thatthe ICFE could be most significant when the electronmotion is restricted to the dimension parallel or anti-parallel to the electric field. As we shall show, this isindeed the case in 1-D quantum wire structures; theICFE becomes significant even under moderate elec-tric field strengths and greatly affects transport char-acteristics such as the drift velocity.
* Corresponding author, e-mail: [email protected], phone: +81-462-40-2616, FAX: +81-462-40-4306t Present address: Institute of Applied Physics, University of Tsukuba, Tsukuba, lbaraki 305, Japan
4 NOBUYUKI SANO and AKIRA YOSHII
QUANTUM KINETIC TRANSPORT EQUATION
Our starting point is the quantum kinetic transportequation (the Barker-Ferry equation) for the one-par-ticle distribution function of electrons fk, which isderived from the quantum Liouville equation for thereduced electron density matrix.[4] Under a constantelectric field F with the non-degenerate condition, theBarker-Ferry equation for electrons is given by
fk(t) eF. )fk(t))t )
2Re f0 dxlMq 12 (gq-f-+)q,,_- (1)q)XA-q 1:)
-e-<*-"q-*+nm)A (t x) },with kF k- eFx. Here, ek is the electron energy withwave-vector k, q the phonon wave-vector, Mq thematrix element for the electron-phonon interaction,
COq the longitudinal optical (LO) phonon energy, and
Nq the phonon occupation number given by the Bose-Einstein statistics, rl (-1) for phonon emission(absorption). Note that we use the units of hthroughout the paper.
Notice that Eq. (1) is very similar in form to thesemiclassical Boltzmann transport equation exceptfor the collision integral [the right-hand side of Eq.(1)]; i.e., the time retardation of the collision dynam-ics with phonons is taken into account. Because ofthis time retardation, a numerical evaluation of Eq.(1) is still a formidable task. Since the electron energydistribution usually spreads over the wide energyranges under high electric fields, the time retardationof the electron distribution function in the collisionintegral may be safely ignored. As a result, the colli-sion integral Ic in Eq. (1) is rewritten as
t 2Eq,__ Uq 2 (Nq + 1/2 + ){S(rl,ke)A-q(t) S(-rl,ke)A(t)}, (2)
where the spectral density S(TI, kF) is given by
S(rl,k) de-rcos[(e+n- ee- rlm)].
(3
Here, Fk is the total phonon scattering rate and shouldbe determined self-consistently with Eq. (1). Recallthat Eq. (1) is derived under the Born approxima-tion,[4] in which only the lowest-order correction forthe electron-phonon interaction is included. There-fore, we put damping factor Fk in the collision inte-
gral and extend the upper limit of the time integral to
infinity because the time correlation would be
destroyed by a consecutive collision with phonons.When the electron lifetime is infinite (Fk 0) and theelectric field is suppressed (F- 0), Eq. (3) correctlyreduces to the energy-conserving delta function andEq. (1) with Eq. (2) becomes identical to the Boltz-mann transport equation. Fortunately, Eq. (3) can be
analytically solved for a fixed Fk and plotted withsolid lines under F 500 V/crn in Fig. 1. The electricfield is assumed to be directed to the negative direc-tion. Notice that the direction to which the spectraldensity is skewed is dependent on the direction of theelectron motion; when the electron propagates againstthe electric field [Fig. (a)], the spectral density shifts
GaAs (a)
F=500
---quantum
’ \/’ -i
(t) GaAs
0 V/cm
j ------quantum
,..._: |-50 0 50 100
ENERGY DETUNING (meV)
75
so
o S
m 0
w -25
-50
100
75
w -25
-50-100
FIGURE Electron spectral density as a function of energy detun-ing for the electrons propagating to the (a) positive and (b) negativedirections under F 500 V/cm (solid lines). The dashed lines rep-resent the approximated spectral densities employed in the presentMonte Carlo simulations
QUANTUM KINETIC TRANSPORT UNDER HIGH ELECTRIC FIELDS 5
to the positive direction, and vice versa. This will beexplained in the next section along with the phononscattering rates for quantum wires.
It is clear from Fig. that the ICFE essentially con-sists of two different effects: (1) the energy-conserv-ing delta function is broadened and (2) the energydetuning (zero-point) is shifted. Both effects aredirectly related to the strength and the direction of theelectric field, and the magnitude of both the broaden-ing and the shift, AF, is approximately given by
A- "2: (4)
where m* is the electron effective mass. We wouldlike to stress that AF is dependent of the direction ofthe phonon wave-vector, or equivalently, of the elec-tron motion. Since the electron motion is strictly con-fined along the electric field in 1-D quantum wires,AF is always finite and, thus, the ICFE is alwayseffective. On the other hand, the ICFE is not alwayssignificant in 3-D bulk because electrons can move inany direction.
QUANTUM TRANSPORT IN 1-D QUANTUMWIRES
The quantum kinetic equation described in the previoussection can be easily applied to ideal 1-D quantumwires in GaAs. For simplicity, we assume that the rec-tangular quantum wire with the size 30 nm is formedby an. infinitely deep potential well and only the low-est subband is occupied. In addition, only the Fr6hlichoptical phonon scattering in bulk mode is considered.Since the emphasis of the present analyses is on howthe dynamical quantum effects affect electron trans-
port in 1-D quantum wires, the details of the scatter-
ing modes related to the low-dimensional structure
itself essentially do not change the present results.Figure 2 shows the Fr6hlich scattering rates (solid
and dashed lines) obtained from the spectral densityEq. (3) under two different field strengths; F 100and 500 V/cm. As well known from earlier studies onthe collisional broadening (CB),[6] the divergence ofthe scattering rate, indicated by dotted lines in Fig. 2,
,o i’ (a) F 100 V/cm
o o
",9.. 40 (b) F 500 V/cm
_z 20
20 40 60 80 100
ELECTRON ENERGY (meV)
FIGURE 2 Energy dependence of the Fr6hlich optical phonon’scattering rates at room temperature under F- (a) 100 V/cm and(b) 500 V/cm. The solid and dashed lines represent the scatteringrates for the electrons propagating to the positive and negativedirections, respectively. The dotted lines show the scattering ratesevaluated by the Fermi golden rule
is associated with the 1-D density of states andremoved by the energy broadening in the spectraldensity. Notice that the divergence is mainly due to
the phonon emission processes so that the divergenceoccurs at the LO phonon energy (-- 36 meV). Themajor feature associated with the ICFE is, however,that the scattering rates become asymmetric with
respect to the direction of the electron motion. This is
closely related to the shift of the energy detuning inthe spectral density, as already seen in Fig. 1, and canbe explained as follows. When an electron movesagainst the electric field, it gains energy from theelectric field during the collision duration with pho-nons. Therefore, it can emit an LO phonon even if theelectron energy is smaller than the threshold energyfor phonon emission. In other words, the phononenergy is effectively reduced when the electronmoves against the electric field, and the scattering rateshifts to the lower energy-side. On the other hand, anelectron loses its energy when it moves along theelectric field and, thus, the phonon energy is effec-
6 NOBUYUKI SANO and AKIRA YOSHII
tively increased. As a result, the scattering rate shiftsto the higher energy-side.
The transport characteristics could be determinedfrom the quantum kinetic equation described in theprevious section. Since the quantum kinetic equationis almost identical to the Boltzmann transport equa-tion, conventional Monte Carlo methods can beapplied to numerically solve it by substituting a
proper spectral density in the collision integral insteadof the energy-conserving delta function.[5] Unfortu-nately, the spectral density shown in Fig. is not pos-itive definite. Therefore, we approximate it by aGaussian form with the width and shift given by Eq.(4). The approximated spectral densities employed inthe Monte Carlo simulations are plotted by dashedlines in Fig. 1.
Figure 3 shows the electric field dependence of theelectron drift velocity at room temperature obtainedfrom the Monte Carlo simulations (dashed line). Forcomparison, the drift velocity evaluated by the con-ventional Boltzmann transport equation is also shown(solid line). The dotted line represents the result whenonly the CB is included. For the CB, we assume aconstant energy-broadening (Fk 2.5 meV), as doneby Leburton and his group.[6] We would like to stress
that the drift velocity is greatly reduced when the
20
(C) 15
0 200 400
semi-classical -Dwith CB-with ICFE
.-" T 300 K,,,I I,..
600 8O0 1000
ELECTRIC FIELD (V/cm)
FIGURE 3 Electric field dependence of drift velocity in the idealI-D quantum wire. The solid line represents the drift velocity withno quantum correction, whereas the dotted and dashed lines showthe respective drift velocities when CB and ICFE are taken into,account
ICFE is included and the reduction becomes greateras the electric field increases. Notice that the CB alsoreduces the drift velocity, but to a much smallerextent. This implies that the reduction of the driftvelocity due to the ICFE is mainly ascribed to the
asymmetric phonon scattering rates. Namely, moreelectrons occupy the states with positive wave,-vector
because the electric field is directed to the negativedirection and, hence, the scattering rate shifted to thelower energy-side (represented by solid lines in Fig.2) is, as a whole, more conducive to electron transportthan that shifted to the higher energy-side (repre-sented by dashed lines in Fig. 2). Furthermore, thelarge peak in the phonon scattering rate indicates theelectron transport characteristics is dominantly con-trolled by electrons with energy below the phonon-emission threshold energy unless the electric field is
extremely strong. Therefore, a slight change in thephonon scattering rates greatly affects the drift veloc-ity and, thus, the ICFE is already significant undermoderate electric fields, which is in marked contrastto bulk cases.
CONCLUSIONS
High-temperature quantum kinetic transport underhigh electric fields has been investigated with empha-sis on the intracollisional field effect (ICFE) in low-dimensional structures. We have shown that the ICFEin GaAs one-dimensional quantum wires is alreadysignificant under moderate electric field strengths (> afew hundreds V/cm) because of the strong restrictionon the electron motion imposed by the low-dimen-sional structures. Employing the Monte Carlo methodincluding the ICFE, it has been shown that the elec-tron drift velocity in quantum wires could be muchsmaller than that expected from earlier investigations.
AcknowledgementsThe authors would like to thank Y. Imamura for hisongoing support during the course of this study. Oneof the authors (N. S.) is also grateful to D. K. Ferryand K. Taniguchi for fruitful conversations.
QUANTUM KINETIC TRANSPORT UNDER HIGH ELECTRIC FIELDS 7
ReferencesFor a review, see L. Reggiani, "Recent Results on Hot-Elec-tron Quantum Transport," Physica, vol. 134B, pp. 123 131,1985. Also, see a series of papers by P. Lipavsky et al, "HighField Transport in Semiconductors," Phys. Rev. B, vol. 43,pp. 4885-48915; pp. 61550-1515154, pp. 61565-151581, 1991.
[2] J.R. Barker, "High Field Collision Rates in Polar Semicon-ductors," Solid State Electron., voi. 21, pp. 2157- 271, 1978;J. R. Barker and D. K. Ferry, "Self-Scattering Path-VariableFormulation of High-Field, Time-Dependent, QuantumKinetic Equations for Semiconductor Transport in the Finite-Collision-Duration Regime," Phys. Rev. Left., vol. 42,pp. 1779-1781, 1979.
[3] C. Jacoboni and L. Reggiani, "The Monte Carlo Method forthe Solution of Charge Transport in Semiconductors withApplication to Covalent Materials," Rev. Mod. Phys., vol. 55,pp. 645-705, 1983.
[4] N. Sano and A. Yoshii, "Energy Broadening in UltrafastRelaxation of Photoexcited Electrons: A Monte CarloApproach," Phys. Rev. B, vol. 48, pp. 142t5-1432, 1993.
[5] L. Reggiani, P. Lugli, and A.-P. Jauho, "Monte Carlo Algo-rithms for Collisional Broadening and Intracollisional Field
Effect in Semiconductor High-Field Transport," J. Appl.Phys., vol. 64, pp. 3072-3078, 1988.
[6] For a recent review, see J. P. Leburton and D. Jovanovic,"Hot-Carrier Dynamics in Mesoscopic Structures," Semi-cond. Sci. Technol., vol. 7, pp. B202-B209, 1992. Also, seeN. Mori and C. Hamaguchi, "Hot-Electron Transport inQuantum Structures," Semicond. Sci. Technol., vol. 9,pp. 941-945, 1994.
Biographies
Nobuyuki Sano is with NTT LSI Laboratories, Kana-gawa, Japan. He is interested in device physics with
emphasis on high-field carrier transport in semicon-ductors.
Akira Yoshii is a Group Leader of Device PhysicsGroup at NTT LSI Laboratories, Kanagawa, Japan.His interests include device simulation, device mode-ling, and device physics including quantum effects.
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