Eur. Phys. J. B (2014) 87: 17DOI: 10.1140/epjb/e2013-40967-3

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Nuclear spin diffusion effects in optically pumped quantum wells

Daniel Henriksen, Tom Kim, and Ionel Tifreaa

Department of Physics, California State University, Fullerton, CA 92831, USA

Received 30 October 2013 / Received in final form 3 December 2013Published online 20 January 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. We studied the influence of the nuclear spin diffusion on the dynamical nuclear polarization oflow dimensional nanostructures subject to optical pumping. Our analysis shows that the induced nuclearspin polarization in semiconductor nanostructures will develop both a time and position dependence dueto a nonuniform hyperfine interaction as a result of the geometrical confinement provided by the system. Inparticular, for the case of semiconductor quantum wells, nuclear spin diffusion is responsible for a nonzeronuclear spin polarization in the quantum well barriers. As an example we considered a 57 A GaAs squarequantum well and a 1000 A AlxGa1−xAs parabolic quantum well both within 500 A Al0.4Ga0.6As barriers.We found that the average nuclear spin polarization in the quantum well barriers depends on the strengthof the geometrical confinement provided by the structure and is characterized by a saturation time of theorder of few hundred seconds. Depending on the value of the nuclear spin diffusion constant, the averagenuclear spin polarization in the quantum well barriers can get as high as 70% for the square quantum welland 40% for the parabolic quantum well. These results should be relevant for both time resolved Faradayrotation and optical nuclear magnetic resonance experimental techniques.

1 Introduction

Electronic and nuclear spins in semiconductor systemshave attracted a lot of attention in connection with newelectronic devices with reduced dimensionality and bet-ter performance [1]. Electronic spins, the key componentof the new spin-based semiconductor devices, are sup-posed to bring new functionality at the same time withnonvolatility, increased data processing speeds, and lowpower consumption [2]. Nuclear spins, on the other hand,have been discussed in connection with quantum com-puting [3–5] or as an effective tool to control electronictransport in spin-based electronic devices [6,7]. In thiscontext, a precise control over the electronic or nuclearspin degree of freedom in semiconductor quantum wellsor quantum dots represents an important task to be ad-dressed by the scientific community. Several experimentaltechniques have been proposed to directly investigate thedynamics of the electronic spin, including time resolvedFaraday rotation (TRFR) and time resolved Kerr rota-tion (TRKR) [8,9]. On the other hand, nuclear spin dy-namics can be directly investigated using optical nuclearmagnetic resonance (ONMR) [10,11] or indirectly usingTRFR [12,13]. The dynamics of the electronic and nuclearspins are not independent, as the electronic and nuclearspins are coupled through the hyperfine interaction. Thisinteraction has major implications, as nuclear spins canbe employed as an efficient method to control the elec-tron spin state, but at the same time can lead to electron

a e-mail: itifrea@fullerton.edu

spin relaxation and dephasing, two possible phenomenathat influence the functionality of spin based electronicdevices.

Here, we will focus on the nuclear spin dynamicsin low dimensional semiconductor systems, with a par-ticular emphasis on semiconductor quantum wells. Nu-clear spins in low dimensional samples are hard to in-vestigate using standard experimental techniques such asnuclear magnetic resonance (NMR). In NMR, an exter-nal magnetic field is used to orient nuclear spins, lead-ing to the polarization of about 1% of the nuclei inthe sample, enough to detect and investigate the nu-clear spin dynamics (a detectable NMR signal requiressamples with a minimum of 1017 nuclei). The numberof available nuclei in quantum wells and quantum dotsis of the order of 1012 and 105, respectively, requiringa modified NMR experimental technique for investiga-tion. To compensate for the reduced number of nuclei inlow dimensional systems, one can use dynamical nuclearpolarization (DNP) to increase the percentage of polar-ized nuclei available for investigation. In DNP an initialnonequilibrium electron spin polarization is transferredto the nuclear system via the hyperfine interaction [14].This technique was successfully used in bulk semicon-ductors [15–18], as well as in low dimensional semicon-ductor nanostructures such as quantum wells [2,10,13],and quantum dots1. At least a couple of different meth-ods can be used to generate the initial electron spin

1 For a review of the current status of nuclear spin dynamicsin quantum dots see [19].

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population. One possibility is to use circularly polarizedradiation to induce spin selected transitions in the elec-tron system, a technique known as optical pumping (whenNMR is used in connection with optical pumping it isknown as ONMR) [10,11,16,20,21]. Alternatively, the ini-tial electron spin imbalance can be created using elec-trical spin injection through ferromagnet-semiconductorinterfaces [22–24].

Nuclear spin dynamics can be also investigated indi-rectly. TRFR, or the similar reflective technique TRKR,measures the Larmor precession of electron spins sub-ject to magnetic fields [12,13]. When used in connectionwith optical pumping, the Larmor frequencies recorded inTRFR experiments are a measure of the external appliedmagnetic field and at the same time of the internal (vir-tual) magnetic field due to the nuclear spin polarizationin the system. Such internal fields are not trivial, as theycan be as high as several teslas [12,13,19].

Nuclear spin dynamics is also influenced by other phys-ical phenomena in low dimensional semiconductor sys-tems. For example, nuclear spins are subject to variousinteractions leading to nuclear spin relaxation. Possibleexamples are the hyperfine interaction and the nuclei in-teraction with phonons [10,25,26]. On the other hand,nuclear spins interact with each other leading to bothrelaxation mechanisms and nuclear spin diffusion [27].Although important in bulk semiconductors [28], vari-ous experimental data indicate the fact that nuclear spindiffusion is strongly suppressed in quantum wells andquantum dots [13,29–32]. The natural confinement pro-vided by the geometry of low dimensional semiconductornanostructures plays an important role in DNP and im-plicitly in nuclear spin dynamics [29,33].

2 Dynamical nuclear polarization in lowdimensional systems

The hyperfine interaction is essential for DNP and de-scribes the flip-flop process that involves both the elec-tronic and nuclear spins:

H =∑

n

8π3μ0

4πg0μBgnμn I · S δ(r − rn) . (1)

Above, r is the electron position, rn is the position ofthe nth nucleus in the sample, μ0 is the magnetic perme-ability of the vacuum, g0 is the bare electron g-factor, gn

is the nuclear g-factor, μB is the Bohr magneton, andμn is the nuclear magneton. I and S are the nuclearand the electron spin operators. The Dirac δ-function re-quires the overlap between electrons and nuclei for an ef-fective hyperfine interaction. The effect of the hyperfineinteraction can be understood if one rewrites the prod-uct I · S = IzSz + (I+S− + I−S+)/2, with Iz and Sz thez-components and I± and S± the creation and annihi-lation operators of the nuclear and electronic spins. Thelast two terms in the expression are responsible for theflip-flop process involving both the electron and nuclear

spins, a process that eventually leads to DNP. In bulkmaterials, when the electron is described by Bloch wave-functions, the overlap of the electron and nuclear spins issmall and implicitly the hyperfine interaction is not veryefficient. This matter is overcome in bulk semiconductorsamples where the presence of donors or acceptors canlead to electron localization and implicitly to an efficienthyperfine interaction [28]. In low dimensional semiconduc-tor systems, however, the geometry of the sample providesa natural confinement for the electrons and implicitly thehyperfine interaction is very efficient [33].

In DNP one initially polarizes the electron spin systemto eventually achieve a large nuclear spin polarization viathe spin flip-flop process governed by the hyperfine inter-action. Although the magnetic field is a good source ofpolarized electrons, the orientation of the electron spinswith a magnetic field is not very efficient as usually oneends with a less than 1% spin polarization when normallaboratory fields are used. Instead, one can use opticalpumping or ferromagnetic spin filtering to provide the ini-tial population of polarized electrons. For example, opticalpumping has been used in connection with DNP in semi-conductor quantum wells and quantum dots to achievenuclear spin polarizations of the order of 6.5% in quan-tum wells [13] and as high as 60% in quantum dots [34].

Theoretically, we can use the standard time depen-dent perturbation theory to estimate the effects of thehyperfine interaction and to compute the nuclear spinpolarization as a result of DNP [14,35]. For a completepicture of how DNP works we have to adjust the dy-namic equation that describes how nuclear spins are re-distributed on various energy levels due to the hyperfineinteraction with a term related to nuclear spin diffusion.If we introduce D = N+ − N− the electronic spin polar-ization (N± represent the number of electrons with thespin oriented along and opposite to the applied externalmagnetic field) and Δ(rn) = Mm+1(rn) −Mm(rn) withMm(rn) the nuclear population on the mth nuclear spinlevel (m = I, I − 1, . . . ,−I; I is the nuclear spin quan-tum number), using the energy and angular momentumconservation we can derive a time dependent differentialequation that governs the flip-flop process due to the hy-perfine interaction and includes nuclear spin diffusion [36]

dΔ(rn)dt

=Δ0(rn) −Δ(rn)

T totn (rn)

+1

(2I + 1)kBT N

D0 −D

T1n(rn)

+Dc∂2Δ(rn)∂r2

n

. (2)

Above, D0 and Δ0(rn) are the equilibrium values forthe electron polarization and the difference in nuclearpopulation between adjacent nuclear levels, kB is theBoltzmann constant, T is the temperature, and N =∫drdεAe(r, ε)f ′

FD(ε) (Ae(r, ε) is the electronic local den-sity of states and fFD the standard Fermi function). Inequation (2) the first term in the right hand side of theequation includes contributions from all interaction mech-anisms that lead to the nuclear spin orientation under anexternal magnetic field, whereas the second term is strictly

Eur. Phys. J. B (2014) 87: 17 Page 3 of 7

related to the hyperfine interaction between electronicand nuclear spins. Accordingly, the total nuclear spin re-laxation time, T tot

n (rn), includes information related toall possible relaxation mechanisms for the nuclear spin.If one separates contributions due to the hyperfine in-teraction and all other interaction mechanisms, we have1/T tot

n (rn) = 1/T1n(rn)+1/T ′n (T1n(r) is the nuclear spin

relaxation time due to the hyperfine interaction; T ′n is

the nuclear spin relaxation time due to other interactionmechanisms).

The position dependent nuclear spin relaxation timedue to the hyperfine interaction, T1n(rn) can be calculatedas [37]:

T−11n (rn) =

512π3(g0μBgnμn)2kBT∫dεA2

e(rn, ε)f ′FD(ε)

3�I(I + 1)(2I + 1).

(3)Its position dependence is justified by the presence of theelectronic local density of states in the above expression,Ae(rn, ε) ∼ |ψ(rn)|2 (ψ(rn) is the electron wavefunctionat the nucleus position).

The third term in the right hand side of equation (2)is due to nuclear spin diffusion and its contribution to theDNP process depends on the value of the nuclear spin dif-fusion constant Dc. For GaAs bulk samples that consist ofthree different nuclear species, 69Ga, 71Ga, and 75As, thediffusion constant is of the order ofDc ≈ 103 A2/s (the ex-act value of the diffusion constant varies among differentnuclear species, however, in bulk GaAs these constants areof the same order of magnitude) [28,38,39]. The standardmodel used to account for nuclear spin diffusion in bulksemiconductor materials was introduced by Blumberg andit relies on the presence of impurities in the system [40].Particularly, this model predicts the existence of a non-diffusive sphere around the impurity sites and an efficientnuclear spin diffusion outside this region.

Nuclear spin diffusion also affects low dimen-sional semiconductor nanostructures. Malinowski andHarley [13] used the TRFR experimental technique todemonstrate and analyze nuclear spin diffusion in dou-ble quantum well structures. In their experiment, opticalpumping was used to polarize nuclei in the first quan-tum well, and 75As nuclear spin polarization in the secondquantum well was probed by TRFR. As the second wellwas not exposed to optical pumping, the only source ofnuclear spin polarization is nuclear spin diffusion throughthe barrier between the two quantum wells. The experi-mental results were modeled using a standard bulk nuclearspin diffusion model. The analysis assumed the presenceof impurities in the system and conclude that for an effi-cient description of the results two different values of thenuclear spin diffusion constant have to be considered; inthe actual quantum well, the nuclear spin diffusion con-stant is the same as in bulk materials, although in thequantum wells barriers the same parameter is one orderof magnitude lower, i.e., Dc ≈ 102 A2/s (the nuclei con-sidered in this analysis was 75As). Such an approxima-tion does not account for the system’s geometrical con-finement and it should be considered with caution. Thesituation is similar in quantum dots where a reduction of

the nuclear spin diffusion constant is supported by boththeoretical models [30] and experimental data [31,41]. Inthe case of GaAs/AlxGa1−xAs quantum dots, Nikolaenkoet al. [31] reported a value Dc ≈ 10 A

2/s, two orders of

magnitude lower than the bulk one. Even in their case,the authors modeled the experimental data using a bulknuclear spin diffusion model, however, they acknowledgethat for a much accurate model the natural confinementof the quantum dot has to be taken into account [31].

The main reason for a reduced nuclear spin diffusionconstant in samples with reduced dimensionality is thesystem’s geometrical confinement that is eventually re-sponsible for a nonuniform polarization of nuclei acrossthe sample. Accordingly, the dipole-dipole spin flip pro-cess is suppressed as it requires energy conservation. The-oretical studies that account mainly for the geometricalconfinement in quantum dots have suggested that the nu-clear spin diffusion constant can be as low as 10% of thebulk value [30]. Although this is a reasonable agreementbetween theoretical and experimental results, a compre-hensive theoretical model of nuclear spin diffusion in semi-conductor quantum dots should include a series of addi-tional parameters, such as shape, composition, or straindistribution.

3 Numerical results for DNPin semiconductor quantum wells

In general, equation (2) is a complicated time dependentdifferential equation that describes the time evolution ofDNP due to the hyperfine interaction. Its general solutionhas two parts, one related to the external magnetic fieldand the other one to the flip-flop process governed by thehyperfine interaction,Δ(rn, t) = Δ0(rn)+Δind(rn, t) [36].The contribution to the nuclear spin polarization due tothe external magnetic field, Δ0(rn), is uniform and it sat-urates fast for standard magnetic fields used in labora-tory to a value that is less than 1%, in contrast withthe contribution attributed to the hyperfine interaction,Δind(rn, t), that develops a position dependence in lowdimensional semiconductor nanostructures and can be sig-nificantly higher than 1% at saturation.

The solution of equation (2) describes the position andtime dependence of the nuclear spin distribution on avail-able levels due to external magnetic fields, hyperfine inter-action under optical pumping conditions, and nuclear spindiffusion. Based on the occupancy of the nuclear spin lev-els we can calculate the induced nuclear spin polarizationof the system as [14,35]:

P(rn) =∑

mmMm(rn)I

∑mMm(rn)

. (4)

We consider two different quantum well structures bothwith a band gap of about 1615 meV: a 57 A GaAs squarequantum well and a 1000 A AlxGa1−xAs parabolic quan-tum well, both within 500 A Al0.4Ga0.6As barriers (thevalue of the Al concentration, x, in the parabolic quan-tum well varies from 0.4 in the barriers to 0.07 in the

Page 4 of 7 Eur. Phys. J. B (2014) 87: 17

Fig. 1. The localiztion probability of the electronic systemmeasured by the modulus square of the electronic wavefunc-tion across the confinement direction in the two quantumwells: (a) the 57 A GaAs square quantum well within 500 AAl0.4Ga0.6As barriers. (b) The 1000 A AlxGa1−xAs parabolicquantum well within 500 A Al0.4Ga0.6As barriers. Vertical linesare for guidance and represent the boundary between the ac-tual quantum well and the barriers.

center of the structure). For a quantum well structure theelectronic dispersion relations are quasi-two-dimensional,and the total electronic wavefunction is a product betweenan envelope function φ(z) and a Bloch function u(r), i.e.,Ψ(r) ∼ φ(z)u(r). For both structures, we evaluated theelectron envelope wavefunction using a 14-band kp calcu-lation2. Figure 1 presents the probability to localize elec-trons along the confinement direction for both structures.The actual width of the barriers is 500 A for each quan-tum well, however, to emphasize the main differences be-tween the two structures, the plot is mainly focused onthe central region of the well (vertical lines are added forguidance – they represent the border between the actualquantum well and the barriers). For the square quantumwell, electrons are localized with a higher probability inthe center of the structure, however, there is a nonzeroprobability that one can find electrons in the side barriersof the structure. The situation is somehow different in theparabolic quatum well, where electrons are strictly local-ized in the central region of the structure, with, basicallya zero probability to be found in the structure’s barriers.These differences between the two considered systems willresult in a different position dependence of the nuclearspin polarization acrosss the wells, and in principle theyshould be measurable in ONMR or TRFR experiments.

We consider the temperature to be T = 5 K, however,recent studies indicated that DNP can be efficient even athigher temperatures [42]. For the nuclear spin relaxationtime due to other mechanisms than the hyperfine interac-tion we use the value T ′

n = 10 min [26]. We consider op-tical pumping to be quasi-continuous leading to approx-imately 50% electron spin polarization in the quantum

2 The 14-band kp numerical code was developed in the groupof M.E. Flatte at University of Iowa, USA.

Fig. 2. The position and time dependence of the in-duced 75As nuclear spin polarization in the presence ofnuclear spin diffusion (Dc = 102 A2/s). (a) The 57 AGaAs square quantum well within 500 A Al0.4Ga0.6As bar-riers. (b) The 1000 A AlxGa1−xAs parabolic quantum wellwithin 500 A Al0.4Ga0.6As barriers.

wells. Our main goal is to analyze the nuclear spin diffu-sion effects on the DNP of nuclei in the two structures.We will focus on 75As nuclei as they are part of both thequantum wells and the barriers.

Figure 2 presents the position and time dependenceof the nuclear spin polarization in the two semiconduc-tor structures in the presence of nuclear spin diffusion(Dc = 102 A2/s). The nuclear spin polarization in the twosamples relies on a position dependent hyperfine interac-tion and is strongly influenced by nuclear spin diffusion,especially when the nuclei in the barriers are considered.As expected, the polarization of the nuclei in the barriers isstronger in the case of the square quantum well as a resultof a less efficient geometrical confinement. Under opticalpumping, DNP is time dependent, and eventually leads toa saturated value of the nuclear spin polarization that isposition dependent across the wells. This is a result of thecompetition between nuclear spin polarization and nuclear

Eur. Phys. J. B (2014) 87: 17 Page 5 of 7

Fig. 3. The position dependence of the saturated in-duced 75As nuclear spin polarization for different values ofthe nuclear spin diffusion constant. (a) The 57 A GaAssquare quantum well within 500 A Al0.4Ga0.6As barri-ers. (b) The 1000 A AlxGa1−xAs parabolic quantum wellwithin 500 A Al0.4Ga0.6As barriers.

spin relaxation in the two samples. Figure 3 showcases theposition dependence of the saturated nuclear spin polar-ization in the two semiconductor nanostructures for dif-ferent values of the nuclear spin diffusion constant. Dif-ferent values of the nuclear spin diffusion constant willresult in different configurations of the nuclear spin polar-ization across the quantum wells. For example, in the caseDc = 102 A2/s (the quantum well reported nuclear spindiffusion constant [13]), for the square quantum well themaximum nuclear spin polarization is about 80% in thecenter of the well, and about 30% at the extreme sides ofthe two barriers. For Dc = 103 A2/s (the bulk materialreported nuclear spin diffusion constant [16]) the nucleiin the square quantum well become quasi-uniformly po-larized due to the strong effects of nuclear spin diffusion.The situation is slightly different for the parabolic quan-tum well due to a different geometrical confinement. WhenDc = 102 A2/s the central region of the well presents amaximum polarization of about 70%, whereas at the ex-treme sides of the barriers the nuclear polarization is aslow as about 3%. For Dc = 103 A2/s, in the case of theparabolic quantum well there are still significant differ-ences between the central region of the well and the ex-treme sides of the barriers. In the absence of nuclear spindiffusion, nuclei in the barriers are polarized only in thecase of the square quantum well, when the less efficientgeometrical confinement allows for a small, but nonzero,probability to find electrons in the quantum well’s barri-ers. The differences between the two structures are clearlyrelated to the different geometrical confinement providedby the two quantum wells. On the other hand, it is clearfrom Figure 3 that the higher the nuclear spin diffusionconstant, the more uniform the nuclear spin polarizationwill be across the semiconductor nanostructure.

Fig. 4. The saturated average 75As nuclear spin polarization inthe barrier as a function of the nuclear spin diffusion constant(black line – 57 A GaAs square quantum well; red line – 1000 AAlxGa1−xAs parabolic quantum well). The inset showcases thedifferences between the two structures in the absence of nuclearspin diffusion.

Nuclear spin diffusion plays an important role in po-larizing the regions of the semiconductor nanostructuresthat are subject to low optical pumping due to the posi-tion dependent hyperfine coupling between electronic andnuclear spins. For our structures, such regions will be rep-resented by the well’s barriers. As the nuclear spin polar-ization in the barriers is position dependent, we can definean average nuclear spin polarization in the barriers as:

Pb =1Lb

∫P(rn) drn, (5)

where Lb is the barrier width. Figure 4 presents the aver-age nuclear spin polarization in the quantum well’s bar-riers at saturation for different values of the nuclear spindiffusion constants. As expected, the higher the diffusionconstant value, the higher the average nuclear polariza-tion in the barriers. Also, one can notice the differencesbetween the saturated nuclear spin polarization in the bar-riers of the square and the parabolic quantum wells asa result of different geometrical confinement in the twostructures. One interesting observation is that for Dc = 0,when the nuclear spin diffusion is completely suppressed,the nuclear spin polarization in the barriers of the squarequantum well is nonzero, due to the small, but finite, prob-ability to find electrons outside the central region of thewell. This result is outlined in the inset of Figure 4 whereone can see that for the case of the square quantum well anaverage polarization of about 4% is present in the barriers.Differently, for Dc = 0 there is no nuclear polarization inthe barriers for the parabolic quantum well, as in this casethe electron confinement is very efficient.

In the case of quantum wells, DNP will result in a po-sition and time dependent nuclear spin polarization across

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Fig. 5. The time dependence of the average 75As nu-clear spin polarization in the barrier for different values ofthe nuclear spin diffusion constant. (a) The 57 A GaAssquare quantum well within 500 A Al0.4Ga0.6As barri-ers. (b) The 1000 A AlxGa1−xAs parabolic quantum wellwithin 500 A Al0.4Ga0.6As barriers.

the confinement direction of the system. By introducing anaverage nuclear spin polarization for the quantum well’sbarriers we eliminated the position dependence; however,this parameter is still time dependent. To characterize hownuclei in the barriers are polarized we can introduce twonew parameters: the rate at which the nuclear polariza-tion is built (Rb = dPb/dt) and the average saturationtime after which the nuclear polarization in the barriersstabilizes (Tb). As the total nuclear spin polarization inthe barriers is difficult to measure, Tb can be an impor-tant parameter to extract information about the nuclearspin diffusion constant. Figure 5 presents the time depen-dence of the average nuclear spin polarization in the bar-riers of the two quantum dots for different nuclear spindiffusion constants. The average saturation time, Tb, de-pends both on the value of the nuclear spin diffusion con-stant and on the geometrical confinement of the struc-ture, as for the same Dc there are significant differencesbetween the square and the parabolic quantum wells. Asa general result, Tb is of the order of few hundred sec-onds in the case of quantum wells, with a larger valuefor the higher confinement situation. The building rate,Rb, characterizes the initial time dependence of the aver-age nuclear spin polarization in the barriers. In the caseof the square quantum well, there is a nonzero nuclearspin polarization in the barriers even in the absence ofnuclear spin diffusion, and we define the building rate asthe first derivative of the average nuclear spin polarizationat t = 0. Rb increases as the nuclear spin diffusion con-stant increases (Rb = 0.37 s−1 for Dc = 0, Rb = 0.46 s−1

for Dc = 102 A2/s, and Rb = 1.17 s−1 for Dc = 103 A2/s).On the other hand, for the parabolic quantum well, thenuclear spin polarization in the barriers is strictly a resultof nuclear spin diffusion. In this case, the building rate ismuch harder to define; for different nuclear spin diffusion

constants we will have a different time at which the av-erage nuclear spin polarization in the barriers is nonzero.However, even in this case, the build rate is higher forlarger nuclear spin diffusion constants.

4 Conclusion

In summary, we addressed the problem of nuclear spindynamics in semiconductor nanostructures. Under opti-cal pumping, DNP is the governing process that leads toa non-uniform nuclear spin polarization across the semi-conductor system. Our calculations include contributionsdue to the hyperfine interaction between nuclear and elec-tronic spins, nuclear spin relaxation, and nuclear spin dif-fusion. In low dimensional semiconductor nanostructures,the hyperfine interaction becomes non-uniform as a re-sult of the natural confinement provided by the system.As DNP relies on the hyperfine interaction, it is naturalthat the resulting nuclear spin polarization will be posi-tion dependent. Hyperfine interaction is also an importantmechanism for nuclear and electronic spin relaxation. Asa matter of fact, at low temperature, the hyperfine relax-ation mechanism is the dominant relaxation mechanismfor nuclear spins, leading to relaxation times of the orderof tens of seconds [10,37]. At the same time, nuclear spinsare subject to other relaxation mechanisms resulting fromvarious other interactions that involve the nuclei. Addi-tionally, nuclear spins interact with each other resulting innuclear spin diffusion. All these components are importantfor DNP, the nuclear spin polarization across the semicon-ductor quantum well being a result of their competition.

Nuclear spin diffusion is an important process in semi-conductor systems [13,16,30]. In bulk semiconductor sam-ples, nuclear spin diffusion is the main source of nuclearpolarization due to the inefficient hyperfine interaction.In low dimensional semiconductor nanostructures, nuclearspin diffusion is less efficient, experimental data in quan-tum well and quantum dot systems leading to the con-clusion that the nuclear spin diffusion constant is at leastone order of magnitude smaller in this case compared tobulk semiconductor systems [13,31]. This result, at leastfor the case of quantum dot systems, is also justifiedtheoretically [30].

Our analysis of DNP in the presence of nuclear spindiffusion proves the importance of the natural geometri-cal confinement provided by quantum well structures. Weconsidered the case of square and parabolic structures andfound that the nuclear spin diffusion is more efficient insquare wells that permit a more even distribution of theelectron states across the quantum well structure. In thecase of a parabolic quantum well, when the geometricalconfinement is more efficient, the effects of the nuclearspin diffusion are smaller when nuclei in the barriers ofthe well are considered. For both structures, the polar-ization of nuclei in the barriers is time dependent andcan be characterized by a building rate and by a satu-ration time. Both these parameters are structure depen-dent, highlighting the geometrical confinement role in lowdimensional nanostructures. Although our examples are

Eur. Phys. J. B (2014) 87: 17 Page 7 of 7

related to quantum well structures, similar results shouldbe true also for quantum dots, although the hyperfinemechanism in this case is somewhat different [34].

The authors would like to acknowledge financial support fromthe Intramural Grant program at CSUF.

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