Josephson Junction

Consider superconducting metals I and II separated by a very thin insulating layer, such that that electron wave functions can overlap between the metals (Josephson junction). A battery V is connected across the junction to ensure an average charge neutrality (see Figure 1.1). This situation can be described by means of the coupled Schrodinger equations:

(i)

Here Ψ₁ and Ψ₂ are the probability amplitudes for an electron in I and II, U₂ and U₂ are the electric potential energies in I and II, K is the coupling constant due to the insulating layer, and KΨ_1­Ψ^*₂/Ψ^*₁ and  KΨ_2­Ψ^*₁/Ψ^*₂ describe the battery as a source of electrons.

a) Show that ρ₁ = |Ψ₁|² and ρ₂ = |Ψ₂|²  are constant in time.

Figure 1.1

b) Assuming ρ₁ = ρ₂ = ρ₀ (same metals) and expressing the probability amplitudes in the form

(ii)

find the differential equations for θ₁ and θ₂.

c) Show that the battery current

(iii)

oscillates, and find the frequency of these oscillations.

SOLUTION

a) Take the first of equations (i),

(1)

and its complex conjugate Ψ^*₁ and Ψ₁ multiply them by and respectively:

(2)

(3)

Subtracting (3) from (2) yields

(4)

Similarly, from the second of (i),

(5)

b) Substituting the solutions and  into (1), we obtain the expression for θ₁:

(6)

Taking (6), the analogous expression for θ₂ gives

(7)

(8)

Subtracting (7) from (8), we obtain

(9)

where δ = θ₂ – θ₁. So

(10)

where eV = U₁ – U₂.

c) The battery current

(11)

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

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