Vol:.(1234567890)

Journal of Computational Electronics (2021) 20:230–236https://doi.org/10.1007/s10825-020-01612-4

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Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode at 300 K

Carl L. Gardner1

Received: 13 July 2020 / Accepted: 24 October 2020 / Published online: 9 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020

AbstractThe smooth quantum hydrodynamic model is an extension of the classical hydrodynamic model for semiconductor devices that can handle in a mathematically rigorous way the discontinuities in the potential energy that occur at heterojunction bar-riers in quantum semiconductor devices. Smooth quantum hydrodynamic model simulations of a GaAs resonant tunneling diode at 300 K with a 5-nm quantum well and 2.5-nm 280 meV double quantum barriers are presented that successfully produce realistic negative differential resistance and hysteresis in the current–voltage curve. This is the first quantum hydro-dynamic simulation of hysteresis in the resonant tunneling diode at 300 K, demonstrating that fluid dynamical concepts can model this fundamental quantum mechanical effect even at room temperature. A dispersive quantum term ℏ2

nuxxx∕(8m) in

the energy transport equation, where n is the electron density, u is the electron velocity, and m is the effective electron mass, is essential here in obtaining a realistic hysteresis loop in the current–voltage curve.

Keywords Resonant tunneling diode · Quantum hydrodynamics · Hysteresis

1 Introduction

Important quantum effects in semiconductor devices (reso-nant tunneling diodes [1], quantum field effect transistors [2, 3], high electron mobility transistors [4], etc.) can be efficiently simulated by means of quantum hydrodynamics. The two most useful quantum hydrodynamic approximations are the original quantum hydrodynamic (QHD) model (see [5] and references therein) and the smooth QHD model [6].

Reference [6] introduced the smooth QHD model emphasizing its derivation from a moment expansion of the Wigner–Boltzmann transport equation, using a smoothed-potential quantum Maxwellian [6] solution of the (thermal equilibrium) Bloch equation to close the first three moments. (See [7] for a review.) The original QHD model [5] used the thermal equilibrium Wigner–Maxwellian [8] to close the first three moments.

Neglecting the optional generalized heat flux � which is derived in the Chapman–Enskog expansion, the original and smooth QHD model equations contain quantum corrections

only in the stress tensor Pij and the energy density W. In the smooth QHD model, these quantum corrections are propor-tional to

where V is the quantum potential defined in equation (8) and V is the classical potential energy. Here, the thermal Planck constant ℏ� is defined by

(with the numerical values for GaAs), where � = 1∕T0 , T0 is the ambient temperature, m is the effective mass of electrons or holes, and Boltzmann’s constant kB has been set to 1.

Thus, the O(

ℏ2⋅ V

)

smooth QHD model, while linear in V, incorporates all orders of ℏ2 . To leading order in ℏ2 , the smooth QHD model reproduces the original O

(

ℏ2⋅ V

)

QHD model, since [6] approximating V ≈ V∕3 in the smooth QHD model Pij and W yields the original QHD model.

The original QHD model succeeded in simulating the effects of electron tunneling through potential barriers in reso-nant tunneling diodes at 77 K including single [5] and multiple

ℏ2�V = ℏ2

�

(

V

3+

2ℏ2�

15∇2V +⋯

)

ℏ2�≡ ℏ2�

4m=

ℏ2

4mT0≈

{

(3.42 nm)2 300 K

(6.75 nm)2 77 K

* Carl L. Gardner Carl.Gardner@asu.edu

1 School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ, USA

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regions [9] of negative differential resistance (NDR)—a region of the current–voltage curve where the current decreases as the voltage is increased—and hysteresis [10] in the current–volt-age curve, although the hysteresis loop was smaller than those experimentally measured. NDR is suppressed in the original QHD model at 300 K [7].

Smooth QHD model simulations of the resonant tun-neling diode were presented in [11] that exhibited enhanced NDR when compared to simulations using the original QHD model. At both 300 K and 77 K, the smooth QHD model correctly predicts significant NDR even when the original QHD model predicts no or little NDR. The smooth QHD model was shown in [1] to give excellent qualitative agree-ment with the current–voltage curves for the resonant tun-neling diode predicted by the industrial simulator NEMO, which solves the full mixed-state quantum mechanics with a nonequilibrium Green’s function technique.

In this investigation, we will demonstrate that the smooth QHD model successfully produces hysteresis in the resonant tunneling diode at room temperature, with a realistic hyster-esis loop. This is the first quantum hydrodynamic simulation of hysteresis at 300 K.

2 Smooth QHD model equations

The smooth QHD model [6] is an extension of the classical hydrodynamic (CHD) model for semiconductor devices that can handle in a mathematically rigorous way the discontinui-ties in the potential energy that occur at heterojunction bar-riers in quantum semiconductor devices. Quantum transport effects including electron or hole tunneling through potential barriers and charge buildup in quantum wells are integrated into the hydrodynamic description of charge propagation in semiconductor devices.

The smooth QHD equations have the same form as the classical hydrodynamic equations, expressing conservation of mass, momentum, and energy, respectively (for simplicity we just treat electrons here):

(1)�

�t(mn) +

�

�xi

(

mnui)

= 0

(2)

�

�t

(

mnuj)

+�

�xi

(

mnuiuj − Pij

)

=

− n�V

�xj−

mnuj

�p

(3)

�W

�t+

�

�xi

(

uiW − ujPij + qi)

=

− nui�V

�xi−

(

Wcl −3

2nT0

)

�w

where m is the effective electron mass, n is the electron den-sity, mn is the electron mass density, ui is the velocity, mnui is the momentum density, Pij is the stress tensor, W is the energy density, Wcl =

3

2nT +

1

2mnu2 is the classical energy

density, T is the temperature of the electron gas, and qi is the generalized heat flux. Indices i,  j equal 1, 2, 3, and repeated indices are summed over. Electron scattering is modeled by the standard relaxation time approximation, with momentum and energy relaxation times �p and �w.

The transport Eqs. (1)–(3) are coupled to Poisson’s equation for the electrostatic potential energy VP:

where � is the electrostatic potential, e > 0 is the electronic charge, � is the dielectric constant, ND is the density of donors, and NA is the density of acceptors. ( NA = 0 for the device simulated here.) The total classical potential energy V consists of two parts, one from Poisson’s equation VP and the other from the potential barriers VB:

VB has a step function discontinuity at potential barriers (Fig. 1).

Quantum effects enter into the expressions for the stress tensor Pij , energy density W, and generalized heat flux qi . The quantum contributions to the stress tensor and energy density are proportional to ℏ2

�V = O

(

ℏ2n�⋅ V

)

, n = 1, 2, 3, ...  The quantum contribution to qi is independent of tem-perature and is proportional to ℏ2.

The stress tensor and energy density are

(4)∇ ⋅

(

�∇VP

)

= e2(

ND − NA − n)

, VP = −e�

(5)V = VB + VP.

Fig. 1 Doping density ND (blue) in 1018 cm−3 and double barriers

VB∕B (cyan), where B is the barrier height, for the resonant tunneling

diode versus x in nm. The channel is 20 nm long, the barriers are 2.5 nm wide, and the quantum well between the barriers is 5 nm wide. There are 5-nm spacers between the barriers and the contacts. In Figs. 1, 2, 3 and 4, dots indicate gridpoint values for 240 Δx (Color figure online)

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where the quantum potential V is given by

Note the double smoothing over both space and (inverse) temperature.

In general, the quantum potential involves a smoothing of VB + VP , but since the contribution of VP to the quantum potential is always much less than the contribution of VB for the resonant tunneling diode simulations presented below, we consider just the smoothing of VB.

Only second spatial derivatives of V appear in Pij and W. For the resonant tunneling diode,

where B is the barrier height, � is the unit step function from 0 to 1, and x1 ( x3 ) and x2 ( x4 ) label the left and right bar-rier edges of the first (second) barrier. The x′ integration for d2VB∕dx

2 can be performed, yielding

The smooth barrier potential combination that appears in the momentum (2) and energy (3) transport equations

(6)Pij = −nT�ij − ℏ2�n

�2V

�xi�xj

(7)W =3

2nT +

1

2mnu2 +

1

2ℏ2�n∇2V

(8)

V(�, �) = ∫�

0

d��

�

(

��

�

)2

×

∫ d3x�(

2m�

�(� − ��)(� + ��)ℏ2

)3∕2

×

exp

{

−2m�

(� − ��)(� + ��)ℏ2(�� − �)2

}

V(��).

(9)VB = B(

�(x − x1) − �(x − x2) + �(x − x3) − �(x − x4))

(10)

d2VB

dx2= −

B√

2� ℏ3�

∫b

0

db b2

(1 − b2)3∕2×

�

(x − x1) exp

�

−(x − x1)

2

2ℏ2�(1 − b2)

�

−

(x − x2) exp

�

−(x − x2)

2

2ℏ2�(1 − b2)

�

+

(x − x3) exp

�

−(x − x3)

2

2ℏ2�(1 − b2)

�

−

(x − x4) exp

�

−(x − x4)

2

2ℏ2�(1 − b2)

��

.

(11)UB = VB + ℏ2�

d2VB

dx2

is presented in Fig. 2 for the double barriers at 300 K ana-lyzed in this investigation. Note that VB , VB , and UB are linear in the barrier height B. UB is two degrees smoother [6, 12] than the barrier potential VB ; i.e., if VB has a step function discontinuity, UB is once differentiable. (See dUB∕dx in Fig. 3 and ℏ2

�d2VB∕dx

2 in Fig. 4.) The effective barrier height in the smooth barrier potential UB is only 40% of the barrier height, enhancing electron transmission through the barrier.

The generalized heat f lux (derived through the Chapman–Enskog expansion for the moments of the Wigner–Boltzmann equation in [12], and earlier through an analysis [9] of the transport equations in terms of the Wigner distribution function)

includes both a classical and a quantum term and incor-porates important effects of the higher moments of the Wigner–Boltzmann equation that are omitted in the fluid dynamical approximation. The dispersive quantum term ℏ2

8m∇ ⋅

(

n∇2�)

in −∇ ⋅ � is essential in obtaining a realistic hysteresis loop in the current–voltage curve for the resonant

(12)� = −�∇T −ℏ2n

8m∇2

�

Fig. 2 Smooth barrier potential UB∕B for the resonant tunneling

diode at 300 K versus x in nm

Fig. 3 First derivative (

dUB∕dx

)

∕B of the smooth barrier potential for the resonant tunneling diode at 300 K versus x in nm

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tunneling diode. (This term also appears in the quantum drift–diffusion model [13].)

We model the relaxation times �p and �w by modified Bac-carani–Wordeman models [14]

and the coefficient � by

where �n0 is the low-field electron mobility, vs is the electron saturation velocity, and 𝜅0 > 0 is a dimensionless phenom-enological constant.

3 Steady‑state upwind method

For the resonant tunneling diode, we solve the 1D steady-state smooth QHD equations:

(13)�p = m�n0

T0

T≡ �p0

T0

T, �w =

�p

2

(

1 +

3

2T

1

2mv2

s

)

(14)𝜅 = 𝜅0𝜇n0nT0 ≡ �̄�n

(15)(mnu)x = 0

(16)(

mnu2 + nT)

x= −n

(

VP + UB

)

x− ℏ2

�nxVBxx −

mnu

�p

(17)

(

u(

1

2mnu2 +

5

2nT

))

x=

− nu(

VP + UB

)

x−

1

2mnu2 +

3

2n(T − T0)

𝜏w+

�̄�(

nTx)

x−

1

2�2𝛽nuVBxxx +

�2

8m

(

nuxx)

x

(18)�VPxx = e2(

ND − n)

using a damped Newton method with pseudo-arclength con-tinuation in the applied voltage Vbias across the device. (We have used (nu)x = 0 to simplify (17).)

The time-dependent 1D smooth QHD equations have two Schrödinger/parabolic modes (the heat conduction term and the dispersive quantum term are intertwined), one hyperbolic mode, and one elliptic (Poisson equation) mode [7]. The Schrödinger/parabolic modes are parabolic if �̄�2 > 3�2∕(4m2) and dispersive if �̄�2 < 3�2∕(4m2) . In Si and GaAs, �̄�2 ≫ 3�2∕(4m2) . Seven physically relevant, well-posed boundary conditions are

where xL and xR denote the left (inflow) and right (outflow) boundaries of the resonant tunneling diode.

The advection terms (ug)x are discretized with the con-servative first-order upwind method (competitive with sec-ond-order schemes)

where

while all non-advected derivatives are discretized with sec-ond-order accurate central differences.

4 Simulation of hysteresis

Smooth QHD simulations with 240 Δx of a GaAs resonant tunneling diode at 300 K with AlGaAs double barriers are presented in Figs. 5, 6, 7, 8 and 9. The barrier height is B = 280 meV. The diode consists of an n+ source (at the left) and drain (at the right), each with doping ND = 1018 cm−3 , and an n channel, with doping ND = 5 × 1015 cm−3 . The channel is 20 nm long, the barriers are 2.5 nm wide, and the quantum well between the barriers is 5 nm wide. There are 5-nm spacers between the barriers and the contacts.

Physical parameters for GaAs are electron effective mass m = 0.063me , saturation velocity vs = 1.5 × 107 cm/s, die-lectric constant � = 12.9 , and relaxation time �p0 = 0.06 ps. The heat flux pre-factor is set to the canonical value �0 = 2.5.

Figure 5 displays the experimental signals of quantum resonance: negative differential resistance and hysteresis in the current–voltage curve. Note that the smooth QHD simulation correctly predicts significant NDR and hyster-esis in the resonant tunneling diode at 300 K, with a more

(19)n(xL) = ND(xL), ux(xL,R) = 0, Tx(xL,R) = 0,

VP(xL) = 0, VP(xR) = −eVbias

(20)(

(ug)x)

i≈(

ui+

1

2

gi+

1

2

− ui−

1

2

gi−

1

2

)

∕Δx

gi+

1

2

=

{

gi (if ui+

1

2

> 0)

gi+1 (if ui+

1

2

< 0), g

i−1

2

=

{

gi−1 (if ui−

1

2

> 0)

gi (if ui−

1

2

< 0)

Fig. 4 Second derivative (

ℏ2

�d2VB∕dx2

)

∕B of the barrier quantum potential for the resonant tunneling diode at 300 K versus x in nm

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realistic hysteresis loop than the original QHD [10] or quantum drift–diffusion models [15, 16] predicted at 77 K.

The resonant peak of the current–voltage curve at 140 mV occurs as the electrons tunneling through the first barrier come into resonance with the energy levels of the quantum well—see Fig. 6 showing enhanced charge in the quantum well. As the voltage bias increases above 140 mV, the resonance effect rapidly decreases and then dis-appears because the effective right barrier height in U in Fig. 7 is progressively lowered.

The physical mechanism for hysteresis is that electrons “see” a slightly different smooth potential energy U as the applied voltage is increasing versus decreasing (Figs. 7, 8 and 9). When the applied voltage is increasing in the hysteresis loop, the effective quantum well is closer to a parabolic well, producing more resonance (the increased electron charge in the quantum well in Fig. 6) and thus amplifying current flow through the device.

Fig. 5 Current in kA/cm2 versus applied voltage in volts for the reso-nant tunneling diode at 300  K with a 5-nm quantum well and 2.5-nm 280 meV quantum barriers. The dots indicate where solutions are compared for increasing (red, relative maximum) versus decreasing (blue) voltage (Color figure online)

Fig. 6 Electron density n in 1018 cm−3 versus. x in nm in the resonant tunneling diode at the resonant peak voltage V

bias= 140 mV (red

increasing/blue decreasing voltage) (Color figure online)

Fig. 7 Close-up of the smooth potential U = VP+ U

B in eV for the

resonant tunneling diode at the resonant peak with Vbias

= 140 mV (increasing voltage) versus x in nm (Color figure online)

Fig. 8 Close-up of the smooth potential U = VP+ U

B in eV for the

resonant tunneling diode with Vbias

= 140 mV (decreasing voltage) versus x in nm (Color figure online)

Fig. 9 Close-up of the smooth potential U = VP+ U

B in eV for the

resonant tunneling diode at the resonant peak voltage Vbias

= 140 mV (red increasing/blue decreasing voltage) versus x in nm (Color figure online)

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5 Conclusion

This is the first quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode at 300 K, dem-onstrating that fluid dynamical concepts can model this fundamental quantum mechanical effect even at room temperature.

Four different versions of the smooth QHD model are considered in Fig. 10: (i) with and (ii) without the optional dispersive quantum term derived in the Chapman–Enskog expansion, and the semiclassical model (iii) with and (iv) without the optional dispersive quantum term. The semiclassical smooth QHD model is obtained by setting all quantum terms to zero except in the smooth poten-tial U = UB + UP . (This model is obtained from the CHD model by the transcription V → U.)

Tunneling and NDR are enhanced at low temperature. NDR is suppressed in the original QHD model at room temperature, but easily calculated in the smooth QHD model. Realistic hysteresis, however, needs the disper-sive quantum term and is more difficult to compute since it requires pseudo-arclength continuation in the applied voltage rather than just the usual simple continuation in applied voltage, which fails. All four versions of the smooth QHD model mentioned above satisfy conservation of mass, momentum, and energy.

The quantum terms −ℏ2�nxVBxx in the momentum trans-

port equation (16) and − 1

2ℏ2�nuVBxxx in the energy transport

equation (17) are essential to generating NDR, while the dispersive quantum term ℏ2

(

nuxx)

x∕(8m) in (17) is also

needed to obtain a realistic hysteresis loop (Fig. 10).To simulate quantum effects like charge buildup in

quantum wells, only the −n(

VP + UB

)

x and −nu

(

VP + UB

)

x

quantum terms in (16) and (17), respectively, are needed

(semiclassical model), but then NDR and hysteresis are not produced at 300 or 77 K.

Two adjustable models enter into the smooth QHD solu-tions and current–voltage curve. (i) In the generalized heat flux formula (12) for � , the classical term can be made more comprehensive (with, for example, a dependence on velocity � ), and the quantum term can be omitted or not; the value for �0 in (14) can be adjusted as well between 0 ≤ �0 ≤ 5 (say); (ii) the Baccarani–Wordeman models (13) for �p and �w can be further generalized.

Additional research into the generalized heat flux model and the relaxation time models needs to be done in order to match specific experimental current–voltage curves. The experimental curves will also depend on the coupling of the resonant tunneling diode to external circuits, which could be modeled by physically motivated boundary conditions.

Numerical methods have been developed by the author (to be submitted) for the time-dependent smooth QHD model for solving the gas dynamical part of the equations with a positivity-preserving third-order WENO method, treating the quantum mechanical terms as source terms; the parabolic heat conduction part using the TRBDF2 method; and the elliptic Poisson equation using Chebyshev SOR. Detailed comparisons will be given of the steady-state first-order conservative upwind solutions and the WENO3 upwinded solutions.

To summarize: the smooth QHD model provides an effi-cient way to incorporate quantum effects like resonant tun-neling, negative differential resistance, and hysteresis into semiconductor device simulation without having to solve the much more computationally expensive Wigner–Boltz-mann transport equation or the full mixed-state quantum mechanics.

Funding Not applicable

Availability of data and material Not applicable

Code availability Custom code

Compliance with ethical standards

Conflicts of interest Not applicable

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Fig. 10 Current in kA/cm2 versus applied voltage in volts for the resonant tunneling diode at 300 K with a 5-nm quantum well and 2.5-nm 280 meV quantum barriers for four versions of the smooth QHD model: with (cyan)/without (blue) the quantum dispersive term, and semiclassical model with (dotted red)/without (red) the quantum dis-persive term (Color figure online)

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