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Interacting electrons in metal nanostructures
I N T E R A C T I N G E L E C T R O N S IN M E T A L N A N O S T R U C T U R E S
D.C. Ralph Physics Department, Cornell University, Ithaca, NY,, USA
In my lectures, I focused primarily on two topics: how interactions affect the "electrons-in-a-box" states in metal nanoparticles, and how a current of spinpolarized electrons interacting with a nanomagnet can apply a torque to the magnet by the direct transfer of spin angular momentum. The spectrum of discrete electronic states inside a nanoscale metal particle can be measured using electron tunneling if the particle is connected to electrical leads in a single-electron-transistor configuration [1,2]. Measurements show that the spectra even in simple metals like aluminum, copper, or gold cannot be described by simple models of free, non-interacting electrons. In fact, all of the different types of forces and interactions that affect the electrons inside a metal affect the discrete spectrum in different ways, so that these spectra can serve as a sensitive probe into the nature of the interactions [3]. The situation is closely analogous to the physics of electrons in semiconductor quantum dots, except that in metals a wider variety of different types of interactions can be explored, including those that give rise to correlated-electron states such as superconductivity and ferromagnetism. My lectures surveyed the consequences of a number of different types of interactions. For example, spin-orbit interactions in heavy metals can lead to a reduction of the g-factors for Zeeman spin splitting below the free-electron value of 2, pronounced spatial anisotropies in the value of individual g-factors as a function of the angle of the applied magnetic field, and avoided crossings between predominantly spin-up and spin-down energy levels [4-8]. When considering electron-electron interactions in metallic quantum dots, theory suggests that the form of the interactions takes a particularly simple form, the "Universal Hamiltonian" [9, 10]. Even relatively weak Coulomb repulsion, which produces exchange interactions with strength less than the Stoner criterion for bulk ferromagnetism, can be capable of producing electronic ground states with non-minimal spin values in quantum dots [11-13]. Attractive effective 599
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D. C. Ralph
electron-electron interactions in materials such as aluminum can lead to a superconducting ground state. Despite the fact that it is not possible to measure either a Meissner effect or a zero-resistance state in a nanoparticle, a pair-correlated superconducting state can still be detected through correlation-induced gaps in the electronic spectra that are sensitive to whether the particle contains an even or odd number of electrons and that are suppressed by spin pair breaking in an applied magnetic field [3]. Strong repulsive Coulomb interactions lead to ferromagnetic electronic states, with rather complicated behavior that is affected by magnetic anisotropy forces as well as the strong electron-electron interaction [ 14, 15]. Some progress in understanding the strongly-correlated electronic states inside a nanoscale ferromagnet has been achieved using both numerical simulations [16] and effective spin Hamiltonians [17-19]. Spin-transfer torques represent a mechanism for manipulating the orientation of the magnetic moments in small magnetic elements that does not involve a magnetic field. The effect originates from the fact that thin layers of ferromagnets can act as filters for spins, producing partially spin-polarized currents. If a spinpolarized current is incident on a ferromagnetic layer, with a spin-polarization angle that is not collinear with the magnetic moment of the layer, then in the filtering process the current can transfer spin-angular momentum to the layer, in this way applying a torque to the layer's magnetic moment [20-22]. This torque has been measured in multilayered ferromagnet/normal metal/ferromagnet samples in which a current flows perpendicular to the layers [23-25]. For samples with a sufficiently narrow diameter, less than about 0.25 microns, the spin-transfer effect provides much stronger torques per unit current than do current-generated magnetic fields. The response of a magnet to the spin-transfer torque can take two forms, depending on the magnitudes of the applied magnetic field and the current. For small fields, the spin-transfer effect can produce reversible switching, in which one sign of current drives two magnetic layers to an antiparallel configuration, and the reversed current can drive them back parallel [25]. This effect is of interest as a means to write information in non-volatile magnetic random access memory elements, more efficiently than is possible using magneticfield writing. The second type of magnetic dynamics that can be driven by the spin-transfer effect is steady-state precession- a DC current of spin-polarized electrons can excite a nanomagnet into a state of dynamical equilibrium in which the magnet precesses at GHz-scale frequencies that are tunable using either the current or by changing the value of an external magnetic field. These oscillations have been measured in the frequency domain [26,27] and also directly in the time domain [28], and they are under investigation for applications such as nanoscale oscillators and microwave sources.
I n t e r a c t i n g e l e c t r o n s in m e t a l n a n o s t r u c t u r e s
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
D. C. Ralph, C. T. Black and M. Tinkham, Phys. Rev. Lett. 74 (1995) 3241. D. C. Ralph, C. T. Black and M. Tinkham, Phys. Rev. Lett. 78 (1997) 4087. J. von Delft and D. C. Ralph, Phys. Rep. 345 (2001) 61. P. W. Brouwer, X. Waintal, and B. I. Halperin, Phys. Rev. Lett. 85 (2000) 369. K. A. Matveev, L. I. Glazman, and A. I. Larkin, Phys. Rev. Lett. 85 (2000) 2789. D. A. Gorokhov and P. W. Brouwer, Phys. Rev. B 69 (2004) 155417. J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 87 (2001) 266801. J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 89 (2002) 156802. I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B 62 (2000) 14886. I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep. 358 (2002) 309. P. W. Brouwer, Y. Oreg, and B. I. Halperin, Phys. Rev. B 60 (1999) R13977. H. U. Baranger, D. Ullmo, and L. I. Glazman, Phys. Rev. B 61 (2000) R2425. G. Usaj and H. U. Baranger, Phys. Rev. B 67 (2003) 121308. S. Gu6ron et al., Phys. Rev. Lett 83 (1999) 4148. M. M. Deshmukh et al., Phys. Rev. Lett. 87 (2001) 226801. A. Cehovin, C. M. Canali, and A. H. MacDonald, Phys. Rev. B 66 (2002) 094430. C. M. Canali and A. H. MacDonald, Phys. Rev. Lett. 85 (2000) 5623. C. M. Canali and A. H. MacDonald, Solid State Comm. 119 (2001) 253. S. Kleff et al., Phys. Rev. B 64 (2001) 220401. J. Slonczewski, J. Magn. Magn. Mater. 159 (1996) L1. L. Berger, Phys. Rev. B 54 (1996) 9353.
[22] [23] [24] [25] [26] [27] [28]
M. D. Stiles and A. Zangwill, Phys. Rev. B 66 (2002) 014407. M. Tsoi et al., Phys. Rev. Lett 80 (1998) 4281. E. B. Myers et al., Science 285 (1999) 867. J. A. Katine et al., Phys. Rev. Lett. 84 (2000) 3149. S. I. Kiselev et al., Nature 425 (2003) 380. W. H. Rippard et al., Phys. Rev. Lett. 92 (2004) 027201. I. N. Krivorotov et al., Science 307 (2005) 228.
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D.C. Ralph Physics Department, Cornell University, Ithaca, NY,, USA
In my lectures, I focused primarily on two topics: how interactions affect the "electrons-in-a-box" states in metal nanoparticles, and how a current of spinpolarized electrons interacting with a nanomagnet can apply a torque to the magnet by the direct transfer of spin angular momentum. The spectrum of discrete electronic states inside a nanoscale metal particle can be measured using electron tunneling if the particle is connected to electrical leads in a single-electron-transistor configuration [1,2]. Measurements show that the spectra even in simple metals like aluminum, copper, or gold cannot be described by simple models of free, non-interacting electrons. In fact, all of the different types of forces and interactions that affect the electrons inside a metal affect the discrete spectrum in different ways, so that these spectra can serve as a sensitive probe into the nature of the interactions [3]. The situation is closely analogous to the physics of electrons in semiconductor quantum dots, except that in metals a wider variety of different types of interactions can be explored, including those that give rise to correlated-electron states such as superconductivity and ferromagnetism. My lectures surveyed the consequences of a number of different types of interactions. For example, spin-orbit interactions in heavy metals can lead to a reduction of the g-factors for Zeeman spin splitting below the free-electron value of 2, pronounced spatial anisotropies in the value of individual g-factors as a function of the angle of the applied magnetic field, and avoided crossings between predominantly spin-up and spin-down energy levels [4-8]. When considering electron-electron interactions in metallic quantum dots, theory suggests that the form of the interactions takes a particularly simple form, the "Universal Hamiltonian" [9, 10]. Even relatively weak Coulomb repulsion, which produces exchange interactions with strength less than the Stoner criterion for bulk ferromagnetism, can be capable of producing electronic ground states with non-minimal spin values in quantum dots [11-13]. Attractive effective 599
600
D. C. Ralph
electron-electron interactions in materials such as aluminum can lead to a superconducting ground state. Despite the fact that it is not possible to measure either a Meissner effect or a zero-resistance state in a nanoparticle, a pair-correlated superconducting state can still be detected through correlation-induced gaps in the electronic spectra that are sensitive to whether the particle contains an even or odd number of electrons and that are suppressed by spin pair breaking in an applied magnetic field [3]. Strong repulsive Coulomb interactions lead to ferromagnetic electronic states, with rather complicated behavior that is affected by magnetic anisotropy forces as well as the strong electron-electron interaction [ 14, 15]. Some progress in understanding the strongly-correlated electronic states inside a nanoscale ferromagnet has been achieved using both numerical simulations [16] and effective spin Hamiltonians [17-19]. Spin-transfer torques represent a mechanism for manipulating the orientation of the magnetic moments in small magnetic elements that does not involve a magnetic field. The effect originates from the fact that thin layers of ferromagnets can act as filters for spins, producing partially spin-polarized currents. If a spinpolarized current is incident on a ferromagnetic layer, with a spin-polarization angle that is not collinear with the magnetic moment of the layer, then in the filtering process the current can transfer spin-angular momentum to the layer, in this way applying a torque to the layer's magnetic moment [20-22]. This torque has been measured in multilayered ferromagnet/normal metal/ferromagnet samples in which a current flows perpendicular to the layers [23-25]. For samples with a sufficiently narrow diameter, less than about 0.25 microns, the spin-transfer effect provides much stronger torques per unit current than do current-generated magnetic fields. The response of a magnet to the spin-transfer torque can take two forms, depending on the magnitudes of the applied magnetic field and the current. For small fields, the spin-transfer effect can produce reversible switching, in which one sign of current drives two magnetic layers to an antiparallel configuration, and the reversed current can drive them back parallel [25]. This effect is of interest as a means to write information in non-volatile magnetic random access memory elements, more efficiently than is possible using magneticfield writing. The second type of magnetic dynamics that can be driven by the spin-transfer effect is steady-state precession- a DC current of spin-polarized electrons can excite a nanomagnet into a state of dynamical equilibrium in which the magnet precesses at GHz-scale frequencies that are tunable using either the current or by changing the value of an external magnetic field. These oscillations have been measured in the frequency domain [26,27] and also directly in the time domain [28], and they are under investigation for applications such as nanoscale oscillators and microwave sources.
I n t e r a c t i n g e l e c t r o n s in m e t a l n a n o s t r u c t u r e s
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
D. C. Ralph, C. T. Black and M. Tinkham, Phys. Rev. Lett. 74 (1995) 3241. D. C. Ralph, C. T. Black and M. Tinkham, Phys. Rev. Lett. 78 (1997) 4087. J. von Delft and D. C. Ralph, Phys. Rep. 345 (2001) 61. P. W. Brouwer, X. Waintal, and B. I. Halperin, Phys. Rev. Lett. 85 (2000) 369. K. A. Matveev, L. I. Glazman, and A. I. Larkin, Phys. Rev. Lett. 85 (2000) 2789. D. A. Gorokhov and P. W. Brouwer, Phys. Rev. B 69 (2004) 155417. J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 87 (2001) 266801. J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 89 (2002) 156802. I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B 62 (2000) 14886. I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep. 358 (2002) 309. P. W. Brouwer, Y. Oreg, and B. I. Halperin, Phys. Rev. B 60 (1999) R13977. H. U. Baranger, D. Ullmo, and L. I. Glazman, Phys. Rev. B 61 (2000) R2425. G. Usaj and H. U. Baranger, Phys. Rev. B 67 (2003) 121308. S. Gu6ron et al., Phys. Rev. Lett 83 (1999) 4148. M. M. Deshmukh et al., Phys. Rev. Lett. 87 (2001) 226801. A. Cehovin, C. M. Canali, and A. H. MacDonald, Phys. Rev. B 66 (2002) 094430. C. M. Canali and A. H. MacDonald, Phys. Rev. Lett. 85 (2000) 5623. C. M. Canali and A. H. MacDonald, Solid State Comm. 119 (2001) 253. S. Kleff et al., Phys. Rev. B 64 (2001) 220401. J. Slonczewski, J. Magn. Magn. Mater. 159 (1996) L1. L. Berger, Phys. Rev. B 54 (1996) 9353.
[22] [23] [24] [25] [26] [27] [28]
M. D. Stiles and A. Zangwill, Phys. Rev. B 66 (2002) 014407. M. Tsoi et al., Phys. Rev. Lett 80 (1998) 4281. E. B. Myers et al., Science 285 (1999) 867. J. A. Katine et al., Phys. Rev. Lett. 84 (2000) 3149. S. I. Kiselev et al., Nature 425 (2003) 380. W. H. Rippard et al., Phys. Rev. Lett. 92 (2004) 027201. I. N. Krivorotov et al., Science 307 (2005) 228.
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