Self-Consistent Scattering Calculation Resonant T nneling Diode …downloads.hindawi.com/journals/vlsi/1998/078412.pdf · 2018. 11. 13. · 86 J.P. SUNandG.I. HADDAD RTDwith Phonon

VLSI DESIGN1998, Vol. 6, Nos. (1-4), pp. 83-86Reprints available directly from the publisherPhotocopying permitted by license only

998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

Self-Consistent Scattering Calculation of ResonantT nneling Diode Characteristics

J. P. SUN and G. I. HADDAD

Department ofElectrical Engineering and Computer ScienceThe University ofMichigan, Ann Arbor, MI 48109-2122

We perform a self-consistent calculation of resonant tunneling diode (RTD) I-V characteris-tics including optical phonon scattering. The self-consistency is obtained by solving theSchr6dinger equation and Poisson’s equation iteratively with the Thomas-Fermi approxima-tion used for the device contact regions. For evaluation of phonon-assisted current density,the optical phonon scattering in the quantum well is modeled using the optical model poten-tial. Electron transverse momentum is also incorporated. The calculated current and electronwavefunction illustrate the optical model and effects of the phonon scattering on the currenttransport. The I-V characteristics we obtain from the model calculation are in good agreementwith experimental results. This work manifests the importance of including self-consistency,optical phonon scattering, and electron transverse momentum in modeling realistic RTDstructures.

1 INTRODUCTION 2 RTD MODEL

Although various models for the Resonant TunnelingDiodes (RTDs) have been developed, research for anaccurate model of RTD suitable for device routine

analysis is ongoing as the results of modeling calcula-tions are generally not very satisfactory for a critical

comparison between calculated and experimental datawith discrepancies typically found in the valley cur-

rent, and in the peak and valley voltages. Within theeffective mass envelope function framework, sev-eral formulations incorporating optical phonon scat-

tering processes have been proposed for simulation ofRTD I-V characteristics with various degrees of suc-

cess. In this report we present modeling results ofRTD I-V characteristics when the self-consistency,optical phonon scattering, and electron transverse

momentum are all taken into consideration.

83

The self-consistent scattering calculation in this workcan be divided into three parts: (1) Thomas-Fermi cal-culation which solves the Fermi distribution and Pois-

son’s equation iteratively and generates self-consistent charge and potential distributions in thecontact regions. (2) The quantum well calculationwhich solves the Schr6dinger equation for the quan-tized charge distribution in the quantum well; thequantum charge is then iterated with Poisson’s equa-tion to obtain self-consistent charge and potential dis-tributions over the entire device. (3) Solution ofscattering states for the Schr6dinger equation whose

potential term consists of both the self-consistentdevice potential and the imaginary optical potentialintroduced for inelastic phonon scattering process 1,2]. To evaluate the phonon-assisted current density,

84 J. E SUN and G.I. HADDAD

for a specified bias voltage and temperature, eigen-state electrons with the Fermi distribution are incident

upon the double-barrier structure, scattered elasticallyby the conduction band profile and inelastically in thequantum well by the optical potential,-iW0. Thedevice structural and doping specifications are incor-

porated in all the three parts. Details for the self-consist-ent solution and electron transverse momentum havebeen described elsewhere [3, 4, 5]. In steady state, theelectron current density is attenuated by the opticalpotential at a rate of 2W0/, corresponding to a scatteringtime " =/2W0. The probability of finding electrons in thequantum well is reduced corresponding to an electronmean free path given by ) aj , where J x/2E/m* isthe electron velocity for an electron state and E is theelectron energy in the growth direction. A damping con-

stant /for the wavefunction in the quantum well canthen be defined accordingly, 3( 1/(2v) m*/2E/.is allows us to express the electron wavefunction inthe well as a linear combination of damping waves

exp[i(k + iy)x] and exp[-i(k + iy)x]. The one-dimensional,time-independent Schr6dinger equation now reads,

2 x ]--

[E’+ 2m*2k(y) V(x)-iWo(E,x)] (x)-O (1)

where m* is the variable electron effective mass, kll iselectron transverse momentum, and x the value of thescattering time of hot electrons in the quantum well.

Sop + exp(mo,/r)-I

E- Eo gop (emission/absorption) (2)

=so[exp(li@op/klT)-1

E-Eol <op (absorption) (3)

where Sop can be calculated for the specific material, h@opthe energy of optical phonons, E0 the resonant energy, kthe Boltzmann constant, and T is the absolute tempera-ture. The inelastic scattering potential is then related to

energy, material and temperature. e total transmission

Tto is the sum of coherent transmission Tco and inco-

herent transmission Tinco, and an attenuation coeffi-

cient is defined as A Tco -Rco, where Rcoh is

the coherent reflection coefficient. After a single col-lision event, the attenuated part is assumed to tunnelin both-x and +x directions in proportion to the trans-

mission coefficient for each barrier, denoted respec-tively as TR for the barrier to the right hand side and

T/ for the barrier to the left hand side of the quantumwell. 5 The transmission from region t to region J is

calculated to be the squared ratio of the transmittedelectron flux to the incident flux,

ICvl 2 kvIA/j k mv

where Cv is transmitted wave amplitude, with associ-ated effective mass mv and wavevector kv while

is incident wave amplitude, with mg and kg in therespective regions. The wave amplitudes are availablefrom the numerical solution of Eq. (1) for a particularself-consistent potential V(x). The total transmission

through the double-barrier structure is then

ATRo roh + 7oh r,oh + (5)TL + TR

The current transport components described as suchare depicted in Fig. 1, the current attenuation and

InGaAs

FIGURE Current transport components in an RTD structure

SELF-CONSISTENT SCATTERING CALCULATION OF RESONANT TUNNELING DIODE CHARACTERISTICS 85

Phonon Scaffering in RTD

2.0

1.5Current Density

0.5 0.5

’,: ,:-o.5 ’,: ’:

o.o oo.o ,oo.o ,oo.o oo.o ,ooo.o

Distance (A)

FIGURE 2 Calculated current density and electron wavefunctionthe RTD at bias voltage of 0.5 V. Self-consistent case

electron wavefunction as calculated for an electronstate due to the phonon scattering are plotted in Fig. 2,which illustrate the optical model. The integration ofthe total transmission over the energy then yields thedevice current.

3 RESULTS

Numerical simulations are performed for an RTDstructure consisting of (20 ) AlAs/(44 ) In.75 Ga.25As/(20/) AlAs double-barrier quantum well with a

contact doping of 1.0 1018 cm-3. The barrier heightis taken to be 1.5 eV between the contact and barrier

and 1.61 eV between the barrier and quantum well,with the electron effective mass being 0.044 m0,

0.092 m0, and 0.034 m0 in the contact, barrier, andwell regions, respectively. Electron transverse

momentum of 3.2 106 cm-1 is used. The opticalphonon energy, hO3op, is estimated to be 32 meV and

Sop 1.25 1013 is calculated using formulas similarto that in [6]. The calculated J-V characteristics of theRTD, at 77 K and 300 K, for cases with and without

the phonon scattering, are plotted in Fig. 3 with the

3.0

2.5 3O0 K

2.0 77 K

1.000 K

0.5

77 K

0.00.0 0.1 0.2 0.3 0.4 0.5

Voltage (V)

FIGURE 3 Calculated RTD J-V characteristics with and withoutphonon scattering. Non-self-consistent case

peak current density and peak-to-valley ratio veryclose to the experimental values at these temperaturesand that the valley current density increases with tem-

perature as expected. Note that for the results in

Fig. 3, self-consistency and electron transverse

momentum were not included. To obtain a goodagreement between calculated and experimental peakand valley voltages, as shown in Fig. 4, the self-con-sistency and electron transverse momentum are incor-

porated to take into account effects of space chargeand effective barrier lowering [5,7].

4 CONCLUSION

For realistic RTD I-V characteristics, inelastic phononscattering, self-consistency and electron transverse

momentum have been incorporated in the presentmodel to give good agreement between calculatedand experimental results. The calculated electron cur-

rent and wavefunction profiles illustrate the effect ofthe phonon scattering in the quantum well. This

approach is conceptually simple and computationallyefficient. On the other hand, as a number of approxi-

86 J.P. SUN and G.I. HADDAD

RTD with Phonon Scaffering

(self-consistent):3o0

:300 K

0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Voltage (V)FIGURE 4

mations is involved, systematic tests on variations ofdevice material and structural parameters are neededto assess the applicability of the present model. For

more microscopic details of RTD device physics, one

may have to invoke models based on quantum kineticformalisms or many-body theories.

AcknowledgementsThe authors wish to thank Dr. R. K. Mains, Dr.J. N. Schulman of Hughes Research Labs, and Dr.Y. Zohta of Tokyo Engineering University, for theirvery helpful discussions, and Dr. W. L. Chen for pro-viding the RTD’s experimental curve. This work wassupported by the ARO-URI program under contractNo. DAAL03-92-G-0109.

References1] A.D. Stone and P. A. Lee, Phys. Rev. Lett., 54, 1196 (1985).

[2] Y. Zohta and H. Ezawa J. AppL Phys., 72, 3584 (1992);Y. Zohta and T. Tanamoto, J. Appl. Phys., 74, 6996 (1993);J. Appl. Phys., 76, 2343 (1994).

[3] R.K. Mains, J. P. Sun and G. I. Haddad, Appl. Phys. Lett., 55,371 (1989).

[4] J. P. Sun, R. K. Mains, W. L Chen, J. R. East andG. I. Haddad, J. Appl. Phys., 72, 2340 (1992).

[5] J. P. Sun, R. K. Mains, K. Yang and G. I. Haddad, J. Appl.Phys., 74, 5053 (1993).

[6] M.A. Littlejohn, T. H. Glisson, and J. R. Hauser, in GalnAsPAlloy Semiconductors, T. P. Pearsall (ed.), ch.10., p. 243,Wiley, 1982.

[7] J.N. Schulman and H. J. De Los Santos, to be presented at1995 International Semiconductor Device Research Sympo-sium, Charlottesville, Virginia.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Documents

Self-Consistent Scattering Calculation Resonant T nneling Diode …downloads.hindawi.com/journals/vlsi/1998/078412.pdf · 2018. 11. 13. · 86 J.P. SUNandG.I. HADDAD RTDwith Phonon