Laser-assisted magnetic resonance Dieter Suter

Laser-Assisted Magnetic Resonance of Semiconductor Nanostructures

Semiconductor nanostructures are used in many different materials, including semiconductor lasers and other optoelectronicdevices. Magnetic resonance is one of the most important techniques for characterizing such structures. However, its limitedsensitivity requires specialized techniques that provide sufficiently high sensitivity and selectivity for structures with lengthscales of the order of nanometers.

Physical Background

Quantum-confined semiconductors

GaAs <001> substrate

GaAs quantum

wells

AlGaAs barriers

790 800 810 820

PL wavelength [nm]

PL in

tens

ity [a

rb. u

nits]

Quantum wells

Bulk

Figure 1: GaAs / AlGaAs multiple-quantum well structure and photolumines-cence spectrum.

Figure 1 shows a typical example of a semi-conductor heterostructure where electronsare confined in quantum wells or quantumdots. The bandgap of the AlGaAs barri-ers is larger than that of the GaAs wells.The charge carriers are therefore confined inthe quantum wells, whose thickness is of theorder of a few nm. At low temperatures,this is less than the coherence length of theelectrons. As a result, the energy levels ofthe electrons become discrete. This can beseen, e.g., in the photoluminescence spec-trum, where every quantum well emits pho-tons at a specific wavelength: In the thinnestquantum wells, the electronic ground stateof the conduction band has the highest energy and the emission wavelength is therefore shifted to shorter wavelengths.

Optical polarization

σ+-light

mJ - 12 +1

2 Conduction band

z ⊥ layer

mJ

- 12 +1

2 Valence band

- 32 +3

2

heavy hole

light hole

Figure 2: Optical pumping in semiconductor nanostruc-tures with direct band-gap.

In thermal equilibrium, the populations of different angular mo-mentum states are almost identical. This can be changed byoptical pumping with resonant polarized light. Fig. 2 shows theprinciple of optical pumping in a semiconductor quantum wellwith a p-type valence band and an s-type conduction band. Theorbital- and spin-angular momentum combine to a total angularmomentum of J = 1/2 in the conduction band and J = 3/2 inthe valence band. The J = 1/2 angular momentum states ofthe valence band are lower in energy (not shown in the figure)and therefore always fully occupied. In the absence of a magneticfield, the four mj susbstates of the J = 3/2 valence band stateare degenerate in bulk crystals. However, in a quantum well,where the symmetry of the Hamiltonian is reduced by the con-finement potential perpendicularly to the layers, the mJ = ±3/2states (known as heavy holes) have a higher energy than themJ = ±1/2 states (the light holes). Optical photons close tothe band edge of the system therefore couple only to the transitions from the mJ = ±3/2 valence band states to themJ = ±1/2 states of the conduction band. Angular momentum selection rules determine which of the two possible tran-sitions, |mJ = �3/2i $ |mJ 0 = �1/2i or |mJ = +3/2i $ |mJ 0 = +1/2i couples to the radiation field. For circularpolarization, the photons therefore create fully spin-polarized electrons in the conduction band and holes in the valenceband.

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Laser-assisted magnetic resonance Dieter Suter

Nuclear spins

Figure 3: Interac-tion of an electronwith nuclear spins viaFermi contact.

Most of the isotopes found in semiconductor materials have non-vanishing nuclear spin, e.g. I = 3/2for 69Ga, 71Ga, and 75As. If these nuclear spins are in contact with an unpaired electron, theyexperience a hyperfine interaction, which can be written as Hhf =

Pi Ai

~Ii · ~S, where Ai quantifiesthe strength of the interaction between the i-th nuclear spin ~Ii and the electron spin ~S. Thisinteraction contains, among others, the so-called flip-flop term I+S�+I�S+, which drives correlatedflips of the electronic and nuclear spins. In thermal equilibrium, the polarization of the nuclear spinsis vanishingly small, while the optical pumping can generate almost completely polarized electronicspins. The flip-flop processes can therefore transfer polarization from the electronic to the nuclearspins.

One effect of the nuclear spin polarization is that the large number of nuclear spins that are coupledto the electron spin starts to affect the evolution of the electronic spin. The overall effect is similarto that of a magnetic field and can therefore change the polarization of the luminescence thatis emitted when a hole and an electron recombine to emit a photon. The polarization of theemitted photoluminescence is therefore one of the possible measures that allow one to monitor thepolarization of the nuclear spins.

Magnetic field [T]Lum

ines

cenc

e po

la-

risat

ion

[arb

. uni

ts]

0 0.2 0.4

75As

69Ga

71Ga

75As 2QT

Figure 4: Optically detected nuclear magnetic resonancefrom a single GaAs quantum well. The system was ir-radiated with a radio frequency field with !rf/2⇡= 5MHz.

If a radio-frequency field is applied to the system, whose frequencyfulfills the condition !rf = !L = �B0, where !rf is the frequencyof the alternating magnetic field, !L is the Larmor frequency ofthe nuclear spin, � it’s gyromagnetic ratio and B0 the flux densityof the static magnetic field, it can generate transitions betweenthe nuclear spin states. These transitions will typically reduce thenuclear spin polarization, as in the example shown in fig. 4. Forthis case, an alternating magnetic field with a frequency of !rf=5MHz was applied to the sample, which was simultaneously subjectto optical pumping with circularly polarized light. As the staticmagnetic field was increased, several resonance conditions werereached, where the polarization of one of the constituent nuclearspins was saturated by the ac magnetic field. These conditionsare marked by the corresponding isotopes. In the case of 75As,the resonance condition is fulfilled for two different fields. At thehigher field, just below 0.4 T, transitions occur between states

whose magnetic quantum numbers differ by �m = ±1, in the region below 0.2 T, for �m = ±2.

Measurement of lattice distortion

Radio frequency [MHz]6.25 6.3 6.35

Lum

ines

cenc

e po

laris

atio

n

75As: I=3/2 m=-3/2

m=-1/2

m=1/2

m=3/2

Corresponding NMR spectrum

B0 = 0.863 T

Figure 5: Details of the 75As resonance measured in a constant mag-netic field by scanning the RF frequency.

If the resonance conditions are analyzed in moredetail, they often consist of several steps. Figure5 shows as an example, the resonance of the 75Asnuclei, for the single-quantum (�m = ±1) transi-tion. In this case, the resonance was measured bykeeping the magnetic field constant and scanningthe frequency of the applied AC-field. The dataclearly show three distinguishable steps of the de-polarisation. As shown in the level scheme in thesame figure, these three steps can be attributed tothe three distinct single quantum transitions thatcan be induced in an I = 3/2 spin. The right-hand panel shows the corresponding NMR spec-trum, which consists of a narrow central line, the�1/2 $ +1/2 transition and two broader ’satellitetransitions’ corresponding to the ± 3/2 $ ±1/2transitions.

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Laser-assisted magnetic resonance Dieter Suter

Bulk GaAs:cubic symmetry

resonance lines degenerate

Strain, fieldslift degeneracy

distinct transition frequencies

3/2

1/2

-1/2

-3/2

Energy

Frequency Frequency

Sign

al

Figure 6: Distortions lower the symmetry. Theresulting electric field gradients lift the degeneracyof the NMR transition frequencies.

The Zeeman energies for these transitions are the same, but an ad-ditional interaction, the nuclear quadrupole coupling HQ = QI2

z shiftsthese two transitions in opposite direction, as shown in fig. 6. In an idealcrystal, the coupling constant Q vanishes at the positions of the nucleidue to the high site symmetry. This means that all three resonancelines should occur at the same frequency. The experimental data show-ing three distinct transition frequencies therefore imply that the crystalis deformed in such a way that the environment of the atomic nucleiis slightly asymmetric. It was actually determined, that the degree ofasymmetry is a linear function of the splitting between the central lineand the satellites [2]. Such deformations occur, e.g., in the vicinity ofdefects. Since the sensitive volumes, from which spectra are collected,typically contain a very large number of defects, they average over manydifferent line shifts, resulting in a broadening of the satellite lines com-pared to the central line. This effect is clearly visible in fig. 5. The factthat not only a broadening is visible, but also a splitting, implies thatsome distortions have correlation lengths larger than the diameter of thesensitive volume, which is determined by the spot size of the laser on the sample (typically of the order of some 10’s ofµm). Such distortions can arise from strain in the substrate but may also be due to the mounting of the sample.

Frequency ω

20 kHz

� � 6 · 10�5

��

2.5

· 10

�5� �

3.5 · 10 �5

� < 0.5 · 10�5

Laser spot Ø 50 µm

5 mm

2 mm

0

1

2

0 1 2 3 40

1

0

40

y-po

sitio

n [m

m]

Qua

drup

ole

split

ting

[kH

z]

Dist

ortio

n [1

0-4]

x-position [mm]

Figure 7: Measurement of lattice distortions at different points on the sample. The left-hand part shows four spectra measured at different positions, indicating different degreesof strain. The right-hand part shows a map of the distortions generated from many mea-surements at different positions on the sample.

The strain is in most cases nothomogeneous over the sample,but varies with position. Fig-ure 7 shows, as an example,in the left-hand part four mea-surements taken at four differ-ent locations on the same sam-ple. Clearly, the splittings vis-ible in the four spectra showsignificant variations. Thesplittings correspond to strainvalues of up to 6 · 10�5, as in-dicated close to the arrows in-dicating the positions at whichthe measurements were taken.The right-hand part of the fig-ures shows the distribution ofstrain over the sample, as eval-uated from a large number ofsimilar measurements.

6.8 nm QW

10.5 nm

11.7 nm

14.5 nm

19.7 nm

0 5 10200

300

400

500

Quadrupole splitting / kHz

Depth/nmGaAs

quantumwells

AlGaAsbarriers

GaAs <001> substrate

Figure 8: Multiple quantum well with quadrupole split-tings measured in different quantum wells.

Electric fields

Splittings in NMR spectra occur not only in the presence of latticedistortions, they can also be induced by electric fields that coupleto the nuclear quadrupole interaction. Such electric fields may beimposed externally, or they can result from internal charges. Inthe example shown in fig. 8, the electric field is generated by theSchottky barrier from the contact with a surface electrode [1].The observed quadrupole splittings increase roughly linearly withdepth, which is compatible with a homogeneous space-chargedistribution due to the Schottky barrier.

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Laser-assisted magnetic resonance Dieter Suter

References

[1] Marcus Eickhoff, Björn Lenzmann, Dieter Suter, Sophia E. Hayes, and Andreas D. Wieck. Mapping of strain and electricfields in GaAs/AlGaAs quantum-well samples by laser-assisted NMR. Phys. Rev. B, 67:085308, 2003.

[2] D.J. Guerrier and Richard T. Harley. Calibration of strain vs nuclear quadrupole splitting in III/V quantum wells. Appl.Phys. Lett., 70:1739–1741, 1997.

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