Terahertz heterodyne mixing with a hot electron bolometer and a quantum cascade laser

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Terahertz heterodyne mixing

with a hot electron bolometer

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Terahertz heterodyne mixing

with a hot electron bolometer

and a quantum cascade laser

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op vrijdag 19 januari 2007 om 10.00 uur

door

Merlijn HAJENIUS

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Toegevoegd promotor: Dr. J. R. Gao

Samenstelling promotiecommissie:

Rector Magnificus Voorzitter

Prof. dr. ir. T. M. Klapwijk Technische Universiteit Delft, promotor Dr. J. R. Gao SRON / TU Delft, toegevoegd promotor

Dr. H. -W. H¨ubers Deutsches Zentrum f¨ur Luft und Raumfahrt, Berlin Prof. dr. J. P. Pekola Helsinki University of Technology, Helsinki

Prof. dr. H. W. M. Salemink Technische Universiteit Delft Prof. dr. C. Walker University of Arizona, Tucson Prof. dr. W. Wild Rijksuniversiteit Groningen

Front: The picture shows a superconducting hot electron bolometer (HEB) for het-erodyne detection of radiation at frequencies above 1.5 terahertz (i.e. equivalent to a wavelength shorter than 200 µm). The center is a niobium nitride (NbN) supercon-ducting bridge with nano/sub-micron dimensions which connects to an on-chip spiral antenna via additional contact pads. The strip covering the bridge is a left-over of the processing.

Casimir PhD Series, Delft-Leiden 2006-17. ISBN-10: 90-8593-024-3

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Introduction

1.1 Why high resolution spectrometry at terahertz

frequencies

1.1.1 Introduction

From early history, mankind has been deeply interested in observing the universe - of which the star spangled sky is known to all of us. One of the best documented an-cient cultures BC, situated between the Eufrates and Tigris river, already addressed astronomy in many of its clay tablets [1]. The observations were likely performed from temple towers (Ziggurats), pyramidical buildings of up to 100 m high with a flat top (like the tower of Babel). The knowledge on star positions served as naviga-tional aid and calender. Apart from these practical motivations, early astronomy was deeply interwoven with religion. The task of the priest-astronomers was to advise the monarch on questions of “war and peace” by reading the will of the gods from the position of celestial objects. In different eras and cultures, the type of questions that astronomy addresses has progressed and changed many times. The culmination of this is that we now hope to find answers about: What is the start of our universe, the so-called “Big Bang”? How do stars and planets form? Which conditions determine whether life can be initiated? Is earth the only life-sustaining planet in the universe? It is partly the fascination for these type of questions that motivates scientists and their sponsors. Up to about six decades ago practically all observations were limited to the visible range because the earth’s atmosphere is largely opaque except for a few frequency windows. However, approximately 98 % [2] of all photons in the universe emitted since the Big Bang are in the infrared wavelength range (300 µm - 700 nm), several orders of magnitude larger than the visible wavelength range (700 - 400 nm). The increased accessibility of space marks the start of a new and exciting era for astronomy. Observations can now be extended into the infrared. Apart from observations of intensity (direct detection), comparable to the familiar pictures from

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telescopes in the visible range, this frequency range holds more promises. At the lower frequency end of the infrared, up to several terahertz (THz), frequency and phase resolved (heterodyne) measurements similar to FM-radio reception becomes possible.

Many molecular species, crucial to astrophysical processes and planetary science, have isolated and bright rotational or vibrational emission/absorption lines at THz frequencies, roughly defined here between 1 THz to 10 THz corresponding to wave-lengths between 300 µm to 30 µm. Thus, by doing heterodyne measurements, relevant molecular species can be identified together with detailed information about veloc-ity (Doppler shift), temperature and pressure (broadness of the line). However, an important part of the total frequency range of interest, i.e. between 1 to 6 THz, is still largely inaccessible because no suitable heterodyne receivers exist. We present several examples (Section 1.1.2) indicating why this range is particularly interesting for astrophysical and planetary science. Not only extraterrestrial objects but also our own planet’s atmosphere can be investigated from high altitude (Section 1.1.3).

To enable such heterodyne observations beyond 1 THz, the major challenge is the development of suitable heterodyne receivers for this frequency range, which is the focus of this Thesis.

1.1.2 Astronomical observations

Recently, the High Elevation Antarctic Telescope (Section 1.3) was proposed. It aims to enable astronomical observations in the THz range. The detectors developed as a result of this research will be used for HEAT. Because HEAT (and other projects) indirectly motivates the detector development we shortly discuss several of its science goals.

Star-formation and molecular cloud formation

From our own Milky Way to the furthest galaxies that are still being formed, the internal evolution is defined by processes depending mostly on the interstellar contents. The evolution of our Galaxy is strongly connected with the life-cycle of the interstellar dust and gas. This life-cycle consists of the following three steps:

1. Transformation of neutral, molecular gas clouds into stars and clusters (star formation).

2. Interaction of the interstellar medium (ISM) with the young stars that are born from it, a regulator of further star formation.

3. Return of enriched stellar material to the ISM by stellar death, eventually to form future generations of stars.

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1.1. WHY HIGH RESOLUTION SPECTROMETRY AT TERAHERTZ FREQUENCIES 3 between ∼ 1 to 6 THz (Table 1.1). Carbon, for example, is found in ionized form (C+_),

eventually becoming atomic carbon (C), then molecular as carbon monoxide (CO) in dark molecular clouds. The formation of interstellar clouds is a prerequisite for star formation, however, the process itself has not yet been observed! High resolution THz spectroscopy on C+_{, C, CO, and N}+_{line emission measurements [3, 4] can provide new}

insight into the relationship between interstellar clouds and star formation, a central component of galactic evolution. Although we are now beginning to understand star formation [5], the molecular cloud’s formation, evolution and destruction still remains shrouded in uncertainty.

Table 1.1: Several molecular emission- and atomic finestructure-lines [6, 7].

Species Frequency (THz) C 0.809 N+ _1.461 CO 1.726 C+ _1.901 N+ _2.495 H2O 2.640 H2 2.670 OH 3.545 O 4.746

1.1.3 Atmospherical observations

Airborne heterodyne THz spectrometry of molecular lines in the upper troposphere and lower stratosphere can provide new insights into the earth’s atmospherical pro-cesses. The detectors developed during this research are planned to be used for the atmospherical “Terahertz LImb Sounder” (TELIS) mission (Section 1.3). Here we shortly describe an important topic that future observations with TELIS may adress.

Atmospherical ozone

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ground level, the stratosphere will cool. It is clear that a cooler stratosphere affects the radiative balance in the rest of the atmosphere, possibly resulting in unforeseen climatic changes. A complete understanding of ozone chemistry is mandatory in order to forecast future changes in ozone levels. The ozone concentration depends strongly on a chemical life cycle involving many gas species, being catalysts and intermediate reaction products.

One of the most important short lived species in ozone chemistry is the hydroxyl (OH) radical [10]. Despite its importance, there are only a few and short measure-ments [11] reported in the literature. Due to the low abundance of OH in the strato-sphere its detection is problematic using current measurement techniques, e.g. laser-induced fluorescence [11]. However, it is one of the main species in the HOx cycle, while also playing an important role in the COz and NOx catalytic cycles [9]. THz spectroscopy of OH lines can reveal the concentration profiles as a function of lati-tude, longilati-tude, altilati-tude, season, and time of day and would substantially improve our knowledge of stratospheric chemistry. This could, in turn, improve future pre-dictions on ozone depletion and climate modelling. Also, the effectiveness of reduced anthropogenic emissions, especially of chlorofluorocarbons (CFCs) and the effects of their substitutes can better be studied.

1.2 Heterodyne receiver system

1.2.1 Principle of heterodyne spectroscopy

Heterodyne detection is a method for detecting radiation by non-linear mixing with radiation at a reference frequency. The mixing down-converts the radiation (astro-nomical signal) at a frequency fstogether with the reference signal, called the local oscillator (LO) signal, at a frequency fLOto an intermediate frequency fIF=|fs−fLO|. Fig. 1.2 schematically illustrates the heterodyne down-conversion process. The IF signal (Fig. 1.2) contains amplitude and phase information of the astronomical signal around fLO. With the help of a spectrometer the spectral components of fs can be analyzed. The fundamental spectral resolution of the IF signal is determined by the LO line-width. The accessible spectral range of fs around fLO is in practice limited and depends on the frequency roll-off of the mixer called the “IF bandwidth” (see Appendix B1 and B3).

As an illustration of the mixing process a simple nonlinear device is considered, characterized by a quadratic current (I) versus voltage (V ) curve, i.e. I = αV2_.

If two signals with slightly different frequencies, VLO = ALOsin(2πfLOt) and Vs = Assin(2πfst) are applied to the device, the total voltage V across the device will be Vs+ VLO. The resulting current through the element becomes:

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1.2. HETERODYNE RECEIVER SYSTEM 5

Optics Mixer IF amplifier Filter Spectrum analyser LO signal

Signal

Figure 1.1: Schematic layout of a heterodyne receiver. The two signals are combined and coupled to the mixer. The signal to be detected has a higher frequency than the LO signal (upper sideband). At the output of the mixer, the intermediate frequency (IF) signal is amplified, filtered and measured by a spectrum analyzer. Intensit y (a.u.) Frequency (GHz) fLO fTHz fIF 3000 3001 1 LO signal astronomical signal IF signal

Figure 1.2: Schematic representation of heterodyne detection. The signal of interest with a frequency fs=3001 GHz is combined with a strong local oscillator signal (fLO=3000 GHz) and down-converted to an intermediate frequency fIF of 1 GHz.

the signal at the original frequency to a lower frequency fIF = |fLO− fs|. By filtering out all higher frequency terms, the output voltage of the device is then:

VIF ∝ αALOAscos(2πfIFt) (1.2)

Inspection of fIFin Eq. 1.2 demonstrates that both the signal at fs=fLO+ ∆f GHz above (upper sideband, example shown in Fig. 1.2) and below fLO - ∆f GHz (lower sideband) are down-converted to to the same IF frequency fIF = ∆f GHz. This is called a double sideband (DSB) receiver. In contrast, a single sideband (SSB) receiver only down-converts the signal either in the upper or the lower sideband to fIF. The transformation of the astronomical signal in one of the sidebands to the IF signal involves a power conversion efficiency that we call Gmix.

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1.2.2 Challenges for terahertz heterodyne receivers

A terahertz heterodyne receiver (Fig. 1.1) consists of a LO source, mixer, optics, amplifiers, filter and a spectrum analyzer. The main challenge for a heterodyne THz receiver is to perform the astronomical observations as time-efficient as possible. In this context, the performance of the receiver is mostly determined by the mixer and, for frequencies above 2 THz, the LO source. Therefore we will focus on the mixer element and (to a lesser degree) on the LO source. We will not address the other elements in this Thesis.

One of the mixing element’s key parameters is its sensitivity. Observation of astro-nomical line emissions requires a highly sensitive receiver system, i.e. with a low Trec. The astronomical sources are very weak and therefore require a long integration time t to achieve an acceptable signal to noise ratio where t is proportional to T2

rec(see ap-pendix B3 and [7, 16]). The operational lifetime of a liquid-helium cooled space-based platform is limited while the number of observation requests is overwhelming. Hence, to facilitate as many observations as possible, the sensitivity of the mixing elements is of key importance. Another important mixer parameter is the IF bandwidth, also relevant to the measurement efficiency. A larger IF bandwidth scales inversely with the required measurement time for spectral line surveys. Also, a large IF bandwidth allows for the simultaneous observation of several lines, or the instantaneous measure-ment of a single broad line. Furthermore, for higher and higher frequencies beyond 1 THz, a large IF bandwidth is a prerequisite for a sufficiently broad velocity coverage (Appendix B3) because of the scaling of the Doppler shift with frequency.

For spectroscopy at a frequency around 2 THz and above, the limited LO power from currently available tunable LO sources [12] is a crucial issue. A mixing element needs a certain amount of LO power to be of practical use. Until recently, no compact LO source existed for frequencies between 2 - 6 THz delivering sufficient LO power. Both the reduction of the LO power requirement on the mixer side as well as the development of new LO sources are needed. The LO output power of currently available tunable solid state LO sources [12] falls off rapidly above 2 THz due to reduced multiplication efficiency. Optically pumped gas lasers can operate at several discrete frequencies in the 2 to 6 THz range, but are in general massive, bulky, and require power in the kW range. Recently, a new type of solid-state THz source was invented called a “quantum cascade laser” (QCL) [13, 14]. This new source holds great promise for application as LO source [15] because of its compactness and high power efficiency.

1.2.3 Mixer types

The variety of devices with nonlinear characteristics that can be used as a mixer is limited and only few are practical for our purposes. We will limit ourselves to a de-scription of the three most practical mixers for THz spectroscopy.

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1.2. HETERODYNE RECEIVER SYSTEM 7 Schottky-mixers use the non-linear current voltage characteristic of a metal - semi-conductor diode for mixing. The non-linearity in a Schottky diode arises from a metal-semiconductor interface where a voltage dependent potential barrier occurs at the boundary layer. Schottky-mixers work over a wide frequency range (up to several THz) and do not require cooling to cryogenic temperatures. They can therefore be used when cryogenic cooling is not possible or unfavorable. The required LO power, however, is rather high, 3-8 mW [17, 19] at the mixer. Their sensitivity is relatively low. At 585 GHz, the best Trec of Schottky receivers has been reported to be 2380 K at room temperature, and 880 K at 4.2 K [18]. At room temperature and at higher frequencies the receiver noise temperature of Schottky receivers increases and reaches ∼ 8000 K at 2.5 THz [19] and 70000 K at 4.75 THz [20]. However, the bandwidth of Schottky-diodes can be large, exceeding 50 GHz [21].

Superconductor-insulator-superconductor mixers

Superconductor-insulator-superconductor (SIS) mixing is based on the principle of photon-assisted tunneling [22] working up to two times the energy gap of the su-perconductor. Beyond that, at a frequency corresponding to 2 to 4 times the energy gap, the photons break Cooper pairs in the electrodes, causing part of the incoming signal (and sensitivity) to be lost. Using superconductors with a high energy gap, like NbN and NbTiN, the upper frequency limit of SIS detectors has been pushed up to 1.2 THz [23]. The required LO power can be relatively low, in the µW range. A key advantage of SIS mixers is their near quantum limited sensitivity, especially at fre-quencies that are low compared to the superconductor’s gap frequency. At 880 GHz best Trec is ∼ 250 K and ∼ 1000 K at 1.2 THz [24], much better than for Schottky diodes. The intrinsic IF bandwidth of SIS mixers can be very large. However, to achieve high IF bandwidth in practice, the capacitance intrinsic to the tunnel junc-tion needs to be tuned out by the geometric design of the device. Using this approach, large IF bandwidths of the order of several tens of GHz have been achieved [24].

Hot electron bolometer mixers

Hot electron bolometers rely on a bolometric effect and use the power-law as the mixing principle. “bolometer” literally means “heat detector”, i.e. V2_{/R ∝T(t).}

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950 K at 2.5 THz [25] and 1050 K at 2.8 THz [26]. The LO power requirement for HEB detectors is very low. A minimal LO power requirement as low as 30 nW at the mixer element has been demonstrated [27].

The IF bandwidth (several GHz) is smaller than for SIS or Schottky mixers. Al-though this is considered low for many applications, it is still enough for practical use (Appendix B3). Overall, HEB mixers are the best alternative for heterodyne receivers for the frequency range between 1 to 6 THz.

Depending on the mechanism controlling the IF bandwidth, a distinction is made between “phonon cooled” and “diffusion cooled” HEB’s. Diffusion cooled (Nb) HEB’s have been introduced by Prober [28] and rely on the out-diffusion of the hot electrons as the energy relaxation mechanism that causes the roll-off of the Gmix at high IF frequencies. Phonon cooled HEB’s, introduced by Gershenzon et al. [29] mainly rely on phonon cooling involving electron-phonon interactions. Experimentally, the IF bandwidth is found to be higher for phonon cooled HEB’s. Therefore we concentrate on phonon cooled NbN HEB’s.

Furthermore, we can distinguish between quasi-optical and waveguide HEB re-ceivers depending on which technique is used to couple the astronomical signal to the mixer. Since waveguide machining becomes challenging for frequencies in excess of 1.5 THz, we have chosen for the quasi-optical approach. As an illustration of their practical relevance we note that such quasi-optical phonon cooled NbN HEB’s have been selected for ground based [30, 31], airborne [32] and spaceborne [33, 34] projects.

1.2.4 Research focus and thesis outline

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1.3. IMPACT ON ASTRONOMY 9 (Chapter 6). The ultimate demonstration of an actual heterodyne receiver for the range between 2 - 6 THz was greatly helped by quantum cascade lasers (QCL) be-coming available. Using the QCL as a LO source we successfully characterized a fully operational heterodyne receiver based on a HEB (Chapter 7).

1.3 Impact on astronomy

As a result of this work, HEB mixers have become technologically much more reliable and have convinced astronomers to such an extent that a new instrument has been proposed [8]. The mixers developed in the course of this research are intended to be used for the High Elevation Antarctic Telescope (HEAT). HEAT is developed by an international consortium led by the University of Arizona and its first observations are hoped to take place within several years. The receiver can operate up to 2 THz by using the HEB together with a solid state local oscillator, similar to the system described in Chapter 5 of this Thesis. HEAT will be located at Dome A on Antarctica which has an exceptionally high atmospheric transmission in the THz range due to its high altitude and dry air conditions. Such transparency promises to enable the first ground based astronomic observation of the C+ _{line at 1.9 THz in addition to}

the detection of C, CO and N+ _{THz line emissions. HEAT will perform pioneering}

surveys of the Galactic Plane and the Magellanic Clouds.

Our HEB mixers are also planned to be used on the second flight of the TEra-hertz and submm LImb Sounder (TELIS) [35]. TELIS will operate from a balloon platform at an altitude of 30 to 40 km and will provide measurements of atmospheric constituents including OH, HO2, O3, N2O, CO, HCl, HOCl, ClO, and BrO that are

associated with the depletion of atmospheric ozone and climate change. TELIS is being developed by a consortium of major European institutes led by the Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR) in Germany.

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AASTINO/PLATO HEAT

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Chapter 2

Physics of hot electron

bolometer operation

∗

2.1 Hot electron bolometer devices

The hot electron bolometer (HEB) consists of a superconducting strip of a thin film of niobium nitride (NbN) that typically covers an area of 1 µm by 0.1 µm up to 4 µm by 0.4 µm. The NbN strip is connected to an antenna of a highly conducting material to minimize conductive losses. The antenna is used to receive the terahertz (THz) radiation from space and to efficiently couple the signals into the superconducting strip via contact-pads. A sketch of the HEB structure is shown in Fig. 2.1. It shows the NbN strip (black), the contact-pads and (part of) the antenna structure. In Fig. 2.2 scanning electron micrograph (SEM) images are shown of the typical types of HEB devices used in this research. Fig. 2.2(a) shows a SEM image of a spiral antenna of which a magnified view of the NbN strip is presented in Fig. 2.2(b), seen protected by negative resist (dark layer). Fig. 2.2(c) shows a SEM image of a twin-slot antenna and a magnified view of the bridge in Fig. 2.2(d). The spiral antenna couples a relatively broad frequency range (non-resonant) and is polarization insensitive, whereas the twin-slot antenna is narrow band (resonant) and polarization sensitive.

The physical properties of the NbN film are essential to the HEB’s operating principle. With THz radiation applied, Cooper-pairs are broken generating quasi-particles. At the very low temperatures at which we are working, the electrons are to a certain degree decoupled from the phonons. The electron gas can therefore be ∗_{The material described in this Chapter has been published in the following papers: M. Hajenius,} R. Barends, J. R. Gao, T. M. Klapwijk, J. J. A. Baselmans, A. Baryshev, B. Voronov, and G. Gol0_{tsman in the IEEE Transactions on Applied Superconductivity, 15, 495 (2005),}

R. Barends, M. Hajenius, J. R. Gao, and T. M. Klapwijk in Applied Physics Letters 87, 263506 (2005),

and as a contribution to the Proceedings of the 15th_{International Symposium on Space Terahertz} Technology, edited by M. Yngvarson, J. Stake, and H. Merkel, (Chalmers University of Technology, G¨oteborg, Sweden, 2005), p. 381, and is authored by M. Hajenius, J. J. A. Baselmans, J. R. Gao, T. M. Klapwijk, P. A. J. de Korte, B. Voronov, and G. Gol0_tsman.

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Si substrate Bridge 10 nm NbTiN NbN thin film 40 nm Au Contact-pads Antenna structure 150 nm Au

Figure 2.1: Sketch of the hot electron bolometer structure. The niobium nitride bridge (black) is in between the contact-pads. The contact-pads can consist of a stack of materials (typically 10 nm NbTiN and 40 nm Au). Also shown is part of the thick Au antenna layer (typically 150 nm) on the NbN film and partly on the contact-pads.

treated separately from the phonon system. The electrons can have an equilibrium temperature that is higher than the lattice (phonon) temperature, hence they are called “hot electrons”. Since the electrons are heated and the heat capacity ce is small, the corresponding thermal response time can be short [1]. However, cooling of the electrons generally involves the transfer of energy to the phonons (hence phonon cooled), which subsequently escape into the substrate. The shorter phonon escape time for thinner films and hence larger IF bandwidth [2] motivates the use of ultra-thin NbN films.

For both metals and semiconductors, the increase of electron temperature leads to an increase in resistance. In contrast to semiconductors, the resistance in normal metals is only weakly dependent on temperature. However, in superconducting metals the resistance arises due to the gradual breakdown of the coherent superconducting state and is very strongly dependent on the electron temperature around the critical temperature Tc. The resistive transition of the NbN film is at the heart of the mix-ing principle. Yet, there is no complete understandmix-ing about the complex physical processes leading to the HEB’s mixer performance.

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2.1. HOT ELECTRON BOLOMETER DEVICES 17 (a) (c) (d) (b) 2 µm 4 µm 20 µm 20 µm

Figure 2.2: (a): Optical microscope image of an Au spiral antenna coupled NbN HEB. The NbN bridge is in the center of the spiral. (b): Scanning electron microscope (SEM) image of the NbN bridge with contacts to the spiral antenna on both sides. The NbN bridge shown here is protected by a resist mask (dark structure). The size of the NbN bridge is 4 µm by 0.4 µm (c): Optical microscope image of a twin-slot antenna coupled HEB. (d): SEM image of the NbN bridge and contacts to the twin-slot antenna on both sides. The bridge shown here is protected by an e-beam resist mask (dark structure). The size of the NbN bridge is 1 µm by 0.1 µm.

formed by the contact. The latter has an important implication: the electron diffusion to the contacts gives rise to a temperature gradient, because of which the highest electron temperature will be in the center of the device while decreasing sideways towards the contacts (Fig. 2.3). The application of both DC and LO bias gives rise to

0.00 0.25 0.50 0.75 1.00 6 7 8 9 Te (K ) x/L

Figure 2.3: Calculated electron temperature profile along the length of the bridge at PLO=90 nW, no DC bias. The metal contacts on both sides of the bridge are assumed to be at a phonon temperature of 4.2 K. The calculation is performed using the heat-balance approach described in Section 2.5.

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each other. The result is that under operating conditions several intertwined processes occur simultaneously, while on top of that there is a spatial variation of parameters (e.g. electron temperature) across the device. It is therefore virtually impossible to learn about the underlying mechanisms based purely on measurements of the device under full operating conditions.

R(T) DC bias LO Bias LO Physics Geometry Material R(T,I) P (t)IF PSignal I(V,P )

Figure 2.4: Diagram of the HEB’s device physics. On the two axis of the diagram are the DC bias (horizontal direction) and LO bias (vertical direction). At the lower left corner of the pyramid is R(T ), which is the resistance as a function of temperature; at the lower right is the R(T, I), the resistance as a function of temperature plus DC current; at the top of the pyramid (under full operating conditions) is the I(V, PLO) which is the current voltage characteristics a function of bias voltage and local oscillator power. With the astronomical (THz) signal applied to the device under full operating conditions, mixing with the LO signal results in an output power, modulated at the IF frequency, containing phase and frequency information of the unknown signal. The relations between the measurable parameters on the pyramid’s corners are defined by the HEB’s physics, material parameters and device geometry, as indicated in the center of the pyramid.

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2.2. MATERIAL PROPERTIES OF THE NIOBIUM NITRIDE THIN FILMS 19

2.2 Material properties of the niobium nitride thin

films

Ultra-thin NbN films are sputtered onto a highly resistive silicon (Si) substrate with native oxide by reactive magnetron sputtering in an Ar-N2 gas mixture. The NbN

films are realized at the Moscow State Pedagogical University in Russia using a Z-400 Leybold Heraeus sputtering system. Based on the sputtering time and rate calibrated for a thick film, a thickness around 3.5 nm is expected. The typical sputtering condi-tions are listed in Table 2.1.

Glue

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NbN

Si

SiO

_x

(2)

(1)

NbN

Glue NbN Si SiO_x (b) (a) Si SiO_x NbN 5 nm 5 nm

Figure 2.5: High resolution transmission electron microscope (HRTEM) micrographs of two thin NbN films grown on a Si substrate with native oxide layer. (a) (Top) shows a film with Tc of 9.4 K. The glue used to prepare the specimen for HRTEM inspection is removed. The magnified view shows a polycrystalline structure of the NbN. (b) (bottom) shows a film with Tc of 9.8 K. In this case, the glue on top of the NbN remains.

The crystalline structure and the thickness of the NbN film are evaluated through high-resolution transmission electron microscopy (HRTEM)1 _{[6]. Fig. 2.5 shows two}

HRTEM micrographs of the NbN film on Si, together with a magnified view of the upper one. We find a lattice constant of 4.39 ˚A for the NbN thin film. As shown by the inset, such films have a polycrystalline structure. The upper part of Fig. 2.5 shows that a NbN film on Si has a thickness around 5.5 nm. The measured sheet

1_{F. D. Tichelaar at Kavli Institute of NanoScience, Faculty of Applied Sciences, Delft University}

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Table 2.1: Fabrication parameters of the NbN films at MSPU, Russia.

Process parameters Values Residual pressure 1.5·10−6 _mbar Ar partial pressure 5·10−3 _mbar N2 partial pressure 9·10−5 mbar

Substrate temperature 800◦_C Discharge current 300 mA

Bias Voltage 300 V

Deposition rate 0.5 nm/s

resistance, defined as the resistance for one square of the film at 300 K (R¤,300K)

equals 620 Ω while the film in the lower part of Fig. 2.5 has a thickness of 6 nm and a R¤,300K=530 Ω. Note that previously a film thickness of 3.5 nm was assumed [7, 8]

based on the sputter rate calibrated for thick NbN films, which clearly differs from the value obtained using HRTEM inspection.

2.3 Intrinsic resistive transition of a niobium nitride

strip

First we consider the dependence of the resistance on electron temperature R(T ), with neither LO nor DC power applied (Fig. 2.4). The dependence of R(T ) of the superconductor around its critical temperature Tc, the so-called resistive transition, is key to the bolometric effect that enables mixing. R(T ) curves of thin superconductors have been studied extensively in the literature. The emergence of resistance in highly resistive 2-dimensional films is controlled by the Kosterlitz-Thouless phase transition Tc,KT [9], which leads to a broadening of the transition for increasing (normal state) resistance of the film. The NbN film used is near the borderline of the superconductor-insulator transition due to the level of disorder and thickness as indicated by R¤,300K

of 550 Ω [10]. A full understanding of such a system has not been reached, although a number of theoretical scenarios have been suggested.

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2.4. INTRINSIC RESISTIVE TRANSITION UNDER DC BIAS 21 6 8 10 12 0.0 0.5 1.0 R /R N T (K)

Figure 2.6: The resistance as a function of temperature of a 1 µm wide NbN strip. The bias current is 1 µA. The resistance is normalized to its resistance at 16 K (RN). A scanning electron microscope image of the four-terminal test-structure is shown as inset, the dark area indicates the NbN film because it is covered by resist.

Note that this is higher than the R2,300K of a large area of NbN film after processing (around 650 Ω). The reason for this is not clear to us.

As a definition for the Tc we take the temperature at which the resistance is half that of the normal state resistance at 16 K. We chose this practical definition because the mean field critical temperature is not straightforward to extract from the resistive transition. The Tc determined from the resistive transition in Fig. 2.6 is 10 K. The Tc of NbN films used by us is typically in the range between 9 to 10 K. We note that the Tc of the upper film in Fig. 2.5 with the thickness of 5.5 nm is 9.4 K. The Tc of the film in the lower part of Fig. 2.5 is 9.8 K but is also slightly thicker (6 nm). The likely reason for this behavior is that thinner films exhibit stronger disorder [11]. The disorder suppresses the superconductivity in morphologically homogeneous superconductors [12] because the diffusive character of the electron motion in dirty systems makes the Coulomb interaction more effective [13]. As a result, the attraction between the electrons in Cooper pairs becomes weaker, and the Tc decreases.

2.4 Intrinsic resistive transition under DC bias

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gen-erating a voltage in the superconductor due to the Josephson relation. However, a current passing through the system contributes to the generation of free vortices by breaking the vortex-antivortex pairs [14]. As a qualitative illustration we consider a simple expression from Kadin et al. [14] for the resistance of a superconducting film:

6 8 10 12 0.0 0.5 1.0 10-3 10-2 10-1 10 -2 10-1 10 I ( µA) 1 5 10 15 20 R/R N T (K) I/IC 1-T_C(I)/T_C(0) 0

Figure 2.7: The intrinsic resistive transition as a function of applied DC bias current of a 1-µm-wide processed NbN film. The steepness changes little while effectively the critical temperature, taken as the midpoint of the transition, shifts with increasing bias. This relation is depicted as inset.

R = RN2πξ2NF(J, Te) (2.1)

in which RN is the normal state resistance, 2πξ2 _{the area of a vortex core and}

NF(J, Te) the density of free vortices, depending on the current density J and tem-perature Te. Although the dissipation leads to a higher temtem-perature of the electron vortex system, denoted by Te, the current breaks the vortex-antivortex pairs by the Lorentz force even in the absence of a temperature rise. This is the key assumption, which we use to understand2 _{the current-voltage characteristics of the hot electron}

bolometer devices. A full theoretical description that can account for experimental observations should include material properties, pinning sites, inhomogeneities, gran-ularity, and finite-size effects. Unfortunately such a description is impractical. Instead we use the independently determined empirical relation for the resistive transition of the NbN film in the presence of a current. The intrinsic transition for different DC current bias is shown in Fig. 2.7 and is measured similar to the one described in Section 2.3. The R(T, I) curve shifts to lower temperatures for increasing current and the apparent downshift of the critical temperature Tc obeys the empirical relation:

2_{In earlier work the current dependence has been taken into account, but in a different way than}

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2.5. HEB’S CURRENT VERSUS VOLTAGE CHARACTERISTICS UNDER LO RADIATION 23 I Ic = (1 −Tc(I) Tc(0)) γ _(2.2)

for γ =0.54, see the inset in Fig. 2.7. For different currents the R(T ) curve measured at small bias is taken with a shifted Tc. We ignore the small change in the steepness of the resistive transition for higher current bias. Thus upon approaching the crit-ical temperature the finite bias current will enhance the resistivity by creating free vortices.

2.5 HEB’s current versus voltage characteristics

un-der LO radiation

As a next step we consider the I(V, PLO), under the full operating conditions, indi-cated at the top of the pyramid in Fig. 2.4. Since fLO À 1/τth where τth is the thermal relaxation time, the electron temperature which determines the device re-sponse cannot follow the LO frequency. This means that for each LO power and current level a time-independent DC resistivity, i.e. I(V, PLO), is established. The I(V, PLO) spans the full range of bias points at which the device can operate. Previ-ous attempts to predict the current-voltage characteristics [5, 15] use a similar heat balance approach (Eq. 2.3) as employed here but utilize a different dependence of the local resistivity on the temperature and applied current. The common feature of these previous models is that their predictions all display a strong disagreement with the actual device response at the optimal operating point. The inset [16] in Fig. 2.10 shows similar calculations using the intrinsic NbN resistivity as input and demonstrates that around optimal operating conditions (100 nW calculated curve and around 0.5 mV DC bias) the calculated curves do not match the measured charac-teristics. We argue that this discrepancy is caused by the absence in the modelling of the current dependence of the resistivity. By introducing a current dependent re-sistive transition as input to the modelling we can for the first time correctly predict the measured I(V, PLO) as will be shown in this section.

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Electrons T e, c e Phonons T ph, c ph Pe-p Pp-e ps P diff, out Substrate, T_b P Electrons Te ce Phonons T (x)p , cp P (x), LO P (x) DC Contact pad Contact pad x=0 x=L dx P diff, in

Figure 2.8: Schematic picture of the power flow in a small segment of the bolometer bridge. In the upper part a schematic top-view of the HEB bridge is drawn with the bridge segment under consideration in gray and contact-pads on either side of the bridge.

equations. In the distributed temperature model the one dimensional NbN bridge is divided into a large number of sections between x = 0 to x = L, each of length dx. For each section, the electrons’ net heat input consists of LO power (PLO) and DC power (PDC). Since hf À 2∆ the PLO power absorbtion is assumed to be uniform along the bridge. The DC power per unit volume is generated locally, depending on the position dependent resistivity and equals PDC = J2_{ρ(x, J). The net power outflow}

is by diffusion (dTe

dx) and through energy transfer between the electron and phonon subsystems Pep (assumed constant). The phonon system can be described similarly. The energy input for each slice is Pep while the energy outflow consists of phonon escape to the substrate Pps (assumed constant) and a diffusion term d

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2.5. HEB’S CURRENT VERSUS VOLTAGE CHARACTERISTICS UNDER LO RADIATION 25 for the temperature Te(x) and the phonon temperature Tp(x) for each section of the bridge of length dx are:

d dx(κe d dxTe(x)) + PDC(x) + PLO− Pep= 0, (2.3) d dx(κp d dxTp(x)) + Pep− Pps= 0 0.00 0.25 0.50 0.75 1.00 6 8 10 12 Te (K ) x/L 0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 R(x)/ R 2 3 4 5 V (mV) 0 0.5 1 N (a) (b)

Figure 2.9: (a) (upper) Local resistivity and (b) (lower) temperature profiles, both at PLO=90 nW (optimal power) and for different DC bias voltages, calculated using Eq. 2.3 and the current-dependent intrinsic resistive transition in Fig. 2.7. The voltage bias for the optimal operating point is around 0.5 mV.

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0 1 2 3 4 5 0 10 20 30 40 50 calculated P LO(nW) 0 80 90 100 110 I ( µ A) V (mV) measured P LO(nW) 0 35 45 60 0 1 2 0 10 20 _calculated PLO (nW) 90 100 110 I ( µ A) V (mV)

Figure 2.10: Calculated current-voltage (I(V )) characteristics (lines) are compared to those of a small 1 µm by 0.15 µm device measured at 4.2 K with several local oscillator power PLO levels at 1.6 THz (symbols). The Tcof the device is 9 K. Using the current-dependent intrinsic resistive transition from Fig. 2.7, the model correctly predicts IV , for high as well as low bias. The LO power is determined using isothermal technique. The difference between pumping power needed for measurements and for calculations is attributed to uncertainties in the input parameters. The inset (extracted from [16]) shows the predicted I(V ) characteristics (lines) by using the measured resistive transition as input without taking into account its current dependence. In the inset the same measured I(V ) curves with the same (smaller) symbols are plotted as in the main Figure.

The voltage V is connected to the bias current through the relation:

V = J Z

ρ(x, J, Te(PLO, PDC))dx (2.4)

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2.6. THERMAL RESPONSE TIMES 27

2.6 Thermal response times

With both LO and the astronomical signal applied to the mixer, beating of both results in the modulation of the absorbed power in the NbN strip (see Section 1.2):

(VLO+Vs)2

R ∝ ALO Ascos(2πfIFt). This, in turn, gives rise to the modulation of the electron temperature (thermal response) at the IF frequency. Thanks to the bias point chosen, e.g. using the steep resistive transition, this translates into a modulation of the device resistance which can be read out as an IF signal (PIF). This is the basic principle of bolometric mixing in the HEB. An important quantity is the maximum bandwidth for the IF frequency, which is experimentally found to be around 3 GHz. The mechanism governing the IF bandwidth is related to the (time dependent) resistive response. The analysis of the DC properties presented in the previous paragraph has identified that the resistivity is due to both electron heating and the DC current (through the vortex-density). Since our LO-frequency is above 1 THz the signal is much faster than the time dependence of the superconducting energy gap, ~/∆. Therefore we assume that the mixing is due to a bolometric effect. At the IF frequency a second order contribution to the impedance, due to the IF-currents, might be present. However, we will assume3 _{that the time dependence is controlled}

solely by the energy-relaxation of the electrons.

In order to model the thermal response we construct a modified heat balance equation compared to the one in the previous Section. Here, we include periodic THz irradiation to represent the IF signal. We take into account the temperature depen-dencies of the relevant parameters. We however assume a constant temperature across the device (lumped element), essentially neglecting out-diffusion to the contacts. The electron out-diffusion is assumed to have a negligible effect on the temporal response. This assumption is supported by the fact that for a long bridge of 400 nm diffusion only4 _{reduces the total thermal relaxation time by about 15 %. A diagram of the}

heat flow is shown in Fig. 2.11.

In previous work on the HEB’s thermal time constants [21] attention was paid primarily to the electron-phonon relaxation times and phonon escape time. As re-ported by us [22] we extend this analysis by including the temperature-dependent heat capacities of the electron and phonon-system at their respective temperatures. This type of heat balance approach was originally introduced by Perrin and Vanneste (PV) [23] allowing for different temperatures for phonons and electrons, which are at elevated temperatures compared to the bath temperature. This two-temperature model has been successfully used to analyze the response of superconducting thin films to periodic optical (high frequency) signals [24, 18].

To make plausible that this model can also be applied to the HEB system we need to discuss one assumption. First, it should be pointed out that the PV model is derived for the case that Te∼ Tph. However, for a HEB mixer operated practically at Te= Tc while the substrate temperature is much lower than Tc, the phonon temperature is

3_{This assumption deserves a more extensive analysis (currently ongoing).} 4_{Using a diffusion constant for NbN of 0.45 cm}2_{/s [19] and the equation τ}

dif f= L

2

b

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Electrons T e, c e Phonons T ph, c ph τ _e-p ps Substrate, Tb Periodic THz irradiation Electrons T (t) ,e c e Phonons T (t) p , c p τ Contact pad Contact pad x=0 x=L τ _p-e

Figure 2.11: Energy flow in a two-temperature model. Periodically modulated THz radiation is absorbed by the electrons (Cooper Pairs), transferred to phonons and subsequently transported to the (electrically insulating) substrate. The symbols Te, Tp, ce, cp, and τ are the electron temperature, phonon temperature, electron specific heat, phonon specific heat, and time constant, respectively.

not necessarily close to the electron temperature. To estimate the difference between Teand Tph under operating conditions, we consider the calculations from Section 2.5 of both Te and Tph presented in Fig. 2.9. In that case Tph is about 0.8 Te in the center of the bridge. In view of the small difference between Te and Tphwe conclude that our assumption is reasonable.

2.6.1 Linearized two-temperature model

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2.6. THERMAL RESPONSE TIMES 29 cedTe dt = p(t) − ce τe−ph (Te− Tp) −ce τd (Te− Tb) (2.5) cpdTp dt = ce τe−ph(Te− Tp) − cp τes(Tp− Tb) (2.6)

where ceand cpare the electron and phonon specific heat, respectively. Tb is the bath temperature, τe−ph the electron-phonon interaction time, τes is the time of phonon escape from the film into the substrate and τd the effective diffusion time. p(t) is the input radiation power absorbed in a unit volume of the film. We are interested in the response of the electron temperature of a superconductor film to a time-dependent power input p(t) since the resistance depends on the electron temperature in the same way as varying the bath temperature. For simplicity we also assume that the input power has the form p0(1 + cos(ωt)) i.e meaning a constant as well as an oscillatory

power input. Here we assume that the PV model can be used to describe the tem-perature response to the IF signal, i.e. the frequency dependent p(t) is assumed to be the power of the IF signal. Note that the temperatures Te and Tph have to be close to Tb, given the linearized form of the energy balance equations. To derive these two equations, the condition: cph/τph−e=ce/τe−ph has been used, resulting from the energy balance [25], with τph−e the phonon-electron scattering time. The diffusion time has been added as a reminder of the fact that in a real system an additional contribution to the energy balance equation can be out-diffusion to the cool reservoirs, the contact-pads.

The solution for the electron temperature modulation ∆Te(ω), which is the AC component of Teand the complex amplitude, as a function of frequency of the exci-tation is:

∆Te(ω) ∆Te,0 =

1 + iωτ3

(1 + iωτ1)(1 + ωτ2) (2.7)

where ∆Te,0 is the solution for the excitation frequency at the limit of ω = 0 and the time constants τi for the case that the diffusion is ignored are defined as:

τ1−1+ τ2−1 = τes−1+ τe−ph−1 ce cph + τ −1 d (2.8) τ−1 1 τ2−1= τes−1τd−1+ τd−1τe−ph−1 ce cph + τ−1 e−phτes−1 (2.9) τ−1 3 = τes−1+ τe−ph−1 ce cph (2.10)

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resistance that equals the impedance of the IF amplifier. As will be shown elsewhere, the gain and IF impedance expressions derived in Ref. [26] describe the measured data reasonably well [28].

2.6.2 Temperature-dependent time constants

It is known that a wide bandwidth, which depends on the energy transfer rate, can only be achieved with superconductors with a strong electron-phonon interaction. In an ideal metal with free electrons, the electron-phonon relaxation rate follows τ−1

e−ph= αTe3, where α is a material dependent constant [29]. This rate is in general relatively low, insufficient for application as a mixer. For thin and dirty metal films, where the electron-mean free path is very short, the electrons mainly interact with transverse phonons due to impurities [30]. This may lead to a quadratic temperature dependence τ_e−ph−1 = αT2

e. Since there is no universally valid relation for τe−ph(Te), one has to rely on experimental values for a specific material of interest. This statement also holds for the process in which the phonons transmit power to the substrate via the interface of film and substrate. The phonon escape time τes can be described as linearly depending on the film thickness d through τes = 4d/ηu, where η is the acoustic transparency of the interface between film and substrate and u is the sound velocity [32]. However, the acoustic transparency has to rely on experimental data. Thin NbN films are currently popular because of the strong electron-phonon relaxation and also the relatively high Tc despite of the small film thickness. From HRTEM studies, reported in Section 2.2, we know that the NbN film has a thickness of 6 ± 1 nm. The fluctuation of 1 nm in the thickness is derived from measurements of different films and on different locations across the same film. We focus on an analysis for devices based on one particular NbN film, although this theory is in principle extendable to other films. Thin films of NbN can have different Tcdepending on deposition conditions, substrate, and thicknesses. These different conditions can influence the mean free path and the dirtiness of the film. Consequently, they influence the strength of the electron-phonon interaction. To understand the time-dependent response one needs to know how the system responds to time-dependent power-inputs at this temperature and one needs to know the temperature dependence of the various quantities appearing in Eqs. 2.7, 2.8, 2.9 and 2.10.

Experimentally the electron-phonon relaxation time τe−ph= α−1_T−n

e in NbN films was reported to be temperature dependent with n=1.6 [33] or 1.5 ≤ n ≤ 2 [34] where n is film dependent. There is no essential difference between the two experimental results, so we use one of the empirical relations, τe−ph ≈ 500T−1.6 _psK1.6 _reported

in Ref. [33]5 _{for our calculations. The escape time for phonons, τes, follows the}

empirical relation τes = 10.9 · d (ps/nm), which was determined indirectly from the measurement of picosecond photoresponse of thin NbN films on sapphire substrates

5_{Note that the paper is supposed to report the measured electron-phonon time τ}

e−ph, but the thermal time constant τthinstead of τe−phwas mistakenly plotted in Fig. 4 of this paper, according to private communication with one of the authors (G. Gol0_{tsman). Note that the τ}

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2.6. THERMAL RESPONSE TIMES 31 4 6 8 10 12 14 10 100 1 3 es e-ph 2 th 6 nm NbN on Si Time c onsta nt (p s) T_c (K) τ τ τ τ τ τ

Figure 2.12: τ1, τ2 and τ3 time constants defined in PV’s model (see Eqs. 2.8, 2.9, and 2.10) as a

function of temperature using the temperature dependencies given in the text. τe−phis the electron phonon time and τesis the phonon escape time. τthis the calculated effective thermal time constant related to the IF gain bandwidth by τth= (2πfIF)−1.

[35]. We assume that the influence of the different substrate (Si) is negligible. The electronic heat capacity [29] is given by the free electron dependence ce= γTe with γ = 1.85 · 10−4_Jcm−3_K−2 _{for a NbN thin film based on the measured resistivity and} diffusion constant [29]. The phonon heat capacity [29] is given by cph= βT3

phfor the 3D case, with β = 9.8 · 10−6_Jcm−3_K−4 _{estimated from the Debye model [29].}

Fig. 2.12 shows the various time constants as a function of temperature. The parameter τe−ph decreases with increasing temperature while τes is temperature in-dependent. The latter is determined by the acoustic mismatch and the film thickness, which in this case is 6 nm. To prepare for the calculation of the spectrum of the gain, the time constants τ1, τ2 and τ3 appearing in Eqs. 2.8, 2.9 and 2.10 are also shown.

For clarity, we note that they are calculated taking both the electron temperature and the phonon temperature equal to the critical temperature. To compare with other time constants, we also introduce the effective thermal time constant τthby relating it to the IF bandwidth via τth= 1/(2πfIF) and plot it in the figure as well. The way to derive the thermal time constant of a HEB is discussed in the following Section.

2.6.3 Mixer gain bandwidth

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moves to higher frequencies for increasing temperature. This temperature response shows up to what frequency the electron gas can follow the time-dependent power-input which is taken as a measure for the IF-roll-off in a heterodyne experiment. Since this is a 3-pole roll off, the dependence is slightly different from a 1-pole roll off (not shown) as observed usually in a diffusion-cooled HEB mixer [36]. The difference is only noticeable at high frequencies. We define the IF bandwidth (fIF) by taking the point at which the gain drops by 3 dB.

0,1 1 10 0,0 0,2 0,4 0,6 0,8 1,0 6 nm NbN on Si T_c = 6 K T_c = 8 K T_c = 10 K T_c = 12 K Relat ive m ixer G ain IF frequency (GHz)

Figure 2.13: Relative mixer gain vs IF frequency for 6 nm NbN films of different Tc. Calculations are based on PV’s two-temperature model (Eq. 2.7)

In Fig. 2.16 we show the predicted temperature dependence of the IF gain band-width for three different film thicknesses. The bandband-width increases with increasing Tc and shows a slightly stronger temperature dependence in the high Tc region than in the low Tc region. Previously an analysis [21] similar to ours has been performed. Those results are also shown in Fig. 2.14 (open circles) and predict a saturation of the bandwidth limited by τes whereas we find that an increase in operating temperature continues to be advantageous. The difference of the results can be attributed mainly to the temperature dependence of the heat capacity.

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2.6. THERMAL RESPONSE TIMES 33 4 6 8 10 12 14 0 2 4 6 8 10 12 Gain bandwidth (GHz) Tc(K) 6 nm NbN/Si 3 nm NbN/Si 3.5 nm NbN/MgO

Figure 2.14: Calculated IF gain bandwidth as a function of Tcfor two NbN thin films with different thickness (solid symbols). For comparison, one of the calculated gain bandwidth curves from Ref. [21] which is for a 3.5 nm NbN film on a MgO substrate, is also included (open circles).

large ce/cphvalue, thereby differing from the temperature dependence of τe−phwhile it becomes close to τe−ph for the smallest values of ce/cph (large phonon heat capacity). Having recognized the importance of the operation at a high electron temperature, we point out that the bandwidth is still strongly in the regime of dependence on the phonon-escape time. Hence reducing the film thickness is still important. Fig. 2.16 shows the dependence of IF bandwidth as a function of temperature for various escape times. The variable τes can be viewed as the variation of the film thickness. Clearly, a variation in escape time has a strong influence on the IF bandwidth.

The calculated IF bandwidth around 9.5 K in Fig. 2.16 for a 6 nm thick NbN on Si in general agrees with the measured values of 2-3 GHz [8, 37] implying that the model works and all the input parameters are reasonably correct. It also explains qualitatively why it increases with increasing the bias voltage as found experimentally [8, 38]. The latter is due to the enhanced electron temperature at the higher bias.

The results in Fig. 2.16 suggest that the IF bandwidth can be improved from the typically measured bandwidth of 3 GHz, at the bias point optimized for the receiver sensitivity, to above 8 GHz if a NbN film of 3 nm with a Tcof 12 K is used. In practice this means that films with a higher critical temperature should be used, which can be achieved by using a lattice matched surface for the growth of the (less disordered) NbN film and by proper tuning of the deposition parameters. Such a film is feasible using new types of buffer layer (SiC or MgO) between film and substrate, which may allow epitaxial growth of thin NbN films. Preliminary6_{experimental results of devices}

based on a NbN film on 3C-SiC buffer layer show a considerably higher Tcof 13 K and

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4 5 6 7 8 9 10 12 14 16 18 20 10 100 0.04 0.1 0.13 0.19 0.30 0.53 1.2 th 3.5 nm NbN/Si for C_e/C_p= e-ph es Time constant (ps) T_c(K) τ τ τ

Figure 2.15: Thermal time constants as a function of Tcfor different ce/cphvalues, which are fixed as temperature independent constants. For comparison, the electron phonon time τe−phand phonon escape time τesare also included. The ce/cphvalues correspond to those at different temperatures, e.g. ce/cph=1.2 for 4 K, 0.19 for 10 K, and 0.04 for 21 K.

4 6 8 10 4 6 0 2 4 6 8 10 12 0.5 es 0.75 es 1.5 es 1.25 es es 6 nm NbN/Si es= 67 ps IF Gain bandwidth (GHz) T_C(K) τ τ τ τ τ τ 2 1 1 1

Figure 2.16: Calculated IF gain bandwidth of a NbN HEB mixer as a function of Tc for different fractions of the escape time. The value τesis 67 ps, based on the 6 nm NbN on Si substrate. The change of τescan be obtained by variation of the film thickness.

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2.7. SUMMARY 35 can be dominated [30, 31, 39] by the interaction with transverse phonons. Scattering with transverse phonons involves impurities, i.e. τe−ph increases for cleaner films. Thus, although τe−ph on the one hand is expected to decrease due to a higher Tc, τe−phon the other hand can increase due to a lower impurity concentration.

2.7 Summary

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Terahertz heterodyne mixing with a hot electron bolometer and a quantum cascade laser

Terahertz heterodyne mixing

with a hot electron bolometer

Terahertz heterodyne mixing

with a hot electron bolometer

and a quantum cascade laser

Proefschrift

Merlijn HAJENIUS

Contents

Chapter 1

Introduction

1.1

Why high resolution spectrometry at terahertz

frequencies

1.1.1

Introduction

1.1.2

Astronomical observations

1.1.3

Atmospherical observations

1.2

Heterodyne receiver system

1.2.1

Principle of heterodyne spectroscopy

1.2.2

Challenges for terahertz heterodyne receivers

1.2.3

Mixer types

1.2.4

Research focus and thesis outline

1.3

Impact on astronomy

Bibliography

Chapter 2

Physics of hot electron

bolometer operation

∗

2.1

Hot electron bolometer devices

2.2

Material properties of the niobium nitride thin

films

Glue

(2)

NbN

Si

SiO

(2)

(1)

NbN

2.3

Intrinsic resistive transition of a niobium nitride

strip

2.4

Intrinsic resistive transition under DC bias

2.5

HEB’s current versus voltage characteristics

un-der LO radiation

2.6

Thermal response times

2.6.1

Linearized two-temperature model

2.6.2

Temperature-dependent time constants

2.6.3

Mixer gain bandwidth

2.7

Summary