Experiment 1: Nanoparticle absorption in a thin film

When condensed matter physicists need to make measurements on materials in extreme environments (low temperatures, high magnetic fields, intense electric fields, etc.), it is often preferable that those systems are either gaseous or solid-like. Traditional condensed matter physics takes place in lattices (i.e., materials with translational invariance), which immediately eliminates the gaseous option (atomic and molecular physicists, however, highly favor this state of matter, since it mitigates inter-atom effects). Up until about two decades ago, producing samples in a lattice form was a no-brainer: nearly everything was either a metal, polymer film, elemental or blended semiconductor, or layered semiconductor (quantum well or quantum wire). However, nanostructures began to appear in the 1990’s when it was realized that very high quantum confinement (geometric dimensions smaller than that spatial extent of a wavefunction) could reveal interesting and new physics. Nanoparticles, such as nanotubes, nanoflakes, quantum dots, nanorods, and 2D materials, now dominate a wide section of physics, chemistry, and materials science.

Given their small size, environmental effects have a disproportionate effect on their physical and chemical behaviors. In particular, the surrounding dielectric environment, surface preparation, and imposed strain are all key considerations for ascertaining the true properties of nanoparticles. For example, strain on semiconducting nanoparticles can change the observed semiconducting bandgap, which not only shifts the absorption peak,

but also augments the linewidth due to inhomogeneous broadening. Therefore, minimizing these effects is often a focus for nanoparticle spectroscopists.

Figure 1: Schematic of the Cary 500 spectrometer showing the double-beam configuration. In this setup, the sample and reference transmission are measured simultaneously in order to determine absorption.

In this lab, you will compare the absorption of nanoparticles in solution (i.e., in a solvent) with those immersed in a polymer matrix. By fitting the absorption peaks of the nanoparticles in both environments, you will empirically show the effect of strain on the bandgap of the immersed semiconducting

nanoparticles. If time allows, you will then measure the absorption temperature dependence of the nanoparticle films down to 2 K a find a Varshni scaling behavior.

Goal 1: Create nanoparticle films using the spin coater.

Goal 2: Measure the absorption of the nanoparticle solution and nanoparticle film in the Cary spectrometer. Absorption, A, is often calculated from measuring the transmission of the reference, Tr, and sample, Ts: A = -log10(Ts/Tr). Although this formula (and method in general) is an approximation, it is fairly decent one when reflection is negligible.

Goal 3: Fit the absorption spectra (the peaks in this case are known as excitonic transitions) of the solution and film with a Lorentzian function to show that the presence of the polymer matrix has altered the semiconducting bandgap and absorption linewidth.

Goal 4 (time permitting): Measure the absorption of the nanoparticles immersed in a thin film in the SpectroMag down to 2 K. Consider making between 10 and 20 absorption measurements in the temperature interval of 2 to 290 K. Choose a specific nanonparticle peak and fit that peak for each spectrum that you take. Plot the center energy (not wavelength) as a function of temperature and fit a Varshni curve to it to determine the Varshni coefficients and the “zero-temperature bandgap”, Eg(0).

Terms that may need clarifying

Excitonic transitions: an exciton occurs when a photon, with an energy greater than the semiconducting bandgap, is absorbed. This process excites an electron from the valence band to the conduction band producing a positively charged hole (absence of an electron) in the valence band. The positively charged hole and negatively charged electron are Coulombically bound in the same manner as hydrogen (one electron and one proton) and thus form a hydrogenic series of orbitals (1s, 2s, 2p, etc.) with energy spacings that are smaller than hydrogen by the reduced mass of the exciton (very roughly, a factor of 2000). Excitons define the absorption spectra of semiconductors, and become more important as dimensionality decreases.

Varshni behavior: The Varshni equation is a semi-empirical equation that describes how the bandgap of a semiconductor changes with temperature: Eg(T) = Eg(0) - T2/(T + ), where Eg(0) is the zero-temperature bandgap and and are fitting parameters. This temperature-dependent behavior is often attributed the increased (decreased) likelihood of carriers (electrons and holes) to have larger (smaller) energy at higher (lower) temperatures: that is, an activation-energy-based model.

Experiment 2: Temperature-dependent conductivity measurements of a semiconductor

One of the most important techniques in condensed matter physics is measuring how carriers (electrons and holes) move through a material. The determination of conductivity, , or resistivity, (where = 1/), as a function of external variables, such as temperature, magnetic field, gate voltage, stress, etc., is a powerful way of gaining insights into intrinsic material properties, such as the carrier effective mass, carrier concentration and sign, formation of quasiparticles (Cooper pairs, polarons, etc.), and effects of lattice vibrations (phonons), dopants, and impurities. For example, the traditional behavior of metals as a function of temperature shows an increase of the conductivity as temperature decreases, which is commonly attributed to a suppression of the phonon density (which goes a Bose-Einstein distribution) at temperatures below the Debye temperature.

If we recall Ohm’s law, V = IR, we can substitute in L/A for R, to obtain an expression for either or . Here, L and A are the length between contacts and the cross-sectional area that the current moves through, respectively. Importantly, to measure , we need to either take into account the resistance of the contacts/wires/etc. to and from the device under test (often abbreviated as DUT) or use a so-called four-wire configuration. In the latter setup, two wires carry the source current, Isource, that flows across the entire DUT and two wires of exactly the same composition and length as the source wires measure the voltage drop, Vlongitudinal, in the direction of the current flow. Since all of the resistances of these four wires that determine source current and longitudinal voltage are exactly the same, we are able to neglect them entirely. (If we are using a two-wire configuration, ignoring the contact and wire resistances is not possible, adding another layer of complication to the measurement.) Thus, a four-wire configuration enables you to unambiguously extract Isource

and Vlongitudinal, and in conjunction with geometric measurements, determine .

Figure 1: Schematic of the double lockin amplifier experimental setup for measuring RDUT described in the manual.

In this lab, you will be creating the programming instructions to properly read Isource and Vlongitudinal, using a double lockin

amplifier. A simple resistor should be used for as a stand-in for the DUT. For this configuration, one lockin amplifier will measure the voltage drop across a large resistor (i.e., much larger than the resistance of the DUT) in series with the DUT. If this large resistance is very well known, then this RMS voltage measurement can be used to determine Isource. This slowly modulated (frequency between 20 and 100 Hz, making special care to only choose odd numbers away from 60 Hz and multiples of 20 Hz), steady-state current creates a Vlongitudinal that is measured by the second lockin amplifier. Critically, this lockin amplifier (“slave”) is locked to the frequency set by the first (“master”) lockin

amplifier, a process which allows both lockins to suppress all other frequencies and thus produce a very high signal-to-noise ratio.

Goal 1: Design and build a program that reads two lockin amplifiers to obtain RDUT and plots it as a function of time. The program should be able to save the data with four columns: time, Isource, Vlongitudinal, and RDUT.

Goal 2: Once you have shown that you can control/read two lockin amplifiers to obtain RDUT, substitute the trial resistor with a semiconductor. Use indium solder to attach the leads to the semiconductor sample. Indium is used for its good adhesion to gold, flexibility at cryogenic temperatures, and relatively small temperature-dependent contraction. Mount the sample on the sample stick with a minimal amount of rubber cement.

Goal 3: Modify your program to add a fifth column that reads the temperature of the sample, which will be reported by the Oxford temperature control system (ITC). Load the stick (with the assistance of one of the graduate students or Prof. Rice) and measure (and plot) RDUT as a function of temperature down to 1.5 K.

Goal 4 (time permitting): Modify your program to add a sixth column that reads the magnetic field. At 5 K, measure RDUT as a function of magnetic field from –7 to + 7 T. Determine the magnetoresistance, MR, of the sample at a fixed temperature using this formula: MR (T, B) = R(T, B)/R(T, B = 0) – 1.

Experiment 3: Second Harmonic Generation in a Crystal

In the small, perturbative limit, a spring obeys Hooke’s law to first order: F = - kx, where k is the spring constant and force, F, is linearly proportional to displacement, x. A more accurate description treats this expression as a polynomial expansion out to an order n, where n > 1 and each coefficient of xn, kn, describes the weight of the nth basis. Although we can often ignore this more-complete description, for certain interactions, these higher-order terms are not only important, but sometimes dominant.

Since the electrostatic force between an electron and a positively charged nucleus can be modeled as spring-like, light-matter interactions in the so-called semi-classical regime are one of the best expressions of the aforementioned higher harmonic modes. In the limit where the time-varying electric field is not too large or too small, the time-dependent polarization, P(t), can be related to this driving electric field, E(t), by a Taylor expansion:

P ( t )=ϵ 0 ( χ (1 )E (t )+ χ (2 )E2 ( t )+ χ (3) E3 (t )+... χ (n )En (t ) ) , (1)where (n) is the n-th order electrical susceptibility and is represented as a (n+1)-rank

tensor. In the simplified case that E (t )=12E0 (e

+i(kx−ωt )+e−i(kx−ωt)) e , the second term in Eq. 1

yields a non-linear polarization density that has angular frequency components at zero (optical rectification) and 2 (SHG). In physical terms, this means that SHG at twice the incident frequency can result if two photons (quantized particles of E2) are incident on the crystal at the same time and place. Or, put in slightly different words, energy conservation must be maintained (Fig. 1).

Figure 1: When two photons at an angular frequency, , are present at the same time and place, a double virtual transition can result, which produces a 2 emission. Classically, we visualize this process as producing a polarization density oscillating at 2 that radiates at this frequency.

The (2), rank-2 tensor adds a further constraint: only certain crystals cut along particular orientations with respect to the incident light polarization (given by e in Eq. 1) have an appreciable nonlinear polarization. This constraint is deepened by bringing in momentum conservation, which is often denoted in nonlinear optics as phase matching. If k2 is the momentum of the emitted photon and k1 is the momentum of one of the incident photons, then k = k2 - 2k1 0. If we provide a full treatment of the topics that we introduced here, and bring in the wave equation, we find that the SHG intensity, I 2ω, can be related to the incident intensity, Iω, using:

I 2ω=ω2

4c3 ϵ 0

¿ χ (2 )∨¿2

nω2 n2ω

Iω2 L2 sinc2( ΔkL2 ) ,¿

(2)

where L is the length of the crystal, the sinc function is maximum at 0, and nω≈n2ω. Finally, we note that modern lasers take significant advantage of second harmonic generation (SHG) to create visible light (400 to 700 nm) from efficiently generated near-infrared light (700 to ~2000 nm). Some common SHG crystals used in industrial applications include potassium dihydrogenphosphate (KDP), potassium titanyl phosphate (KTP), beta barium borate (BBO), lithium triborate (LBO), and lithium niobate (LNB). In fact, several of the lasers that we commonly use in the lab, such as the 532 nm pump laser, utilize non-linear crystals inside the laser housing.

In this lab, you will determine the major component of the (2) for BBO at 1035 nm by using a pulsed, amplified laser. **This laser is extremely dangerous, since it operates at very high fluences (energy/area) in the near infrared (i.e., outside the visible range). Laser safety goggles must be worn at all times. No exceptions!!** For these lasers, we are relying on the very high number of photons per pulse to produce a situation where multiple photons are in the same location at the same time. Although this laser is excellent for nonlinear optical investigations, it also means that all of the energy is concentrated into very short (10-13 sec) bursts.

Goal 1: Design and build a system to measure the incident power (i.e., power at 1035 nm) and the emitted power at 2f.

Goal 2: Measure emitted 2f power as a function of incident power (at f) from 0 to 10 mW. Plot and fit P2f as a function of P2

f. Remember that P (power; W) is related to I (intensity; W/m2) via dividing by area.

Goal 3: Either measure or look up the relevant numbers to determine the non-linear coefficient for BBO using Eq. 2. You should make yourself familiar with the meaning and mathematics behind Eq. 2 before you begin (Boyd’s “Nonlinear Optics” book is a good place to start).

Goal 4 (time permitting): Measure 3f.

Terms that may need clarifying

Semi-classical regime: We treat the material (atom or crystal) as a quantum mechanical entity, but the electrical (or magnetic field) is considered to be classical. A fully quantum mechanical treatment of both the field and matter differs from this semi-classical model in that the field is fully quantized (using a harmonic oscillator treatment with raising/lowering operators). Although this treatment can be more accurate in the few photon limit, it is often much more mathematically cumbersome. Plus, it converges (because of the correspondence theorem) to the semi-classical/classical solutions in the limit of large photon number.

Momentum: For light, momentum is very small, since p= kℏ , where k=2πλ and is the

wavelength. Also, it’s worth noting that (angular frequency) = 2f and that fλ=ωk=c .

Experiment 4: Building an Autocorrelator to Measure Ultrafast Pulses

A typical photodetector, i.e., one that relies on photogenerated carriers, has a response rate that ranges from 1 to 100 ns. Although these time scales may seem fast, they pale in comparison to modern ultrafast lasers, whose pulse durations range from 1 to 300 fs. More exotic detectors exist, but the time scales that they operate at are still too slow to properly resolve features below a picosecond. If we try to skirt the problem by moving to the frequency domain and employing a Fourier transform to relate time and frequency, we can obtain a decent estimate of the temporal pulsewidth. However, it is quite easy to show that this condition only exists if we make a rather large assumption that our laser pulse is Fourier transform-limited: that is, the time-varying electric field that actually forms the pulse has no dispersion. This assumption is typically incorrect, which leads us back to finding a way to directly measure the temporal pulsewidth.

A more direct way of obtaining the pulsewidth of an ultrafast laser is to use the technique of autocorrelation, which is a subset of a second-order correlation function, G(2)(), where is the delay time between events. In an autocorrelation (which is a generalized method used across many fields), the time domain of one pulse train from the laser is varied over the time domain of a second pulse train (Fig. 1). When the two pulse trains are coincident, the electric field is twice the magnitude of the two pulse trains when they are completely offset. This electric field enhancement can be ascertained by directing both pulse trains onto the same spot of a non-linear crystal. For this lab, you will be using a second-harmonic generation (SHG) crystal known as beta barium borate (BBO). For the traditional (intensity) autocorrelation, each pulse train generates a second harmonic signal because of the high electric field magnitude of a given pulse. However, when both pulses are present at the same time (direct overlap of the pulse trains: = 0), the field (intensity) is doubled (quadrupled) giving a large increase in the SHG. Put differently, the SHG intensity, ISHG, is a composite of the self-SHG terms from each pulse plus the SHG generated when the pulse trains are increasingly overlapped with one another: I SHG (τ )≈2∫ I 2 ( t ) dt+4∫ I ( t ) I ( t−τ )dt , (1)where t is time and is the time delay between the two pulse trains. Since the first term acts like a constant background, only the second term in Eq. 1 is relevant for our purposes. By scanning over from large negative values to positive ones (or vice versa), we obtain a -dependent ISHG whose shape allows use to extract the pulse width.

Figure 1: Time ordering of the two pulse trains. For the red pulse train, the time difference, , between it and the fixed pulse train (blue) is negative. A negative time delay occurs when the red pulse train travels a farther distance from the beam splitter to the nonlinear crystal (thus, arriving at the crystal later). For the green pulse train, , is positive (i.e., it arrives at the crystal before the fixed pulse train).

Figure 2: Schematic of the intensity autocorrelator. Note that function generator both drives the delay stage and triggers the oscilloscope. When aligning this system, the distance that the fixed and the variable (the one sent to the delay stage) beam paths travel from the beam splitter to the nonlinear crystal should be exactly the same.

In this lab, we will construct an intensity autocorrelator (Fig. 2) using a mirror mounted to an audio speaker as our delay stage retroreflector. This system, which costs >$10,000 commercially, will be able to resolve pulsewidths down to 10 fs. **Please note that the laser you will be using is extremely dangerous, since it operates at very high fluences (energy/area) in the near infrared (i.e., partially outside the visible range). Laser safety goggles must be worn at all times. No exceptions!!**

Goal 1: Using the schematic shown in Fig. 1, build an intensity autocorrelator. Put the audio speaker/retroreflector combination on a manual, linear translation stage and move the micrometer that controls this stage to the center of its travel. When you are aligning this system, use a laser power below 10 mW. Do not put in the nonlinear crystal, Schott glass filter, or detector yet.

Goal 2: At the position of the focus of the two beam paths, place a pinhole mounted in a lens holder (afixed atop a post and a postholder). Please place this ensemble on a two-dimensional translation stage. Align both beams through this pinhole (note, this is the hardest part of this lab). Move the translation stage for the delay stage back and forth. Both beams should still go through the pinhole regardless of the translation stage position.

Goal 3: Replace the pinhole assembly (but not the 2D translation stage) with the BBO crystal. Make sure that the crystal is in the exact same position as the pinhole was. Put in the filter and silicon detector. Move the delay stage translation stage so that you can observe SHG (SHG should appear very briefly as you turn the micrometer of the delay stage translation stage). You may need to increase the laser power (never exceed 100 mW).

Goal 4: Move the delay stage translation stage to where the SHG signal (as seen by your eye) is maximized. Rotate the BBO crystal until the SHG signal is maximized. Turn on the function generator, oscilloscope, and silicon detector after you ensure that SHG light is hitting the silicon detector. The trace on the oscilloscope is an autocorrelation. Move the delay stage translation stage by a known amount, which should shift the trace. This movement will allow you to calibrate the x-axis of the oscilloscope in units of time (fs).