Transport Properties of a Multichannel Kondo Dot in a Magnetic Field

Preliminary results of the overscreened Kondo model out of equilibrium in a magnetic field, presented at the march meeting of the Deutsche Physikalische Gesellschaft (DPG) in Berlin, 2012.

Results have been published in Physical Review B, http://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.134413.

Transcript

1. 27.03.2012 Dpg Berlin ‘12 Transport Properties of a
   Multichannel Kondo Dot
   in a Magnetic Field Preprint: http://arxiv.org/abs/1202.4558 Christoph B. M. Hörig and Dirk Schuricht Institute for Theory of Statistical Physics • RWTH Aachen • Germany 1
2. • isotropic spin-½ Kondo dot • fixed charge with S=½ on dot • K independent reservoirs at different chemical potentials • in magnetic field Model h0 2 S=1/2 +V 1 / 2 +V 2 / 2 +V K / 2 -V 1 / 2 -V 2 / 2 -V K / 2 J J J J J … … ! µi L µi R = Vi H = X i↵k ✏k c† i↵k ci↵k + h0 Sz + J0 2⌫0 X i↵↵0kk0 0 ~ S · ~ 0 c† i↵k ci↵0k0 0
3. • real-time renormalization group method in frequency space [H. Schoeller, Eur. Phys. J. Special Topics 168, 179 (2009)] • formally solve von Neumann equation • integrate out the reservoirs & resum self-energy insertions • calculate with RTRG-FS approach • solve quantum kinetic equation Method ˙ ⇢(t) = i [H, ⇢(t)] = iL⇢(t), L = [H, ·] 3 d dt ⇢S(t) + i LS⇢S(t) = i Z t t0 dt0 ⌃(t t0) ⇢S(t0) ⌃(z) ⇢S(t) = Trres ⇢(t) ) ⇢S(z) = i z LS ⌃(z) ⇢S(t0)
4. • real-time renormalization group method in frequency space • high energy cutoff • fixed point ➡ stops flow to strong coupling at • expansion around • Kondo temperature (scaling invariant) Scaling equation 4 J⇤ = 1 K ⇤ = 0 ⇤ ⇤ d d⇤ J = (J) = 2J2(1 KJ) 0.6 0.8 1 0 2 4 6 8 10 J(Λ) / J* Λ/ TK K= 5 Jc = J ⇤c = max {h, V } TK = ⇤ ✓ eJ J⇤ J ◆K/2 e 1/2J
5. • renormalized magnetic field, associated with g-factor • screening of dot spin results in renormalization of h0 ➡ increased compared to single-channel (first order) Results 5 • relaxation rates • appear naturally and prevent divergencies in logarithms ➡ decreased compared to single-channel (second order) 1 = ⇡  h + 1 2 |V h| 2 + V + h KJ2 c , 2 = ⇡ 2  V + h + 1 2 |V h| 1 + V + h KJ2 c h = (1 KJc)h0, g = 2 @h @h0 = 2(1 KJc) Jc ⇠ 1 K
6. • identical bias voltages Vi =V: • logarithmic enhancement + broadening in RTRG • elastic cotunneling for • inelastic cotunneling for Results: Differential Conductance 6 V V h h V > h V < h 0 0.2 0.4 0.6 0 1 2 3 G / G∆ V / h K= 20 PT RTRG
7. • identical bias voltages Vi =V: • logarithmic enhancement + broadening in RTRG • elastic cotunneling for • inelastic cotunneling for Results: Differential Conductance 6 V V h h V > h V < h 0 0.2 0.4 0.6 0 1 2 3 G / G∆ V / h K= 20 PT RTRG
8. • identical bias voltages Vi =V: • sharper features with increased channel number K • renormalized magnetic field h nearly channel independent • power-law behaviour only found for h=0 ➡ [A. Mitra and A. Rosch, PRL 106, 106402 (2011)] Results: Differential Conductance 7 0 0.2 0.4 0.6 0 0.5 1 1.5 2 2.5 3 3.5 Gi / G∆ V / TK K= 10 K= 20 h0 = 0 TK h0 = 1 TK h0 = 2 TK h0 = 3 TK 0 1 2 3 V / h PT RTRG Gi = 8 < : ⇡ 4 J 2 c ⇥ 1 + 2Jc L2 V h ⇤ for V < h ⇡J 2 c ⇥ 1 + 2⇡Jc L2 V h ⇤ for V > h L2 x = ln ⇤c p x 2 + 2 2 V = h
9. Results: Differential Conductance 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 G6... 10 G11... 15 G16... 20 0 1 2 3 4 5 Gi / G∆ Vi / h 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h • different bias voltages • inelastic cotunneling most prominent • additional feature from other reservoirs • coupling due to dot magnetization • e.g. black curve: @M @V own resonance: V = h other resonances: V = 2 h V = 3 h V = 4 h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 V, V/2, V/3, V/4 V V/4 V/3 V/2 for different leads:
10. Results: Differential Conductance 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 Gi / G∆ V / h h0 = 100 TK K= 20 G1... 5 G6... 10 G11... 15 G16... 20 0 1 2 3 4 5 Gi / G∆ Vi / h 1×10-3 2×10-3 3×10-3 0 1 2 3 4 5 6 7 8 ∂M / ∂V TK V / h h0 = 100 TK K= 20 0 -0.5 0 0 5 10 15 20 M V / h • different bias voltages • inelastic cotunneling most prominent • additional feature from other reservoirs • coupling due to dot magnetization • e.g. black curve: @M @V own resonance: V = h other resonances: V = 2 h V = 3 h V = 4 h V1…5 =V V6…10 =V/2 V11…15 =V/3 V16…20 =V/4 V, V/2, V/3, V/4 for different leads:
11. • derived an analytic solution of a spin-½ multichannel Kondo dot out of equilibrium up to • differential conductance shows typical features of inelastic cotunneling at voltages much smaller than the applied magnetic field • non-trivial coupling between conductance channels for different bias voltages Conclusions O(J3 c ln Jc) Preprint: http://arxiv.org/abs/1202.4558 9