Teaching Nanomagnets New Tricks v / Igor Zutic R. Oszwaldowski, J. Pientka, J. Han, University at Buffalo, State University at New York A. Petukhov, South Dakota School Mines & Technology P. Stano, RIKEN Mean Field Theory: Friend and Foe in Nanostructures simple, inexpensive, but often misleading Magnetic Ordering T C Mean Field Fluctuations nanomagnet bulk magnet no phase transitions Temperature
Nanomagnetism in Quantum Dots? Dilute Magnetic Semiconductors: (II,Mn)VI, (III,Mn)V, carrier-mediated magnetism, controlling magnetism by changing number of carriers (photo-excitations, gating, ) T. Dietl and H. Ohno, Rev. Mod. Phys. 86, 187 (2014) Differences as Compared to Bulk? Controlling Magnetism at Fixed Number of Carriers Magnetic Order Enhanced by Heating Unexpected Magnetism in Closed-Shell Systems Small Systems: Mean-Field Theory?
Magnetism in Quantum Dots epitaxial or colloidal QDs Experiments: Many or 1 Mn in QDs Photoluminescence (PL) (Cd,Mn)Te/ZnTe QDs Mn T=5 K Large B eff ~ 100 T R. Beaulac et al., Science 325, 973 (2009) Early Theories: J. Fernandez-Rossier, L.Brey, PRL 93 (2004) A. O. Govorov, PRB 72 (2005) Al. L. Efros et al., PRL 87 (2001) fingerprint for Mn location & QD confinement Y. Leger et al., PRL 95, 047403 (2005) @ Grenoble
/ Model: N electrons, N m Mn-impurities (spin S) (II,Mn)VI Mn-isolectronic with group-ii elements carriers controlled by chemical doping or gate electrodes H = H e + H Mn - J ex S s. S d(r i R I ) i,i H e = S [ -1 i 2m* D 2 + U QD (r i ) + V ee (r i ) ] e-mn interaction S=5/2 U QD (r ) = V(x,y) + 1 2 m*w 2 z 2 weaker harmonic in-plane (x,y) confinement V ee (r i ) = e 2 e S 1 j r i - r j e-e interaction H Mn = S I,I J I,I S S. AFM Mn-Mn interaction R. Abolftah, A. Petukhov, I. Zutic, PRL 101, 207202 (2008) R. Abolftah, P. Hawrylak, I. Zutic, PRL 98, 207203 (2007) N m =10 6 10 6 10 7 # of configurations (Mean Field needed for Mn)
Piezomagnetism in QDs? R. Abolftah, A. Petukhov, I. Zutic PRL 101, 207202 (2008) T x T Gate Bias-Induced Confinement Deformation T Gates T T T T T Experiments (no Mn): Tarucha group, U. Tokyo G. Austing et al., PRB (1999) N=6 electrons Spin=0 d p s Interaction ~ s. S Spin > 0 Mn-spins Zero Mn-Magnetization Large Mn-Magnetization! Controlling Magnetism at Fixed Number of Carriers
Magnetic Polarons (MPs) Mn-spins (1) (2) known for 50+ years in the bulk but still not fully understood degennes, Phys. Rev. 1960 carrier spin MP formation 1 Energy 2 E MP - binding energy of MP MP formation -lower energy -> Red Shift in the optical data Reduced Dimensionality (Confinement) Higher MP Stability
Magnetic Polaron Formation (Type-I QDs) almost all experiments in Type-I QDs red shift in PL peak up to T ~ 30 K E MP < 20 mev (Zn,Mn)Se CdSe Type-I CB (Zn,Mn)Se VB Problems (low temperature effect): carriers (excitons) quickly recombine small overiap of Mn & carriers J. Seufert et al., Phys. Rev. Lett. 88, 027402 (2002)
Magnetic Polaron Formation (Type-II QDs) (Zn,Mn)Te Type-II CB ZnSe Desirable Properties: Large overlap between Mn and holes ZnSe Small overlap between electrons and holes (long recombination times) VB Ising Exchange Splitting (Holes): Max = x Mn N 0 = 1.05 ev N 0 S x Mn ~ 5-10% Mn, ~100 Mn experiments at U. Buffalo samples: W.-C. Chou (NTCU) I. R Sellers et al., PRB 82, 195320 (2010)
Unconventional Magnetic Polarons? ZnTe experiments @ Buffalo B. Barman et al., to appear in PRB (Zn,Mn)Te CB (Zn,Mn)Se VB E MP (mev) 30 B = 0 T 20 10 0-10 0 10 20 30 40 50 Temperature (K) Type-II E MP (mev) CB ZnSe VB 60 40 20 0 0 20 40 60 80 100 120 140 Temperature (K) B = 0 T Temperature Helps? 30 T = 7 K 60 T = 7 K E MP (mev) 20 10 0 E MP (mev) 40 20-10 0 1 2 3 4 Magnetic Field (T) 0 0 1 2 3 4 Magnetic Field (T)
Magnetic Order Enhanced by Heating? Model: QD WL ~! CB VB QD WL E CV ~! " 0 T chemical potential Wetting Layer (WL) µ Photo-created holes in the WL which is a particle reservoir with quasi-fermi Distribution of 2D hole gas A heavy hole gets trapped in the QD and interacts with Mn-spins (Ising Interaction due to strong spin-orbit coupling) J. Pientka, R. Oszwaldowski, A.G. Petukhov, J. E. Han, and I. Zutic PRB 86, 161403 (R) (2012)
QD States and Free Energy Zero Occupancy Single Occupancy Double Occupancy E 1 =0 E 2,3 = " ± E 4 =2" + U 2 " 2 e, U Confinement, Coulomb Energy Thermally Changing Occupancy Alters Mn-Ordering MP Gibbs Free Energy ( Hole, Mn ): G sys ( ) =F h ( )+G Mn ( ) F h ( ) =F 0 ( )+F 1 ( )+F 2 ( ) = max Max = x Mn N 0 S J. Pientka et al., PRB 86, 161403 (R) (2012)
Mean-Field Description Minimize G sys ( ) to obtain MF (the most probable ) Calculate the average exchange energy to obtain E MP E MP mev 30 20 10 b 0.75T C3 T C3 a c T C1 G sys Ξ a.u. p 4 10 9 cm 2 p 5 10 9 cm 2 1.25T C3 0.1 0.1 0.0 Ξ p 9 10 10 cm 2 0.75T C1 T C1 G sys Ξ a.u. 1.25T C1 0.5 0.5 0.0 Ξ p = hole densities Multiple T C Phase Transitions (b) 2 nd (c) 1 st Order R T C1 R T C2 0 0 5 10 15 20 25 R T C3 30 T K Reentrant MP Formation green curve: magnetic order enhanced by temperature T C3
Reentrant MP Reentrant MP Formation T<T R C1 Note: Hole Energy increases downwards m T R C1 apple T<T R C2 low-t single occupancy & Mn-alignment 1 st Transition due to decrease in E MP m T R C2 apple T<T R C3 2 nd Transition due to decrease in m m
Fluctuations Approach (FA) 6 a 35 28 b p 5 10 9 cm 2 MF p 5 10 9 cm 2 FA Magnetization Fluctuations E MP mev 4 2 21 14 7 p 9 10 10 cm 2 p 4 10 11 cm 2 FA FA 0 0 5 10 15 20 25 0 0 50 100 150 200 250 300 T K Reentrant MP Formation beyond Mean- Field J. Pientka et al., PRB 86, 161403 (R) (2012) Fluctuation- Dissipation Tm. hm 2 i hmi 2 = µ o k B T
1 3 Experimental Probe: PL Spectrum PL Spectrum: I total (w) = I 1->0 (w) + I 2->1 (w) a p =9.0 10 10 cm 2 2 1 10 T K 1 0 60 I tot a.u. 100 Reentrant MP Signature 190 40 1K 3K 10 K 60 K 100 K 190 K 30 20 10 0 10 20 30 X mev 1 0 20 b 0 I tot a.u. 20 filled blue circles overall PL peak X mev Nonmonotonic E MP (T) exp. @ U. Buffalo X(!) =~! E CV " nomonotonic T-dependence for I total u peak position (red & blue shifts) and for I total peak intensity, consistent with nonmonotonic E MP (T)
Magnetism in Closed-Shell QDs? Shell structure in Fermionic systems: atoms, nuclei, QDs, Closed-shell systems, like noble gases, known for stability and 0 total spin Wigner Theorem (Lieb & Mattis, Phys. Rev. 62) The ground state of any nonmagnetic 2-electron system is a spin singlet It seems no magnetic ordering in closed-shell QDs doped with Mn But: Pseudosinglet (0 total spin) can have lower energy! 2 different Bohr radii R. Oszwaldowski, I. Zutic, A. G. Petukhov, PRL 106, 177201 (2011)
Schematic Description of Two-Fermion States state orbital spin wavefunction Singlet (S) symmetric antisymmetric Triplet (T) antisymmetric symmetric Pseudosinglet (PS) not separable R. Oszwaldowski, I. Zutic, A. G. Petukhov, PRL 106, 177201 (2011)
R. Oszwaldowski et al., PRL 106, 177201 (2011) Proposal for Magnetic Bipolarons Competition! no MP (0 holes) MP (1 hole) 2 holes No Order? R. Oszwaldowski et al., PRB 86, 201408(R) (2012) Spontaneous Symmetry Breaking! PseudoSinglet Core-Halo
Spin Corral ρ PS,min 0 ρ PS,max Experimental realization? - self-assembled (Zn,Mn)Se/CdSe - colloidal QDs with radial segregation N. S. Norberg et al., JACS 40 13195 (2006) Ferromagnetic alignment in a closed-shell system - STM to assemble Mn atoms 1 by 1: C. F. Hirjibehedin et al., Science 312, 1021 (2006) D. Kitchen et al., Nature 442, 436 (2006) 2-carrier QD (analog of He) R. Oszwaldowski, I. Zutic, A. G. Petukhov, PRL 106, 177201 (2011)
Wigner Crystallization & Wigner Molecules correlations, Coulomb/Kinetic Energy ~ r s correlation driven quantum phase transition elusive in 3D, density too low (r s >100 ) E. Wigner, PR 46, 1002 (1934) Wigner Crystal observed in QDs (r s ~ 1) C. Ellenberger et al., PRL (2006) A. Singha et al., PRL 104 (2010) John Trail www.tcm.phy.cam.ac.uk Wigner Molecules: 2 electron QDs, classical limit V(x) V(x) Top View no Coulomb repulsion x x rotating dimers Wigner Molecules
Hartree-Fock Identifying Wigner Molecules? charge density Exact charge density not enough to identify Wigner molecules in nonmagnetic QDs Hartree-Fock conditional probability density g(x) pair correlation function Restricted HF Exact Exact x - radial coordinate D. Pfannkuche et al., PRB (1993) C. Ellenberger et al., PRL (2006)
R. Oszwaldowski et al., PRL 106, 177201 (2011) Magnetic Bipolarons (Revisited) Competition! no MP (0 holes) MP (1 hole) Spontaneous Symmetry Breaking! PseudoSinglet 2 holes No Order? R. Oszwaldowski et al., PRB 86, 201408(R) (2012) Core-Halo Which Configuration Wins? Spin Wigner Molecule
Magnetic Patterns and Correlation Effects 2 hole Stability Diagram: exact diagonalization T Triplet (T) vs Pseudosinglet (PS) x Mn (%) PS PS Violet Mn up Yellow Mn down spin density R eff (nm) spin-wigner molecule imprinting carrier correlations on Mn impurities constant Mn doping for 0 < r < R eff R. Oszwaldowski, P. Stano, A. Petukhov, I. Zutic, PRB 86, 201408(R) (2012)
Robustness/Detection of Spin Wigner Molecules? confinement w x,y = w 0 /(1 + d) 1/2 holes separated along the softer potential Detection: Interband Photoluminescence two emission lines: 2 to 1 (PseudoSinglet) and 1 to 0 (MP) QD occupancy
Conclusions & Future Directions Heat-Enhanced/Reentrant Magnetism Closed-Shell Magnetism Probing Correlation Effects With Mn-Impurities Detecting Fluctuating Nanomagnetic Patterns? Spin-Polarized STM? Talk R. Wiesendanger NV Centers? Talk N. Mizuochi L. Molenkamp Beyond mean field: Fluctuation Approach, Monte Carlo, Non-interacting vs Interacting Problem E ex E ex 2 polaron: rigid hole-mn coupling bipolaron: hole in a polarized medium that mediates magnetic interaction