Can a Non-Ideal Metal Ferromagnet Inject Spin Into a Semiconductor With 100% Efficiency Without a Tunnel Barrier?

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Can a Non-Ideal Metal Ferromagnet Inject Spin Into a Semiconductor With 100% Efficiency Without a Tunnel Barrier?
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   6   0   2   3   3   5  v   1   [  c  o  n   d  -  m  a   t .  m  e  s  -   h  a   l   l   ]   1   4   F  e   b   2   0   0   6  Can a non-ideal metal ferromagnet injectspin into a semiconductor with 100%efficiency without a tunnel barrier? J. Wan, M. CahayDepartment of Electrical and Computer Engineering and Computer ScienceUniversity of Cincinnati, Cincinnati, Ohio 45221S. BandyopadhyayDepartment of Electrical and Computer EngineeringVirginia Commonwealth University, Richmond, Virginia 23284 ABSTRACT Current understanding of spin injection tells us that a metal ferromagnet caninject spin into a semiconductor with 100% efficiency if either the ferromagnet is anideal half metal with 100% spin polarization, or there exists a suitable tunnel barrierat the interface. In this paper, we show that, at absolute zero temperature, 100% spininjection efficiency from a  non-ideal   metal ferromagnet into a semiconductor quantumwire can be reached at certain injection energies,  without   a tunnel barrier, providedthere is an axial magnetic field along the direction of current flow as well as a spinorbit interaction in the semiconductor. At these injection energies, spin is injected only   from the majority spin band of the ferromagnetic contact, resulting in 100%spin injection efficiency. This happens because of the presence of antiresonances inthe transmission coefficient of the minority spins when their incident energies coincidewith Zeeman energy states in the quantum wire. At absolute zero and below a criticalvalue of the axial magnetic field, there are two distinct Zeeman energy states andtherefore two injection energies at which ideal spin filtering is possible; above the1  critical magnetic field there is only one such injection energy. The spin injectionefficiency rapidly decreases as the temperature increases. The rate of decrease isslower when the magnetic field is above the critical value. The appropriate choice of semiconductor materials and structures necessary to maintain a large spin injectionefficiency at elevated temperatures is discussed.2  I. INTRODUCTION The problem of spin injection across a ferromagnetic/semiconductor (Fe/Sm) in-terface has received increasing attention over the last ten years with the advent of spintronics. Several recent experimental investigations have shown successful spin in- jection into semiconductor heterostructures from ferromagnetic metals using tunnelbarriers in the form of Schottky contacts [1, 2, 3], thin metal oxides [4, 5, 6], or AlAsbarriers [7]. A spin injection efficiency of about 70 % has been recently demonstratedusing a CoFe/MgO tunnel contact to a GaAs layer [6]. Simultaneously, several the-oretical models have been developed to understand the mechanisms controlling spininjection across specular and disordered Fe/Sm interfaces, some of which have beenbased on a simple Stoner model of the ferromagnetic contact [8, 9, 10] while othershave included the full electronic band structure of the contact [11, 12].It is currently believed that if the ferromagnet is metallic and there is no tunnelbarrier between the ferromagnet and the semiconductor, then the infamous “resis-tance mismatch” problem will preclude a high spin injection efficiency unless theferromagnet is an ideal half metal with 100 % spin polarization [9, 13]. Here we showthat 100 % spin injection efficiency is possible at absolute zero temperature from ametallic ferromagnet with less than 100 % spin polarization into a semiconductorquantum wire, in spite of a resistance mismatch and in spite of the absence of anytunnel barrier, as long as there is an axial magnetic field along the wire and a Rashbaspin orbit interaction [14] in the semiconductor. The 100% efficiency is obtained onlyfor certain injection energies. At temperatures  T >  0 K, thermal smearing will causethe energy-averaged spin injection efficiency to be much less than 100%. This prob-lem can be mitigated by injecting through a double barrier resonant tunneling diodewhose transmission peak is narrow and matched to the required injection energy. Thisapproach results in a nearly monochromatic injection even at elevated temperatures.Accordingly, a high injection efficiency can be maintained even for  T >  0 K.3  This paper is organized as follows. In Section 2, we calculate the spin-dependentcontact conductance between a non-ideal metallic ferromagnetic contact and a quasione-dimensional semiconducting wire formed using a combination of mesa etching andelectrostatic confinement as shown in Fig. 1. This problem is of relevance consideringthe many experimental efforts to realize Spin Field Effect Transistors [15]. 1 Spin dependent interface conductance The structure that we consider is shown in Fig. 1. The ferromagnetic contact thatinjects electrons into the quantum wire (which we call the source contact) is quasione-dimensional, but the extracting contact (which we call the drain contact) is atwo-dimensional electron gas to which the quantum wire opens up on the right. Thequestions we address are: in the linear mode of operation (i.e., for a small biasbetween the source and drain contact) how is the contact resistance of the Fe/Sminterface affected by the presence of the Rashba interaction due to the heterostructureinterface formed between materials I (narrow bandgap) and II (wide bandgap) in thesemiconducting channel? How is that conductance affected by the presence of amagnetic field applied in the direction of current flow? The magnetic field can beeither an external magnetic field or the stray magnetic field that exists in the vicinityof the Fe/Sm interface because of the magnetized source contact. In this paper, weconsider an external magnetic field which has uniform strength along the length of the wire.In ref. [16], we found that the energy dispersion relationships (E- k x ) for thelowest energy bands in the semiconducting channel have the general shape shown inFigure 2(a). As was shown in [16], close to the bottom of the lowest subband, theE- k x  relationship has a camel-back shape. Here we will show that this is strictly validfor a magnetic field below some critical value  B c . Above that threshold, the E- k x relationships are as shown in Fig. 2(b) where the camel-back feature of the bottom4  subband disappears.If the potential applied to the gate in Fig. 1 is changed, the potential step betweenthe bottom of the conduction band in the ferromagnetic layer and the bottom of theconduction band in the semiconductor quantum wire far from the interface (∆ E  c )will change. As a result, the Fermi level can be swept from below the energy level  E  1 (bottom of the lower subband) to above  E  3  (bottom of the upper subband).Depending on whether the magnetic field is below or above the threshold value B c , the following two possibilities will occur. Referring to Fig. 2(a) ( B  < B c ), if we start with a value of ∆ E  c  such that  E  F   is above  E  3 , there will be initially twopropagating modes in the semiconductor channel. When  E  F   is in the range [ E  2 ,E  3 ],the upper band will become an evanescent channel with a wavenumber which is equalto zero when  E  F   is exactly equal to  E  3  or  E  2 , whereas the lower subband is stillconducting. In the energy range [ E  1 ,E  2 ], the two channels are conducting again andthere should be a rise in conductance until  E  F   reaches  E  1 . At that point, wavevectorscorresponding to the Fermi energy in both subbands will be real, equal and  finite  .With a slight increase in ∆ E  c , the Fermi level will fall below  E  1  and both subbandswill be evanescent. We therefore expect a sharp drop in the conductance as the Fermilevel falls below  E  1  and the semiconducting channel is completely pinched-off.For the case where the magnetic field is above the critical value  B c , the interfaceconductance should show a kink as the Fermi level is swept from above to below thethreshold energy  E  3 . Since the upper mode stays evanescent after  E  F   drops from  E  3 to  E  1  and the lower mode has a propagating wavevector which gradually shrinks tozero as  E  F   approaches  E  1 , the conductance of the interface should smoothly approachzero as the channel is being pinched-off, contrary to the previous case.The above discussion pertains to temperature  T   = 0 K. At elevated temperatures,the effect of thermal averaging will smooth out any of the abrupt features in theconductance versus gate voltage characteristics discussed above. At finite temperature5
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