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Quantum transport in RTD: asymptotic models and simulations

• f Schrödinger–Poisson systems.

“The unreasonable effectiveness of semiclassical analysis.”

Francis Nier

Francis.Nier@univ-rennes1.fr

IRMAR, Univ. Rennes 1 Joint work with V. Bonnaillie and A. Faraj.

QUATRAIN, Rennes 19-05-08 – p.1

Model

Quantum wells in a (semiclassical) island Injection from the left (and right). Applied Bias B.

i

λ1

λ1

λ2

λ2

λ2

a c1 c2 b Λ

λ

g (k2

x

QUATRAIN, Rennes 19-05-08 – p.2

Model

H = −h2∆ + ˜ V0(x)−W h(x) + V h

NL(x)

−W h(x) = −

N

• i=1

wi x − ci h

• [H, ̺] = 0

̺

F .F .E.S.S (Landauer-Büttiker)

−∆V h

NL = n

V (a) = V (b) = 0

• n(x)ϕ(x) = Tr [̺ϕ]

h = heff after rescaling

QUATRAIN, Rennes 19-05-08 – p.2

Model

Limit h → 0 → finite dimensional system: “The phenomena are governed by a finite number of resonant states”

−

Asymptotical model Ei c c c c

i i i i+1 i+1 + −

c

+

QUATRAIN, Rennes 19-05-08 – p.2

Model

Limit h → 0 → finite dimensional system: Easy to solve numerically after some adaptations

h = heff ∼ 0.1 or 0.3

Imaginary parts of resonances e−

dAg h

∼ 0 OK

Comparison of tunnel effects: e− A(h)

h

≥

≤ e− B(h)

h

A(h = 0), B(h = 0), h = O(1).

Charge in a well ∼ δc?. Feynmann-Hellman interpolation when V h

NL small.

QUATRAIN, Rennes 19-05-08 – p.2

First results

Bonnaillie-Nier-Patel JCP-2006

Fast numerical simulation (about 1 minute to solve 100 nonlinear problems) − > allows to study the effect of changing the values of the size of the barriers, the donnor density. . .

QUATRAIN, Rennes 19-05-08 – p.3

First results

Bonnaillie-Nier-Patel JCP-2006

Fast numerical simulation (about 1 minute to solve 100 nonlinear problems) − > allows to study the effect of changing the values of the size of the barriers, the donnor density. . . Provides complete bifurcation diagrams and explains the possibility of hysteresis effects.

QUATRAIN, Rennes 19-05-08 – p.3

First results

Bonnaillie-Nier-Patel JCP-2006

Fast numerical simulation (about 1 minute to solve 100 nonlinear problems) − > allows to study the effect of changing the values of the size of the barriers, the donnor density. . . Provides complete bifurcation diagrams and explains the possibility of hysteresis effects. At first sight, good agreement with other numerical simulations (Pinaud for Ga-As and Kumar-Laux-Fischetti for Si-SiO2).

QUATRAIN, Rennes 19-05-08 – p.3

Comparison

Full Schrödinger-Poisson and asymptotic model tested with the same numerical data. Check that the asymptotic model helps to guess all the possible nonlinear solutions. Check that the solutions with interaction of resonances detected with the asymptotic model make sense. Dynamics: beating effect?

QUATRAIN, Rennes 19-05-08 – p.4

GaAs, 1 well, 1 resonance

I-V curve

1e+09 2e+09 3e+09 4e+09 5e+09 0.05 0.1 0.15 0.2 0.25 0.3 I (A m-2) B (eV) Diagramm current-voltage Current BNP Current AF Current AF retour Current AF aller

QUATRAIN, Rennes 19-05-08 – p.5

GaAs, 1 well, 1 resonance

Nonlinear potential

• 0.005

0.005 0.01 0.015 0.02 0.025 5.5e-08 6e-08 6.5e-08 7e-08 7.5e-08 8e-08 potential (eV) position (m) B=0.05 eV BNP potential AF potential

QUATRAIN, Rennes 19-05-08 – p.5

GaAs, 1 well, 1 resonance

Eres − V curve

• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 0.05 0.1 0.15 0.2 0.25 0.3 Energy (eV) Bias (eV) Energy on the well according to the bias E1 BNP E1 AF E1 AF retour E1 AF aller

QUATRAIN, Rennes 19-05-08 – p.5

Si-SiO2, 1 well, 2 resonances

I-V curve

2e+11 4e+11 6e+11 8e+11 1e+12 1.2e+12 1.4e+12 0.5 1 1.5 2 2.5 3 I (A m-2) B (eV) Diagramm current-voltage Current BNP Current AF Current AF retour Current AF aller

QUATRAIN, Rennes 19-05-08 – p.6

Si-SiO2, 1 well, 2 resonances

Nonlinear potential

• 0.001

0.001 0.002 0.003 0.004 0.005 0.006 0.007 9e-09 1e-08 1.1e-08 1.2e-08 1.3e-08 1.4e-08 1.5e-08 potential (eV) position (m) B=2.04 eV BNP potential AF potential

QUATRAIN, Rennes 19-05-08 – p.6

Si-SiO2, 1 well, 2 resonances

Eres − V curve

• 1.5
• 1
• 0.5

0.5 1 0.5 1 1.5 2 2.5 3 Energy (eV) Bias (eV) Energy on the well according to the bias E1 BNP E2 BNP E1 AF E2 AF E1 AF retour E2 AF retour E1 AF aller E2 AF aller QUATRAIN, Rennes 19-05-08 – p.6

GaAs-2 Wells-2 resonances

Possible interaction of resonances

2

c c

1

QUATRAIN, Rennes 19-05-08 – p.7

GaAs-2 Wells-2 resonances

Asymptotic model, Eres − V curve

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Energie (eV) DiffØrence de potentiel (eV) Energie dans le puits selon la diffØrence de potentiel Energie Energie

• En. crit.

—: Energy crossing. —: The resonant energies are equal.

QUATRAIN, Rennes 19-05-08 – p.7

GaAs-2 Wells-2 resonances

Is it realistic ? Does it occur in the S.P . model ?

QUATRAIN, Rennes 19-05-08 – p.7

GaAs-2 Wells-2 resonances

Is it realistic ? Does it occur in the S.P . model ? Crossing or E1 = E2

h>0

→ Avoided crossings

E2 E1 E2 E1 E1 = E2

OR ?

QUATRAIN, Rennes 19-05-08 – p.7

GaAs-2 Wells-2 resonances

Is it realistic ? Does it occur in the S.P . model ? Crossing or E1 = E2

h>0

→ Avoided crossings

E2 E1 E2 E1 E1 = E2

OR ?

Wells [5nm, 3.5nm] Barriers [5nm, 5nm, 6.5nm] Previous bifurcation diagram sensitive to a a few Angström !!!.

QUATRAIN, Rennes 19-05-08 – p.7

2 Wells - 2 Resonances, comparison

Continuation by increasing the bias

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Energy (eV) Bias (eV) Continuation by increasing the bias E1 AF aller E2 AF aller

The second case is realized.

QUATRAIN, Rennes 19-05-08 – p.8

2 Wells - 2 Resonances, comparison

Continuation by in- and de-creasing the bias

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Energy (eV) Bias (eV) Continuation by increasing and decreasing the bias E1 AF retour E2 AF retour E1 AF aller E2 AF aller

For some bias, both solutions coexist !!

QUATRAIN, Rennes 19-05-08 – p.8

2 Wells - 2 Resonances, comparison

Comparison of the bifurcation diagrams

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Energy (eV) Bias (eV) Energy on the well according to the bias E1 BNP E2 BNP E1 AF retour E2 AF retour E1 AF aller E2 AF aller

Remember the strong sensitivity to the data and all the ap- proximation process.

QUATRAIN, Rennes 19-05-08 – p.8

2 Wells - Beating effect

At time t = 0, the bias goes from 0V to 0.08V

5e+08 1e+09 1.5e+09 2e+09 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 Current density (Am-2) time (s) Current w.r.t time

Is it a (damped) beating effect ?

QUATRAIN, Rennes 19-05-08 – p.9

2 Wells - Beating effect

2e+15 4e+15 6e+15 8e+15 1e+16 1e-12 2e-12 3e-12 4e-12 5e-12 6e-12 Density (m-2) time (s) Charge densities in the first and second wells

And T = hphys

∆Egap up to 10%.

QUATRAIN, Rennes 19-05-08 – p.9

Conclusion

In spite of several “rough” approximations, the numerical results of the asymptotic model and the complete Schrödinger-Poisson model agree very well.

QUATRAIN, Rennes 19-05-08 – p.10

Conclusion

In spite of several “rough” approximations, the numerical results of the asymptotic model and the complete Schrödinger-Poisson model agree very well. All the semi-algebraic complexity of the bifurcation diagrams of the asymptotic model actually occurs in the Schrödinger-Poisson model.

QUATRAIN, Rennes 19-05-08 – p.10

Conclusion

In spite of several “rough” approximations, the numerical results of the asymptotic model and the complete Schrödinger-Poisson model agree very well. All the semi-algebraic complexity of the bifurcation diagrams of the asymptotic model actually occurs in the Schrödinger-Poisson model. The faithful and fast computations of the asymptotic model allow to design interesting test cases for the Schrödinger-Poisson problem.

QUATRAIN, Rennes 19-05-08 – p.10