LandauerBüttiker approach for hyperfine mediated electronic transport in the integer quantum Hall regime
Abstract
The interplay of spinpolarized electronic edge states with the dynamics of the host nuclei in quantum Hall systems presents rich and nontrivial transport physics. Here, we develop a LandauerBüttiker approach to understand various experimental features observed in the integer quantum Hall set ups featuring quantum point contacts. The approach developed here entails a phenomenological description of spin resolved interedge scattering induced via hyperfine assisted electronnuclear spin flipflop processes. A selfconsistent simulation framework between the nuclear spin dynamics and edge state electronic transport is presented in order to gain crucial insights into the dynamic nuclear polarization effects on electronic transport and in turn the electronspin polarization effects on the nuclear spin dynamics. In particular, we show that the hysteresis noted experimentally in the conductancevoltage trace as well as in the resistively detected /media/arxiv_projects/145801/NMR lineshape results from a lack of quasiequilibrium between electronic transport and nuclear polarization evolution. In addition, we present circuit models to emulate such hyperfine mediated transport effects to further facilitate a clear understanding of the electronic transport processes occurring around the quantum point contact. Finally, we extend our model to account for the effects of quadrupolar splitting of nuclear levels and also depict the electronic transport signatures that arise from single and multiphoton processes.
I Introduction
Nuclear spintronics concerns the manipulation of nuclear spins by means of hyperfine interaction between the host nuclei and the itinerant electrons and their read out using electronic transport Hirayama et al. [2009] or optical Urbaszek et al. [2013] measurements. Quantum Hall geometries in both the integer Wald et al. [1994]; Machida et al. [2002, 2003]; Würtz et al. [2005]; Ren et al. [2010]; Dean et al. [2009]; Song and Omling [2000]; Kawamura et al. [2007]; Keane et al. [2011a]; Hirayama et al. [2009] and the fractional regime Hashimoto et al. [2002]; Kou et al. [2010]; Akiba et al. [2011, 2013]; Hatano et al. [2015] featuring gated quantum point contacts (QPC) offer a viable method for controlling the spin polarization of the electronic edge channels. This in turn facilitates the manipulation of the nuclear spins via a hyperfine mediated interplay between the spinpolarized edge states and the dynamics of the host nuclei. Such an interplay has revealed rich and nontrivial transport physics in the form of hysteresis in the observed conductancevoltage traces and nontrivial lineshapes in the resistively detected /media/arxiv_projects/145801/NMR (RD/media/arxiv_projects/145801/NMR) traces Wald et al. [1994]; Machida et al. [2002, 2003]; Kawamura et al. [2007, 2010]. Despite several advancements in the transport experiments involving such set ups, theoretical models for hyperfine interaction mediated edge transport through the QPC in the Hall geometry are clearly missing in the current literature. The object of this work is hence to develop transport models that couple the dynamics of the host nuclei with edge channel electronic transport as an attempt to fill this gap and theoretically interpret various experiments with specific focus on the conductancevoltage traces Wald et al. [1994] and the RD/media/arxiv_projects/145801/NMR lineshapes Gervais [2009]; Bowers et al. [2010]; Keane et al. [2011a]; Tiemann et al. [2014]; Yang et al. [2011]; Kodera et al. [2006]; Tracy et al. [2006]; Desrat et al. [2002]; Gervais et al. [2005] .
We develop our transport models based on a modified LandauerBüttiker formalism that includes a spinflip transmission coefficient, which is nuclear polarization dependent and describes the rate of electronnuclear spin flipflops per unit energy around the QPC region. Using this approach, we show that the hysteresis noted in both the conductance and the RD/media/arxiv_projects/145801/NMR traces Wald et al. [1994]; Gervais [2009]; Bowers et al. [2010]; Keane et al. [2011a]; Tiemann et al. [2014]; Yang et al. [2011]; Kodera et al. [2006]; Tracy et al. [2006]; Desrat et al. [2002]; Gervais et al. [2005] results from a lack of steady state between electronic transport and nuclear polarization evolution and can be explained by taking into account the finite rate of electronnuclear spin flipflops in a source limited channel in addition to a finite nuclear spinlattice relaxation time. The selfconsistent simulation framework between the nuclear spin dynamics and the edge state electronic transport developed here offers crucial insights into the dynamic nuclear polarization effects on electronic transport and in turn the electronspin polarization effects on the nuclear spin dynamics. In addition, we present circuit models to emulate such hyperfine mediated transport effects for a clear understanding of the phenomena occurring near the QPC. Finally, we also address the effects of quadrupolar splitting of the nuclear levels and depict the electronic transport signatures that arise from single and multiphoton absorption processes Kawamura et al. [2010].
This paper is organized as follows. In Sec. II, we briefly detail the experimental set up and features that form our current focus after which we spell out the generic formalism. Specifically, in Sec. II.2, a phenomenological model for hyperfine mediated transport through the QPC is developed in detail. Section III elucidates the results from the simulation framework developed with the specific focus on explaining the various experimental trends noted. Specifically, Sec. III.1 is devoted to the understanding of the hysteritic conductance voltage traces noted for different filling factors and Sec. III.2 deals with the RD/media/arxiv_projects/145801/NMR lineshape features in great detail.
Ii Experimental details and theoretical description
In the schematic of the experimental set up shown in Fig. 1(a), an appropriately gated single QPC is utilized to selectively filter out a single spin channel into the region beyond the QPC thereby creating an imbalance between the upspin channel and the downspin channel. The principal experimental signature here is the change in conductance with voltage sweep near Wald et al. [1994] as shown in Fig. 1 (b) as well as the change in Hall resistance with RF frequency sweep (also known as resistively detected /media/arxiv_projects/145801/NMR or RD/media/arxiv_projects/145801/NMR) Keane et al. [2011a] as shown in Fig. 1 (c). Along with the change in the conductance, another feature which has attracted significant attention is the hysteresis in the conductance plots during forward and reverse voltage or RF frequency sweep as shown in Fig. 1 (b) and (c) respectively. A compact theoretical model to elaborate the physics of such a conductance modulation as well as hysteresis occuring in the gated QPC set up forms the primary focus of this work.
An accurate modeling of such phenomena involves taking into account the details of wavefunction correlations via the density matrix approach Zaletel et al. [2015]. Mathematical modeling of such hyperfine mediated electron transport process selfconsistently with evolution of the nuclear polarization from the density matrix formalism Zaletel et al. [2015] is complicated and computationally heavy. In this paper, we thus adapt a computationally efficient phenomenological model to account for such hyperfine mediated electronic transport through the QPC.
We now provide a theoretical description of the nuclear spin dynamics coupled to the electronic transport following which we focus on how to apply this to our specific set up. We begin with the description of the nuclear spin dynamics by formulating a master equation in the nuclear spin space followed by the description of the extended LandauerBüttiker formalism for the edge state electronic transport.
ii.1 Description of scattering processes
In order to describe the electronnuclear hyperfine interaction, we start with the Fermi contact hyperfine interaction Hamiltonian for the case with nonvarying electronic density of states in space, given by Urbaszek et al. [2013]
(1) 
where represents the electron wavefunction at the point , with representing the effective electron density per unit volume at the point , is the effective hyperfine coupling constant, are the operators representing the component of the electronic spin and the nuclear spin respectively, with representing a unit cell volume. The operator represents the tensor product between the electron spin and the nuclear spin spaces. The operators, and are respectively the corresponding spin raising (lowering) operators for the electron and the nuclear spins respectively. The above equation assumes , where ’’ and ’’ represents the eigen states in the electron spinspace. In the quantum Hall regime, however, the eigen states are localized in space along the transverse direction. In this case, and thus the Hamiltonian should be recast in the form Sakurai and Napolitano [2011a, b]; Griffiths [2005]:
where and with and belonging to the electron spinspace and and belonging to the nuclearspin space in the presence of a magnetic field pointing along the direction. When the coupling constant is small, the effect of the two terms in (1) can be separated. The first term results in an effective magnetic field and the second term within the curly brackets represents the electronnuclear spin flipflop processes. The first term in (1), which also corresponds to and in (LABEL:eq:hyperfine_ham) introduces an additional shift in the electronic energy levels as well as between states of different nuclear spins. The electronic energy difference between the upspin channel electrons and the downspin channel electrons is given by Slichter [1990]
(3) 
where , and are the effective Lande factor of the electron in GaAs, the Bohr magneton and the applied magnetic field respectively. The above expression is obtained by assuming an isotropic nuclear spin distribution. Similarly, the energy difference between the adjacent nuclear spin states differing by a magnetic quantum number of is given by
(4) 
where is the effective Lande factor of the nuclide in consideration and is the nuclear Bohr magneton. The effective coupling constant Das Sarma et al. [2003]; Slichter [1990]; Urbaszek et al. [2013], where is the electronic carrier density. The magnetic quantum number for the GaAs nuclei varies from to in steps of . In standard literature, the second terms in (3) and (4) represent the Overhauser shift and the Knight shift respectively. However, for practical purposes, and and hence may be neglected with respect to and the nuclear spin flips may be considered elastic.
The electronnuclear spin flipflop processes are described by the second term in the Hamiltonian in (1), which also corresponds to and in (LABEL:eq:hyperfine_ham) and the scattering rates are evaluated via the Fermi’s golden rule Buddhiraju and Muralidharan [2014]; Siddiqui et al. [2010], typically related to the densities of the initial and the final states. In this case, the electronic wavefunction distribution and overlaps with the nuclear wavefunction on each site differently, and this effect is accounted for via the overlap terms for up to down or down to up electronic spin transitions. The procedure for a selfconsistent description of electronic transport coupled to hyperfine spin dynamics then entails the timedependent simulation of the nuclear spin dynamics, with the electronic transport processes in steady state. This is because the nuclear spin dynamics are typically slow due to slow relaxation rates, slow diffusion rates as well as longer flipflop times in comparison with the electronic transport velocities. The nuclear spin dynamics at each point are dictated via the electronnuclear hyperfine flip rates calculated from the Fermi’s golden rule Buddhiraju and Muralidharan [2014] given by:
where () represents the up to down (down to up) nuclear spin transition rate at the nuclear coordinate between magnetic quantum numbers that differ by () in the nuclear spin space due to flipflop transitions of electrons at energy . The quantities and ( and ) denote the densities of filled (vacant) states per unit energy per unit area at the point . \colorblack The up to down (down to up) electronic spin transition rate based on (LABEL:eq:scattering_rates) depends not only on the the availability of electrons in the up (down) spin density of states and the vacancy in the down (up) spin density of states , but also on the spatial overlap of the corresponding density of states. The total rate of electronnuclear spin flipflop now depends on the integral of () over the spatial coordinates.
The spatial dynamics of the nuclear spins can be described by the following master equation:
where is the probability column vector representing the probability of occupancy of the nuclear spin levels, and is a phenomenological nuclear spin relaxation time, which is typically a very slow process. The matrix takes into account the transition between the individual nuclear spin levels. The vector denotes the probability of occupation of the nuclear spin levels in equilibrium. The above equation also includes nuclear spin diffusion described by the last term, where is the phenomenological diffusion constant. In this paper, we neglect the exact spatial distribution of nuclear spins due to diffusion and approximate the effects of nuclear spin diffusion by incorporating a larger number of nuclei. The equation governing the dynamics of the nuclear spins is then given by:
(8) 
The transition probability matrix may be specifically cast for the spin case in the current study in terms of the spinflip rates defined in (LABEL:eq:scattering_rates2) as:
where and are defined in (LABEL:eq:scattering_rates2). An additional constraint used to solve (8) using (LABEL:eq:matrix) is that of the normalization of the nuclear state probabilities, i.e., , where is the occupation probability of the nuclear density of states with spin .
The temporal evolution of the average electronic polarization and the average nuclear polarization at the QPC are calculated selfconsistently by solving (8) and (3) via the relations:
(10) 
where are obtained by solving the master equations. The matrix comprises the row vector of the spin magnetic quantum numbers of the GaAs nuclei. The procedure for transport calculations follows solving (3), (8), and (10) sequentially in a selfconsistent loop with the electronic transport to be described now.
ii.2 Electronic edgestate transport in the QPC region
While the dynamics of the nuclear spins simply follow the master equation (8) described above, a description of electronic transport involves transport currents due to the source and drain reservoirs held at electrochemical potentials and respectively. From a LandauerBüttikker perspective, a consistent description of transport currents in our case demands the use of both a) direct transmission and b) spinflip transmission. The need to include spin flip transmission follows from the interaction between the edge channels of different spins that gives rise to nuclear polarization which in close proximity of the QPC region determines the electronic transport. Near the QPC, the forward propagating edge channels and the backward propagating edge channels come in close proximity and hence spinflip scattering can occur to the forward propagating as well as to the backward propagating edge channels Wald et al. [1994].
The Landauer direct transmission denotes the tunneling probability of the up (down)spin electrons through the QPC. We model the spinsplit edge states in the device by a continuum of density of states as in a ballistic conductor Salahuddin [2006]; Kawamura et al. [2015]; Wan et al. [2009]; Côté and Simoneau [2016] with the region of the QPC being represented by a Gaussian potential barrier, as shown in Fig. 2 (a) and (b) along with the model used for simulation of electronic transport shown in Fig. 2 (c). The direct transmission coefficients and are then calculated using the nonequilibrium Green’s function (NEGF) method applied to the barrier described above using a atomistic tightbinding Hamiltonian Datta [1997, 2005]. The pertinent details of the approach used here have been briefly discussed in Appendix B.
In our scheme, we only consider the number of transmitted modes and the filling factor at the QPC for the electronic transport which is related to the geometry of the set up that is ascertained apriori. Near the vicinity of at the QPC, the downspin edge channel at the QPC is almost empty in the energy range between and , resulting in a considerable simplification of the transport equations.
We begin with the case where the filling factor , i.e., , where only the upspin edge channel originating from the source contact contributes to the total current terminating in the drain contact. The electrons in the forward propagating upspin edge channel originating from the source contact can tunnel through the QPC to the upspin edge channel terminating in the drain contact with a probability while the forward propagating downspin edge channel originating in the source contact is completely disconnected from the forward propagating downspin edge channel terminating in the drain contact, as depicted in Fig. 2(a) and (b) respectively.
A few upspin electrons at the QPC in the forward propagating edge channel terminating in the drain contact can however scatter to the forward propagating downspin edge channel terminating in the drain contact with a spinflip process as shown in Fig. 3 (a). This gives rise to the spinflip scattering current , where the superscript ’’ denotes the flow of current due to spinflip scattering at the QPC and the subscript denotes the current flow from the up (down)spin to down (up)spin edge channel via electronic spinflips. Assuming that the direct transmission coefficients ( and ) depend on the nuclear polarization only via the Overhauser field (3), for a system with four nuclear spin levels, the spinflip transmission coefficient at the QPC from the forward propagating upspin channel terminating in the drain contact to the forward propagating downspin channel terminating in the drain contact is given by (details given in Appendix A):
Note that depends on the spatial overlap of the density of states of the upspin and downspin edge channel at the QPC between the energy range and (details given in Appendix A, Eq. 32 and 37). We approximate as a constant. Therefore, the downspin current recorded just outside the QPC relies entirely on such spinflip processes and hence is simply the spinflip current while the upspin current in the edge channel just outside the QPC is reduced by . Based on the above discussions, the up and down spin channel currents are given by
(11) 
The subscripts and the source and drain contacts respectively, with denoting the FermiDirac distribution in the source (drain) contact held in quasiequilibrium at . The parameter takes into account the spinflip scattering of electrons at and around the QPC from the forward propagating upspin edge channel terminating in the drain contact to the forward propagating downspin channel terminating in the drain contact, with the superscript denoting spinflip scattering to a forward propagating channel. It must be noted that the edge channels in the quantum Hall arrangement are unidirectional and hence the expressions for the current in (11) depend on the factors and only and not on the factors and as expected in a typical Landauer type scattering treatment.
Turning our attention to the case when the filling factor , i.e., , the downspin electrons in the edge channel originating in the source contact are partially transmitted through the QPC to the downspin edge channel terminating in the drain contact, as depicted in Fig. 3(b). In this case, the spinflip scattering at and around the QPC can occur from the forward propagating upspin channel terminating in the drain contact to the forward propagating downspin channel terminating in the drain contact as well as from the forward propagating downspin channel terminating in the drain contact to the backward propagating upspin channel originating from the drain contact. Again, assuming that the direct transmission coefficients and depend on the nuclear polarization only via the Overhauser field (3), the spinflip currents in this case are given by (details in Appendix A)
where superscript denote spinflip scattering to a backward propagating edge channel while the superscript has the same meaning as described previously. The spinflip current flows from the forward propagating upspin channel terminating in the drain contact to the forward propagating downspin edge channel terminating in the drain contact while the spinflip current flows from the forward propagating downspin edge channel terminating in the drain contact to the backward propagating upspin channel originating in the drain contact. Hence, causes a change in the total output current since the spinflip scattering occurs to a backward propagating edge channel. It however does play a role in the nuclei polarization near the QPC. The current in the upspin and downspin channel terminating in the drain contact just outside the QPC is then given by:
(12) 
(13) 
From the above discussion, the generalized equations for the upspin, downspin and spinflip currents through the QPC are given by:
(14) 
where and are the direct transmission coefficients between the forward propagating edge channels originating and terminating in the source and drain contacts respectively through the QPC in the absence of electronnuclear spin flipflop scattering. As already discussed, the term and characterize spinflip scattering from a forward propagating edge channel terminating in the drain contact to a forward propagating edge channel terminating in the drain contact at the QPC, while and characterize spinflip scattering at the QPC from a forward propagating edge state terminating in the drain contact to a backward propagating edge channel originating in the drain contact at the QPC. The terms , , and , being the probability of electronnuclear spin flipflop processes, are dependent on the nuclear polarization (details given in Appendix A).
The spinflip currents and give rise to nuclear polarization at and around the QPC region.
Turning our attention to the selfconsistent solution of the electronic transport and the temporal evolution of the nuclear polarization, the electronic transport is influenced by the nuclear polarization via the Overhauser field while the evolution of nuclear polarization is determined by the spinflip current and the nuclear spinlattice relaxation time (). The matrix , which determines the temporal evolution in nuclear polarization is hence related to the spinflip currents and . A schematic diagram on selfconsistency involved in the temporal evolution of nuclear polarization and electronic transport phenomena is shown in Fig. 4.
For a system with quad nuclear spin levels as in GaAs, it can be shown that and with and (details given in Appendix C), being the number of nuclei that are being influenced by spin flipflop processes at the QPC. We can hence rewrite the expression for as (details given in Appendix C):
(15) 
Let us now consider the experimental features on a case by case basis.
Iii Results
iii.1 Conductance hysteresis with voltage sweep
We first reproduce some trends noted in the conductance plots of a recent experiment Wald et al. [1994] where a change in the conductance along with hysteresis in the conductance was noted in the vicinity of with positive and negative source to drain voltage sweep. We explain the possible phenomena giving rise to such experimental trends.
Case I:
A schematic of the scattering processes in this regime is shown in Fig. 3(a) while the upspin and downspin edge current paths and equivalent circuit models for the phenomena occuring around the QPC are shown in Fig. 5(a) and (b) respectively. In this case, the following points are to be noted:

Only the upspin channel is transmitted through the QPC.

The downspin channel originating in the source contact is totally reflected at the QPC.

Some upspin electrons in the edge channel terminating in the drain contact can scatter at the QPC to the downspin edge channel terminating in the drain contact via electronnuclear spin flipflop scattering. Such a scattering decreases the current in the upspin channel just outside the QPC and increases the current in the downspin edge channel outside the QPC. However, the total current remains proportional to .
Reason for an increase in near .

Near V=0, the nuclear polarization cannot be maintained.

Nuclear polarization drops due to spin lattice relaxation.

A drop in nuclear polarization results in an increase in the direct transmission coefficient of the upspin channel due to a decrease in the Overhauser field as well as an increase in the spinflip transmission coefficient .
Equivalent circuit model:
A schematic of the edge channel path in this case is shown in Fig. 5 (a) while the equivalent circuit in this case is detailed in Fig. 5 (b) to aid a visualization of the various transport phenomena inside the device. The circuit model can be described as follows:

The upspin edge channel is represented by a conductance .

The uptodown spinflip scattering can be represented by an equivalent current source from the upspin channel. .

The change in transmissivity of the upspin channel due to the Overhauser field is represented by a by an equivalent conductor in series with a voltage dependent voltage source . The current change due to the Overhauser field is represented by .

The nuclear polarization is represented by the voltage across the capacitor.

The resistance in parallel with the capacitor represents nuclear spinlattice relaxation.
Case II
A schematic of the scattering processes in this regime is shown in Fig. 3(b) while the upspin and downspin edge current paths and equivalent circuit models for the phenomena occuring around the QPC is shown in Fig. 5 (c) and (d) respectively. In this case, the following points are to be noted:

The upspin electrons in the edge channel originating in the source contact are fully transmitted through the QPC to the upspin edge channel terminating in the drain contact.

The downspin electrons in the edge channel originating in the source contact are partially transmitted through the QPC to the downspin edge channel terminating in the drain contact.

Two kinds of electronnuclear spinflip scattering dominate at the QPC in this case:

Electrons from the forward propagating upspin channel terminating in the drain contact can undergo spinflip scattering to the forward propagating downspin channel terminating in the drain contact which is almost empty in the energy range between and . Such scattering at and around the QPC results in a positive nuclear polarization.

Electrons in forward propagating downspin channel propagating through the QPC can undergo a spinflip scattering to the backward propagating upspin channel (which is totally empty in the energy range between and ) terminating in the source contact. Such scattering results in a negative nuclear polarization in addition to decreasing the total current through the QPC.

Out of these two processes, the former process dominates at the QPC due to the presence of more upspin electrons compared to downspin electrons resulting in a net positive nuclear polarization at the QPC.

Reason for a decrease in near :

Near , the nuclear polarization cannot be maintained.

Nuclear polarization drops due to spinlattice relaxation.

A drop in polarization results in an increase in the uptodown spinflip rate as well as a decrease in downtoup spinflip rate in addition to a decrease in the direct transmission coefficient of the downspin channel due to decrease in the Overhauser field.

The decrease in transmission coefficient of the downspin channel decreases the conductance of the QPC in the vicinity of .
Equivalent circuit model:
The equivalent circuit in this case is detailed in Fig. 5(d). The circuit model can be described as follows:

The up spin edge channel originating in the source contact is fully transmitted through the QPC and hence is represented by a conductance .

The downspin edge channel originating in the source contact is partially transmitted through the QPC to the downspin edge channel terminating in the drain contact and hence is represented by a conductance .

The uptodown spinflip current at the QPC from the forward propagating upspin channel terminating in the drain contact to the forward propagating downspin channel terminating in the drain contact is represented by a current dependent current source .

The downtoup spinflip current from the forward propagating downspin channel originating in the source contact to the backward propagating upspin channel terminating in the source contact is represented by a current dependent current source .

The change in the transmission coefficient of the downspin channel due to the Overhauser field is represented by a by an equivalent conductor in series with a voltage dependent voltage source (). The current change due to the Overhauser field is represented by .

The nuclear polarization is represented by the voltage across the capacitor.

The resistance in parallel with the capacitor represents nuclear spinlattice relaxation by causing charge leakage from the capacitor.
The simulated results of the change in conductance with source to drain voltage sweeps are shown in Fig. 6. The parameters and in the simulations are calculated directly via a nonequilibrium Green’s function (NEGF) method using an atomistic tightbinding Hamiltonian Datta [1997, 2005, 2012] while the parameters and are calculated using (37). The parameters in the above illustration for the circuit diagrams in Fig. 3 are chosen to match the simulated result of the change in conductance with source to drain voltage sweep. The maximum change in the conductance due to a difference in the Overhauser field between the fully polarized nuclei and the nonpolarized nuclei is less than . However this maximum change can be enhanced due to spin flipflop tunneling as noted experimentally Wald et al. [1994]. \colorblack The hysteresis in the curves in Fig. 6 near occurs only when the nuclear spin relaxation time () is of the order of the voltage sweep time. This results in a lag between the applied voltage and nuclear polarization near thereby resulting in the hysteresis. The hysteresis in plots disappear when the is very large such that the change in nuclear polarization is negligible during the time of voltage sweep. The hysteresis in vs plots also disappear when is very small compared to the voltage sweep time because the nuclear polarization is always in a steady state with the applied voltage.
iii.2 Resistively detected nuclear magnetic resonance (RD/media/arxiv_projects/145801/NMR)
We begin our analysis with (14), where the nuclear polarization in the vicinity of the QPC is perturbed by an externally applied alternating magnetic field in the radio frequency (RF) range resulting in the Zeeman split nuclear levels to interact with each other. Near the frequency corresponding to the difference in energy between the two spin split nuclear energy levels (), precession of the nuclear spins accompanied by a rapid decay in the nuclear polarization occurs. To model such processes, we model the spin split nuclear energy levels by a broadened normalized density of states Datta [1997, 2005]; Slichter [1990].
where is the free variable denoting energy in the nuclear spin space, is related to the amount of broadening of the nuclear spin levels and is the energy level of the nuclear spin in the absence of broadening. Broadening might be a result of thermal motion of the nucleus Slichter [1990], hyperfine interaction mediated electronnuclear spin exchange Slichter [1990] as well as nuclear dipoledipole exchange interaction Slichter [1990] which is the causative agent for nuclear spin diffusionSlichter [1990]. We take broadening to be . We simulate the case of quad nuclear spin levels, as in GaAs, separated in energy due to Zeeman splitting. The rate equations in this case are given by:
(17) 
where is the row vector denoting four nuclear spin levels in GaAs and . is the probability of occupancy of the density of states of the nuclear spin level at energy . The matrix is the diagonal matrix representing the nuclear density of states at energy given by:
At low temperatures, is a boxcar function. Assuming that the average value of and in the energy range between and are and respectively and the spinflip transmission coefficients, and , are constant in the range of energy between and , the above equation can be simplified to,
where . The set of equations (LABEL:eq:nmr2)(LABEL:eq:nmr5) have to be solved selfconsistently to calculate the temporal evolution of the nuclear polarization. We now turn our attention towards (LABEL:eq:nmr2). The first term on the right hand side of (LABEL:eq:nmr2) is given by:
(21) 
where is the identity matrix of the fourth order and [], [], and are given by:
(22) 
(23) 