\section{Introduction}
Student engagement is a major issue / goal in education. Student engagement is thought to improve achievement outcomes on one hand, and on the other, where engagement occurs thought to entify good teaching: therefore high engagement reflects both good teaching and good learning. Thus, engagement is a high priority. Engagement can be thought about as something students do (engage with learning), or teachers do (engage students in learning). Our aim is to treat feedback (plus other resources) as an engagement challenge - i.e. something we provide - which can evoke an engagement response as well as develop students' skills, satisfaction and outcomes.

The hypothesis driving this research is that feedback improves engagement and so feedback is a necessary part of good teaching and effective learning.

The focus in this paper is on a situation where some feedback is automated-delivered (automated feedback from a learning environment) and other feedback is human - delivered by teaching staff face-to-face in co-located learning settings. Our interest is multi-fold: how these two forms of feedback differ, and how they are complementary. How are they experienced by students and what is their value? How do students get meaning out of feedback which improves their outcomes?

\section{Background}
\subsection{Engagement}
Engagement has been defined by \cite{trowler} as ``concerned with the interaction between the time, effort and other relevant resources invested by both students and their institutions intended to optimise the student experience and enhance the learning outcomes and development of students and the performance, and reputation of the institution''. This definition implies that students are partners in learning who make investment of time and resources to improve outcomes.

In our view engagement has three dimensions: first, behavioural, concerning e.g. attendance and time on task; second, cognitive, to do with students' understandings of their learning as well as metacognitive skills including reflection and self-regulation; and third, affective, to do with interest, and e.g. belonging. Importantly, this suggests that engagement is something both students and universities are involved in, with both parties contributing.

The importance on interactive engagement such as viewing, responding, changing or constructing is highlighted in \cite{Hosseini}. Engagement is defined as the extent of a student's active involvement in a learning activity. It can involve behavioural engagement, affective, cognitive and academic engagement \citep{Christenson}. Deep engagement improves learning. Authors of \cite{Hosseini} show that time spent on a learning task is positively correlated with learning and better performance. Engagement is enhanced with engaging programming examples. Learning from these engaging examples had a positive effect on student's engagement and learning.

\subsection{Feedback}
There are two types of feedback: formative and summative. Summative feedback is often comments on assessment scripts returned to students. Formative feedback can be around formative assessment or be ongoing: comments etc., written or spoken. \cite{Shute2008} defines formative feedback as ``information communicated to the learner that is intended to modify his or her thinking or behavior to improve learning''. There are various types: ``verification of response accuracy, explanation of the correct answer, hints, worked examples) and can be administered at various times during the learning process'' (e.g., immediately following an answer, after some time has elapsed).

These characterizations of feedback reflect a constructivist perspective on learning \citep{Laurillard2002}, which assumes that learners are not passive receivers of information but agents who actively make sense of their own learning and seek meaning. On this view students can have different understandings of what learning is, with different experience and skills. This approach assumes that depending on this, feedback messages can be complex and need to be deciphered by students, rather than being something that can be assumed to straightforwardly be received and understood.

The authors of \cite{Sergio} describe how learning to program evolves. It includes exploring different strategies and techniques in order to identify a programming solution. Support is needed during the learning process. Typically this support is provided by lecturers and teaching assistants. The support tries to help students to overcome misconceptions. We are interested to explore how students seek help in order to perform different programming tasks with a learning environment.

The author of \cite{sadler} proposes a `feedback gap'. Students need to have a concept of the goal of their learning; be able to compare their current performance with that goal; and take action to close the gap. However, this assumes that students must be able to self-assess in order to evaluate the gap and the actions that need to be taken; in turn, they should be able to make sense of and use feedback to improve performance. The implication of this is that feedback should support students in developing self-assessment skills in terms of closing the gap between performance and goal.

There has been research into types of feedback and effectiveness \citep{Shute2008}, often aimed at producing guidelines. Other research concerns students' response to and feelings about feedback. Authors of \cite{Higgins} show how students value feedback which promotes `deep learning' and critical thinking rather than `surface' feedback. \cite{Weaver} argues that the value of feedback depends on individual students' conceptions, depending on how far they share understandings of academic practice and discourse with tutors.

\section{Codio}
As described by \cite{David} Codio is a learning platform where teachers prepare Codio units which include learning material and tasks that can provide formative feedback. Figure \ref{fig:codioTasks} shows a typical Codio unit. You can see the learning material on the right hand side and a program file with Linux terminal on the left. Students can enter their code and run it within the terminal. They can also use the check it button on the right to receive feedback.
\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{codio_02.png}
  \caption{Codio learning environment.}
  \Description{The Codio learning environment.}
  \label{fig:codioTasks}
\end{figure}

\subsection{Automatic feedback and Codio}
Two different types of automatic formative feedback are available to students. First there is the feedback from the Linux terminal and second the feedback from the check it button inside the Codio unit. The feedback from within the Codio unit is developed by the instructor and is based on automated unit and system tests. Figure \ref{fig:CodioFeedback} shows typical feedback provided by the check it button. Students can check their code by clicking on the check it button. It will produce a green tick if the code is correct otherwise it will provide a red cross and some feedback if the code does not pass the tests.
\begin{figure}[h]
  \centering
  \includegraphics[width=\linewidth]{codioFeeback_02.png}
  \caption{Formative feedback from within the Codio Unit.}
  \Description{Formative feedback from the Codio Unit.}
  \label{fig:CodioFeedback}
\end{figure}

\section{Method}
\subsection{Participants}
We interviewed 27 undergraduate computer science students. The students were in their first year and had just completed their Python programming module that used Codio.
\subsection{Study design}
This was a mixed-method study. There was automated data capture giving quantitative data on attendance (regarded as a broad - and flawed - index of engagement) and achievement (total marks for the module). This data was used to split students into 6 categories (see below).
Following the end of the module, we ran 20-minute interviews with students where we attempted a representative sample by split. The purpose of this was to gather qualitative data on students' experience of the module, particularly their perceptions on the nature and usefulness of human and automated feedback. A key focus was on whether and in what way these perceptions might differ by which of the 6 categories they were in and what are the further implications for the design and delivery of effective feedback.

\subsection{Category split}
The students were grouped into the following categories: Attendance was split into high (above 60\%) and low (below 60\%). Achievement was split into low (0-50\%), medium (50-70\%) and high (above 70\%).
Attendance was measured by students' swiping their ID as they enter the room (as used across the university). Achievement was the overall module score.
Table \ref{tab:cat} shows the number of students per category.
\begin{table}
  \caption{Number of students per category}
  \label{tab:cat}
  \begin{tabular}{rccc}
    \toprule
    & &Achievement\\
    Attendance& low&medium&high\\
    \midrule
    low&  1 (LL)& 3 (LM)& 10 (LH)\\
    high& 2 (HL)& 4 (HM)&7 (HH)\\
  \bottomrule
\end{tabular}
\end{table}
\subsection{Interview design}
Interviews were run face-to-face with one student and two staff. These were run in quiet rooms, with audio recording, note-taking, and screen / person recording with screencapture, which captures interaction with Codio.

The structure of the interview was to ask what students thought about the module and allow interviews to follow student concerns. We wished to avoid leading questions on our research interests, but to progressively focus questions according to students' interests. Given this approach, a skeleton structure for the interview was: (1) broad questions; (2) detailed questions on issues based on student interests; and (3) an exploration of a Codio lesson chosen by the student where they talk through what they did and how they experienced it, with a focus on feedback and engagement.

This produced 27 datasets of both audio and video, which were analysed thematically, with an attempt to map themes to categories to expose differences; but also to see what themes are general.

\section{Findings and Analysis}
The quantitative data for the module allows a split (see Table \ref{tab:cat}). The table shows that the largest category was LH, i.e. low attendance and high achievement. This is followed by HH (high attendance and high achievement). The other 4 categories are smaller. LM means that low attendance students had medium achievement. This leaves the HM, high attendance and medium achievement group; HL, high attendance but low achievement; and LL: low attendance and low achievement.

Students in the LH group all had previous programming experience.  3 out of the 10 students would not approach the lecturer as they felt they did not need help other than the automatic feedback; 4 reported that they would only approach the tutor if they did not understand the automatic feedback. Another 3 reported that they would ask a friend for help before approaching the tutor.

The students in the LM group all had previous experience with programming. They did not have any problems to understand the automated feedback. One student reported that he would contact the tutor when they got stuck.

In contrast, the student in the LL group did not have any programming experience. The student said the attendance could have been better but felt sometimes demotivated. the student had problems to understand the automated feedback although the feedback from the tutor was perceived as more effective than the automatic feedback. However, as the student showed low attendance there was less experience of both automated and human feedback.

In the HH group 3 students reported that they did not have any previous programming experience. All students in this group made good use of the tutor feedback which they found very important as the tutor was able to explain the automatic feedback when it was difficult to understand. Interestingly nobody in this group reported that they would ask a friend for help. They asked the tutor directly when a problem was unclear.

All students in the HM group had previous experience with programming. 2 out of the 3 students reported that they would not ask the tutor for help as they were shy. One reported that they would ask a friend for help.

Again, all students in the HL group had previous experience with programming. Both reported that they struggled at times with the programming tasks. They said that when there was a problem they got in touch with the tutor for help.

The LH group still engaged with the module without high attendance, through the remote use of the Codio lessons. These students were frequently of high ability due to previous experience and attainment, where the automated feedback on detailed issues was more interpretable than for lower ability (no prior experience and / or attainment) students where issues were more conceptual, to do with basic difficulties with fundamental programming concepts. These students were represented by the HL group, and the LL group. The HL group are of particular interest because there is an implication that feedback was less effective for students with low experience / attainment.
\subsection{General themes}
Feedback was useful. Automated feedback was seen as different to human feedback. This shows expectations on what kind of feedback comes from each source, and where to look.
Codio was seen as a useful (maybe essential) resource. It should be noted that this is despite the fact that the delivery method and interaction pattern for a lab session was on-Codio for at least 90 minutes, with lecturer feedback and solutions at the end.

\subsection{Differences }
The HH group were able to talk clearly about the role and value of human and automated feedback. They made good use of the tutor in the programming sessions.

The LH group also talked about peer support and peer feedback. Talking to friends about programming challenges was seen as useful, and could be a way of reducing the time asked of staff. For example some people said:
`If we can work it out on our own, there's no need to bother staff'.
This group experienced a rich set of feedback resources and were able to manage and sequence each type to achieve outcomes.

Some students in the HM group reported that they were to shy to ask the tutor but also asked their friends for help, perhaps reflecting embarrassment about engaging with staff who taught this module. This group will not have experienced the valuable feedback from tutors but they found a way of dealing with problems by trying to resolve them via peer feedback.

In contrast the LM group showed less need than either HH or HL groups for feedback particularly peer-to-peer or staff. They gave nuanced accounts of how to use Codio feedback particularly error messages.
The HL group reported that they relied on tutor help in order to understand the automatic feedback. However this group did not seem to be able to make use of the feedback provided as effectively as the students in other groups.
The student in the LL group talked about issues regarding understanding Codio feedback. This student as well as the HL group were less able to either elicit or make use of feedback in the nuanced way the HH group did, implying that students are more or less expert not just in the domain but in how they learn and make use of feedback.

\section{Discussion and Conclusion}
All groups saw feedback as valuable, both human (staff) and automated. Therefore the issue is how to design feedback so that it engages effectively with all six types of students.
The findings show that attendance and engagement are not the same thing. For the LH group, students can be engaged (via working on Codio at home) despite low attendance. Here the quality of automated feedback is paramount. Since Python interpreter feedback is a given, this needs no further attention, but unit testing becomes highly important.

This is not true of the LL student, where there is a much greater need for basic conceptual guidance and arguably greater staff attention. They can be difficult to engage when they do attend, perhaps not wishing to be `bothered' by staff due to embarrassment about level and this being public. Some may be feedback resistant, i.e. not necessarily interested in their studies or engaging with staff. This is the most challenging group, where engagement faces basic issues with motivation and willingness. There may also be other issues about being comfortable in class, and feelings about self and their own abilities.
 The HL group were better able to deal with the social nature of the experience but less able to act on feedback, which implies that its design and delivery were less useful for this group. The implications of this work are that a blended approach to feedback which recognises the usefulness and complementarity of different types is essential for learning programming. However there are clear differences between students with different feedback needs and different ability to make use of it. This suggests that
Staff responsiveness, i.e. awaiting student requests, is not enough. While this works for HH, LH, HM and LM it is less appropriate for HL and LL where the approach needs to be proactive.
This implies the nee for the development of tools to help staff be more aware of students' performance in class. But human feedback is often seen as something high-value and not to be sought until other attempts exhausted. However some students may not like the interaction and in this case it is important to develop kinds of automated feedback which could help this group. The difference seems to be about hints and the general shape of the program.

There is also a need to recognise that the different groups are an outcome: what group students fall into is not necessarily predictable. This has implications for the value feedback can add; as well as where it is failing. This implies being able to track students in terms of how often they attend, how often they use feedback of what type, and how their use of, request for and experience of feedback may change. This also has a negative aspect where feedback is ineffective, and then stops, i.e. disengagement following attempts to engage. There are students who are disengaged from the outset and do not seek to study much if at all and these are the most difficult.
A further important dimension is affective. It seems that low-achieving students are frequently confused and conflicted and may even dislike their experience and thus avoid it. This means that type and quality of feedback is not enough to consider: we also need to be aware of the affective context in which feedback is delivered \citep{umuai}. Thus feedback should engage not just with the cognitive but also the affective.
This moves into not just needing awareness tools of how students are doing cognitively but also affectively. There is evidence that affective state influences the cognitive, and this has implications for what use can be made of feedback, and what type of feedback should be given when a student is in a particular affective state.
\bibliographystyle{ACM-Reference-Format}


\section{Introduction}
According to theoretical considerations, a~field-effect transistor (FET) can act as a detector of the electromagnetic radiation\cite{Dyakonov96}. Depending on material parameters, the length of the channel and the electron scattering time, the detection can be resonant or non-resonant. In the resonant case, plasma waves excited by the incident radiation reflect from the drain and source ends of the channel and create a~standing wave in an analogy with sound waves in an acoustic resonator. In theory, the frequency of the resonance depends on the electron concentration and can be tuned with the gate polarization. In the non-resonant case, the scattering time is too short to allow for generation of standing plasma waves and damped plasma oscillations develop only at the vicinity of the FET source. A~detailed discussion of these issues  has been recently presented in Ref.~\onlinecite{Knap10}.

The theoretical proposal \cite{Dyakonov96} stimulated a~large number of experiments starting from the first demonstration of THz detection with a~FET a~decade ago \cite{Knap02nonresonant} (for a~review, see \onlinecite{KnapComtR}). Although the general concept of a~FET as a~THz detector was confirmed in a~large number of experiments, neither of known types of  FET showed, up to now, plasma resonances  at room-temperature. This concerns also a~recently investigated antenna-coupled graphene state of the art transistor\cite{Vicarelli12}. A~a~resonant detection was found only in some cases at liquid helium temperatures \cite{Knap02resonant, Elfatimy06, Dyer12}. On the other hand, a~strong non-resonant signal was observed on  transistors of different types, including Si metal-oxide-semiconductor FETs (Si-MOSFETs) \cite{Knap04, RabihAPLNEPSi}.

The current interest in room-temperature applications of FETs in THz detection seems to be split into two main research directions. On the one hand, technological and engineering efforts led to fabrication of multipixel cameras \cite{Pfeiffer12} based on a~non-resonant FET response. On the other, one tries to construct resonant antennas which -- due to their frequency characteristics -- could shrink a~non-resonant FET response into a~narrow-band one\cite{HustonPK, HustonJL}.

The research on low-temperature detection with FETs is far less developed. There exists a~huge volume of theoretical and experimental results related to a~two-dimensional (2D) plasma investigation which started in 1977 with a~low-temperature Fourier spectroscopy on a~Si-MOSFET \cite{Allen77}. No doubt, from this perspective, different types of plasmon resonances in 2D systems seem to be perfectly well understood and analyzed. A~comprehensive review on this topic can be found in Ref.~\onlinecite{Kushwaha01}. However, developments of THz techniques, concepts and applications which have been created during last 20 years for room-temperature FET-based devices do not find their equally abundant counterpart in a low-temperature research.

The present work aims to answer a~few questions which are related to the construction and performance of THz detectors fabricated for a~low-temperature THz magnetospectroscopy. The basic idea of the study was to investigate devices in which plasmon resonances could give a~strong and tunable signal. This led us to fabrication of FET-like devices with the design of the gate metallization formed as a~meander and an ungated reference sample of the same dimensions. Next, the detectors were tested in two different experiments: a~Fourier magnetospectroscopy and a~THz-laser magnetospectroscopy.
Let us note that the magnetic field is a~natural tuning parameter at low-temperature studies and is a~powerful tool allowing to interpret experimental results. In the Fourier spectroscopy, a~spectrum is acquired as a~function of the energy of photons at different magnetic fields. In the laser spectroscopy, the spectrum is measured as a~function of the magnetic field at a~constant wave length of incident THz photons.
Testing the same detector in these two different experimental systems is not an usual approach since typically only one of these techniques is used. We show that carrying out these complementary experiments leads to a~much deeper characterization of the detector's performance.

The paper is organized as the following. Section \ref{SamplesandExperiments} describes the samples and experimental techniques. Section \ref{Results} is split into two parts. The first one presents results of spectroscopic measurements obtained with monochromatic THz sources as a~function of the magnetic field. The second one is devoted to an~analysis of the Fourier spectroscopy measurements. The results are discussed in Section~\ref{Discussion}.

\section{Samples and Experiments}\label{SamplesandExperiments}
\begin{figure}
  \includegraphics[width=8.5cm]{layers.eps}
 \caption{\label{Structure} A~scheme of the GaAs/AlGaAs heterostructure. The density of $\delta$ dopings is 7.5$\times$10$^{11}$~cm$^{-2}$.}
\end{figure}
\begin{figure}
 \includegraphics[width=8.5cm]{uklad.eps}
 \caption{\label{lamp}Right: a~scheme of the experimental setup. Top inset: the spectrum of THz radiation incident on the sample. Bottom inset: the shape of a~meander-like gate on a~Hall bar sample; the photovoltage is measured between S and D contacts; G -- the gate electrode.}
\end{figure}
The samples were processed on a~high electron mobility GaAs/Al$_{0.36}$Ga$_{0.64}$As heterostructure with a~layer sequence shown in Fig.~\ref{Structure}. The electron mobility ($\mu$) in a~two-dimensional electron gas (2DEG) at 4~K was estimated by the Hall effect measurements to be about 6$\times$10$^5$~cm$^{2}$/Vs. The electron concentration ($N_s$) in ungated and gated samples was about 3.1$\times$10$^{11}$~cm$^{-2}$ and 2.65$\times$10$^{11}$~cm$^{-2}$, respectively. In gated samples, $N_s$ was typically about 15\% lower than in ungated ones. The thickness of the barrier $d$=48~nm will be used in the following paragraphs as one of parameters which determine the spectrum of plasmons.

Two samples used for spectroscopic measurements were fabricated with an electron beam lithography on a~chip with dimensions of 4~mm~$\times~$4~mm. The mesas in the shape of Hall bars of dimensions of 1.300~mm~$\times$~0.065~mm were defined by a wet etching and were separated by about 1.5~mm (see Fig.~\ref{lamp} for a~scheme of the samples). Ohmic contacts were prepared by deposition of a~160~nm-thick layer of Au/Ge/Ni and heating the sample to 430$^o$C for a~few minutes. Connections of the four voltage contacts with the bar were done with 20~$\mu$m-thick channels.
The gate was evaporated on one of the mesas by deposition of a~15~nm-thick layer of Au/Pd alloy in a~meander form. The width of both gated and ungated stripes was equal to 2~$\mu$m. Let us note that due to the meander shape the period of the grid is equal to $a$~=~8~$\mu$m. This sample acted as a~grid-gated field-effect transistor. The chip was attached to a~metallic surface of a~14-pin dual-in-line support and both samples were bonded with Au wires.

The chip was placed in a~cryostat in a~center of a~superconducting coil and cooled down to 4.2~K with an exchange gas. The THz radiation was guided to the chip with an oversized waveguide made of a~stainless steel tube. The signal measured was a~photovoltage (PV) generated between the current-supplying contacts of a~Hall bar. During measurements, one of ohmic contacts was grounded, while Hall contacts were on a~floating potential. Measurements were carried out for both gated and ungated samples

Two types of experiments were carried out. The first one was performed with a~monochromatic radiation from a~molecular THz laser pumped with a~CO$_2$ laser. Spectra were measured as a~function of the magnetic field. We used a~lock-in technique with the reference signal generated by a~mechanical chopper modulating on/off the laser beam. This type of measurements were carried out for a~few laser lines. In this setup we have also measured transport to determine the electron concentration and mobility.

The second one was a~Fourier spectroscopy experiment which gave the PV signal as a~function of the THz photon frequency at a~constant magnetic field. In this case, the measured signal was directed to a~differential amplifier with a~band-pass filter enabled and next to Bruker 113v Fourier spectrometer set to the spectral range of 20 to 80~cm$^{-1}$. The scheme of the experimental system as well as the radiation spectrum are shown in Fig.~\ref{lamp}. We collected Fourier spectra changing the magnetic field typically by 0.1~T. For gated samples we also measured spectra at a~constant magnetic field $B=3.0$~T and $5.2$~T, changing the gate polarization $U_G$ with 0.01~V steps. These values of the magnetic field were chosen because of a~strong PV signal and essentially different values of the conduction band electron filling factor $\nu$ at $U_G=0$~V.

\section{Results} \label{Results}
\subsection{Laser spectroscopy}
\begin{figure}
 \includegraphics[width=8.5cm]{gated-ungated-laser-norm.eps}
\caption{\label{wyniki-laser}Normalized photovoltage spectra obtained at four THz laser lines (with the wavelength indicated in the figure) for gated (bottom) and ungated (top) samples at 4.2~K. Plasmon resonances are marked with bars. $B_{CR}$ is indicated with an arrow.}
\end{figure}
\begin{figure}
 \includegraphics[width=8.5cm]{disp-H2-grating-8um.eps}
\caption{\label{dyspersja-gated}Experimental (points) and theoretical (lines) dispersion of plasmon excited in the gated sample.}
\end{figure}
\begin{figure}
 \includegraphics[width=8.5cm]{disp-H1-20um.eps}
\caption{\label{dyspersja-ungated}Experimental (points) and theoretical (lines) dispersion of plasmon excited in the ungated sample.}
\end{figure}

Let us discuss results of the molecular laser spectroscopy first. Typical spectra are shown in Fig.~\ref{wyniki-laser}. For both for gated and grid-gated sample, a~cyclotron resonance (CR) magnetic field $B_{CR}$ is preceded by a~series of maxima at lower magnetic fields. We interpret these peaks as magnetoplasmon resonances which are numbered with an integer $n=1,2,...$ counting from $B_{CR}$ toward lower fields. The frequency of magnetoplasmon follows the relation \cite{Theis77}:
\begin{equation}
 \omega^2=\omega_c^2+\omega_{p,n}^2,
 \label{mps}
\end{equation}
where $\omega$ is the magnetoplasmon frequency, $\omega_{p,n}$ is the frequency of $n$-th plasmon mode at zero magnetic field and $\omega_c=eB/m^*$ is the cyclotron  frequency; $e$ and $m^*$ are the electron charge and effective mass, respectively. The dispersion relation of plasmons for a~2D system at zero magnetic field is given by\cite{Stern67}:
\begin{equation}
 \omega_{p,n}=\sqrt{\frac{e^2N_s}{2m^{*}\epsilon_0}\frac{k_n}{\epsilon(k_n)}},
 \label{dispplasmon}
\end{equation}
where $N_s$ is the electron concentration, $k_n$ is the wavevector of the $n$-th plasmon mode and $\epsilon(k_n)$ is the effective dielectric function.

Depending on whether the sample is covered with a~metal gate or not, the effective dielectric function takes the form\cite{Eguiluz75}:
\begin{equation}
 \epsilon_{gated}(k)=\frac{1}{2}(\epsilon_{GaAs}+\epsilon_{GaAlAs}\coth(kd)),
 \label{dispgated}
\end{equation}
or\cite{Popov05}
\begin{equation}
 \epsilon_{ungated}(k)=\frac{1}{2}(\epsilon_{GaAs}+\epsilon_{GaAlAs}\frac{1+\epsilon_{GaAlAs}\tanh(kd)}{\epsilon_{GaAlAs}+\tanh(kd)}),
\label{dispungated}
 \end{equation}
where $d$~is the barrier thickness. In the case of grid-gated samples, a~weighted average of expressions given by Eqs.~\ref{dispgated} and~\ref{dispungated} is assumed\cite{KarolPhD}:
\begin{equation}
\epsilon(k)=\alpha\epsilon_{ungated}(\omega,k)+(1-\alpha)\epsilon_{gated}(\omega,k),
\label{alpha}
\end{equation}
where $\alpha$~is a~parameter interpolating between two functions.

In Fig.~\ref{dyspersja-gated} we show experimentally determined frequencies of subsequent plasmon modes shifted to the zero magnetic field with Eq.~\ref{mps}. Theoretical plasmon dispersion curves, calculated at different values of the parameter $\alpha$ with Eqs.~\ref{dispgated}~--~\ref{alpha} are also presented. Calculations were carried out with the electron concentration $N_s = 2.65\times 10^{11}$~cm$^{-2}$ and $N_s = 3.1\times 10^{11}$~cm$^{-2}$ for gated and ungated sample, respectively. The effective mass $m^*= 0.0707m_e$ was determined from Fourier spectroscopy data described in the next section. Static dielectric constants, equal to 12.9 and 11.9 for GaAs and Al$_{0.36}$Ga$_{0.64}$As respectively, were used in calculations. With $k_n=2\pi n/a$ we get the best fit with the value of $\alpha\approx1.0$.

In the case of the ungated sample to fit the experimental data with the Eq.~\ref{dispungated} we had to assume that observed magnetoplasmons were excited in the paths connecting the sample's channel with voltage side contacts. The width of these paths is $W_H=20$~$\mu$m. Due to the identical boundary conditions along both sides of the path, only wave vectors satisfying $k_n=(2n-1)\pi/W_H$, where $n=1,2,3,...$, modes can be excited under uniform spatial radiation distribution over the sample\cite{Alsmeier89, Kukushkin06, Mikhailov04}.
Under these assumptions we get a~good quantitative description of experimentally observed resonances shown in Fig.~\ref{dyspersja-ungated}. Generation of plasmons in samples without a~grid seems to contradict the general idea that a~grid coupler (i.e., a periodic structure) is necessary to couple incident photons with 2D plasma excitations, but the phenomenon is know for more than 20 years\cite{Alsmeier89, Vasiliadou93}. Let's note that the role of the grid is to diffract the radiation, so an
electromagnetic wave acquires an in-plane wave vector component which allows to fulfill the momentum conservation law during excitation of a~plasmon with a~photon. Similarly, in grid-free samples, diffraction of photons on sample's or mesa's borders allows for plasmon generation.

\subsection{Fourier Spectroscopy}
\begin{figure}
 \includegraphics[angle=-90,width=8.5cm]{notgatedabs.eps}
\caption{\label{H1-abs} Fourier spectra of PV measured  on the ungated sample at every 0.1~T between 2.8~T (bootom) to 6.1~T (top) at 4~K. Bars in the bottom inset show the maxima appearing due to interferences in the substrate.}
\end{figure}
\begin{figure}
 \includegraphics[angle=-90,width=8.5cm]{gatedabs.eps}
\caption{\label{H2-abs} Fourier spectra of PV measured  on the gated sample between 1.2~T (bootom) to 6.4~T (top) at 4~K. Bars in the top inset mark the maxima appearing due to substrate interference.}
\end{figure}

Fourier spectroscopy experiments were carried out for both gated and ungated sample. The resulting spectra collected at different magnetic fields for the ungated and gated samples are shown in Fig.~\ref{H1-abs} and Fig.~\ref{H2-abs}, respectively. Let us note that no PV signal was observed for radiation power below about 0.2~arb.u. of the intensity scale shown in the top inset in Fig.~\ref{lamp}. As one can notice, the spectra are dominated with a~symmetric peak of the full width at half maximum (FWHM) of about 5~cm$^{-1}$.
The peak is moving with the magnetic field towards higher frequencies and corresponds to the cyclotron resonance transition modified by magnetoplasmon excitations. The peak is broad enough to observe subsidiary maxima which appear with the period of (7.24$\pm$0.22)~cm$^{-1}$ (see the insets to Figs.~\ref{H1-abs} and \ref{H2-abs}). Since the period of these oscillations corresponds to interferences in the GaAs substrate, they are neglected in the~further analysis. For each spectrum, peaks were fitted with Lorentzian function to give estimations of their positions and amplitudes. The insets to Figs.~\ref{H1-abs} and~\ref{H2-abs} give examples of the fitting procedure.

\subsubsection{Frequency}
\begin{figure}
 \includegraphics[angle=-90,width=8.5cm]{frequency.eps}
 \caption{\label{frequency}Dependence of the magnetoplasmon-shifted cyclotron resonance peak frequency as a~function of the magnetic field.}
\end{figure}
\begin{figure}
 \includegraphics[angle=-90,width=8.5cm]{resolution.eps}
 \caption{\label{resolution}Resolution limits of Fourier spectrometer for gated and ungated samples measurements and an evolution of a~difference in energy between plasmon mode and CR with the magnetic field for magnetoplasmon in both samples.}
\end{figure}

Data plotted in Fig.~\ref{frequency} show the position of the magnetoplasmon-shifted CR peak. We observe that the resonance frequency of the gated Hall bar is higher than that of the ungated one by a~few cm$^{-1}$. This results from the excitation of a~magnetoplasmon which blue-shifts the resonance frequency in the gated sample. Fitting Eq.~\ref{mps} to the gated sample data gives the plasmon frequency at zero magnetic field equal to (13.2$\pm$0.5)~cm$^{-1}$ which coincides with the frequency of $n=1$ mode  obtained from the THz laser experiment, shown in Fig.~\ref{dyspersja-gated}. Contrary to the THz laser spectroscopy results, we do not observe higher ($n > 1$) magnetoplasmon modes in the Fourier spectroscopy experiment which is caused by a~much smaller THz radiation power incident on the sample and its wide spectrum.

We do not observe signatures of magnetoplasmon excitations in the ungated sample. This is related to the spectral resolution of the measurements which is discussed in Fig.~\ref{resolution}. The experiments were carried out with the resolution equal to 0.2~cm$^{-1}$ and 0.5~cm$^{-1}$ in the case of ungated and gated samples, respectively. We assume that the least distance between two maxima that can be resolved are twice these values which are represented with horizontal lines in Fig.~\ref{resolution}.
Curved lines show an~energy difference $\Delta E$ between the first magnetoplasmon mode and the cyclotron resonance: ($\sqrt{\omega_{p,1}^2+\omega_c^2} -\omega_c$) with $\omega_{p,1}=13$~cm$^{-1}$ and $\omega_{p,1}=7$~cm$^{-1}$ for gated and ungated sample, respectively (see Figs.~\ref{dyspersja-gated} and \ref{dyspersja-ungated}). One can notice, that with given experimental resolution, it is not possible to discriminate the plasmon from the CR for magnetic fields higher than about 4.5~T and 6.5~T for ungated
and gated samples, respectively. This result means that at higher $B$~the peak observed in Fourier spectra coincides with the CR. The value of the cyclotron mass obtained from this data is equal to $0.0707m_e$. It is slightly higher than the value of $\approx$0.069$m_e$ expected in GaAs/AlGaAs heterojunction\cite{Batke88}.
The right scale in Fig.~\ref{resolution} approximately recalculates $\Delta E$ into the magnetic field scale $\Delta E = e/m^* \Delta B$ which shows that the resolution of 0.005~T in the laser experimental system allows to discriminate between the CR and plasmon peaks even at high magnetic fields

\subsubsection{Amplitude}\label{FSamp}
\begin{figure}
 \includegraphics[angle=-90,width=8.5cm]{notgatedfit.eps}
 \includegraphics[angle=-90,width=8.5cm]{gatedfit.eps}
 \includegraphics[angle=-90,width=8.5cm]{gatedfitUg.eps}
 \caption{\label{fits}In the top and middle plots solid line show a~theoretically predicted PV including an~electron temperature dependence on the magnetic field. Dashed lines in these two plots show predictions with the constant electron temperature. In the bottom plot lines show predicted PV for a~constant $\Delta T_e$ in case of $U_G$ sweeps at a~constant $B$.}
\end{figure}

The amplitude of the resonance present in Fourier spectra show a~non-monotonic dependence on the magnetic field. To analyze this dependence we refer to a~model described in Refs.~\onlinecite{Thiele89}. Its essence lies in an assumption that absorption of a~THz radiation changes the electron distribution function. This can be approximately described as a~change of the electron temperature $T_e$, which becomes higher than the lattice temperature $T_L$. In a~gated sample, there is a~strict correspondence between the gate potential and the electron distribution function (it just reflects the fact that the electron concentration can be changed with the gate polarization). For this reason, changes in the electron temperature can be observed as changes of the gate -- to -- channel potential, as it was done in Ref.~\onlinecite{Thiele89} where the gate -- to -- channel photovoltage was investigated. In our case, typically a~drain -- to -- source photovoltage was measured (see the inset to Fig.~\ref{lamp} to recall
the names
of the sample's contacts) which was proved to be essentially identical to that of a~gate -- to -- source, as Fig.~\ref{PVUdsUg} shows. This similarity must exists since in gated samples gate -- to -- source and drain -- to -- source potentials are dependent under the open drain conditions, as it is the case of a~PV measurements. To explain the origin of a~PV signal in an ungated sample we note that an asymmetry must be present in the sample which allows for a~charge separation and generation of a~PV under the incidence of THz radiation. Since the process of charge separation depends on the electron diffusion, it is influenced by the electron distribution function, hence $T_e$.

The model proposed in Ref.~\onlinecite{Thiele89} allows to estimate the value of the photovoltage for magnetic fields corresponding to electron filling factors close to even values, $\overline\nu = 2, 4, ...$:
\begin{eqnarray}
\label{wzor}
\nonumber
\Delta V(\nu)=A\{T_e\mathrm{arcsinh}(\frac{\nu-\overline\nu}{4}\exp[\frac{\hbar\omega_c}{2kT_e}])\\
-T_L\mathrm{arcsinh}(\frac{\nu-\overline\nu}{4}\exp[\frac{\hbar\omega_c}{2kT_L}]
)\},
\end{eqnarray}
where $A$ is a~fitting parameter and $T_L$ = 4.2~K. For the gated sample, the filling factor $\nu$~was calculated as a~function of $B$~and the gate voltage $U_G$:
\begin{equation}
\nu=\frac{h}{e}N_s\frac{U_G-U_{th}}{U_{th}}\frac{1}{B},
\label{ff}
\end{equation}
where the fraction ($U_G-U_{th})/U_{th}$, with $U_{th}$ denoting a~threshold voltage, gives a~fraction of free electrons with respect to their number at zero gate polarization. The value of $U_{th}$=-0.323$\pm$0.006~V was obtained from a~dependence of the electron concentration on the $U_G$ determined by an analysis of the period of SdH oscillations; for the ungated sample, this fraction was $1$. The correspondence between $\nu$~and $B$~allows to express $\omega_c$ as a~function of $\nu$~in Eq.~\ref{wzor}. In the experiment, only the amplitude of the photovoltage was measured. For this reason, the modulus of the function~\ref{wzor} was fitted to the data. Our measurements were done for magnetic fields corresponding, approximately, to $1.5 < \nu < 4.5$. For this reason, the fitting function was composed of a~sum of two contributions, one corresponding to $\overline\nu = 2$ and the other -- to $\overline\nu = 4$.

We estimated $T_e$ in two ways. First, we assumed that $T_e$ is constant, i.e., it does not depend on the magnetic field. This is an approximation since heating of the electron gas is the most efficient with photons of the energy close to that of the CR and the spectral intensity of the THz source in the Fourier spectrometer is not constant. That is why, we assumed that $T_e(B) = T_L +\Delta T l(B)$, where $l(B)$ is a~normalized power of the THz source, shown in Fig.~\ref{lamp}, with the argument $B$~determined according to $E = \hbar \omega_c$, where $E$~is the photon energy. This model gives the maximal heating efficiency at the magnetic field equal to about 3.7~T which corresponds to the CR at $E=45$~cm$^{-1}$.

Finally there were two fit parameters only: $\Delta T_e$ -- the difference between temperatures of electrons and the lattice (4.2~K) and $A$ -- a~scaling factor for the function (\ref{wzor}). Results of both fitting procedures are shown in Fig.~\ref{fits}. One can notice that the model describes the experimental data with a~reasonable accuracy. Also, a~better fitting results are obtained with an assumption of a~$B$-dependent electron temperature (dashed lines) than with a~constant one (solid lines). Fitted values of $\Delta T$ are equal to 21.0~K and 3.0~K for the ungated and gated samples, respectively. The same value of $A$~was used for both samples. Taking into account that the PV signal was observed for the source intensity higher than 0.2 of the maximum value, the electron temperature changes between (4.2 + 0.2*21.0)~K = 8.4~K to 25.5~K for the ungated sample and between 4.8~K and 7.2~K for the gated one.

An analysis of Eq.~\ref{wzor} shows that a~sharp feature in $\Delta V$ should be expected at $\nu= 2, 4, ...$. In fact, such abrupt changes are clearly visible in the data presented in Fig.~\ref{fits}. Since transport measurements were not done during the Fourier spectroscopy measurements, we estimated the value of the filling factor by assuming that $\nu = 2$ corresponds to $B$ = 5.6~T and 5.0~T in the case of ungated and gated samples, respectively. This leads, correspondingly, to $N_s = 2.65\times10^{11}$~cm$^{-2}$ and $N_s=2.4\times10^{11}$~cm$^{-2}$. These values are by about 10-15\% less than obtained at the THz laser spectroscopy and magnetotransport measurements ($3.1\times10^{11}$~cm$^{-2}$ and $2.65\times10^{11}$~cm$^{-2}$) which can result from the dependence of $N_s$ on a~particular cooling cycles in these two separate experiments.

\section{Discussion} \label{Discussion}
\begin{figure}
 \includegraphics[width=8.5cm]{PVUdsUg.eps}
 \caption{\label{PVUdsUg}Photovoltage measured on gate and drain contacts with monochromatic sources: 19.4~cm$^{-1}$ (left) and 84.2~cm$^{-1}$ (right).}
\end{figure}

The most evident application of the investigated samples as low-temperature tunable THz detectors results from the Fourier spectroscopy measurements. Neglecting the interference in the substrate (which can be eliminated by wedging the sample), a~linear dependence of the resonance energy on the magnetic field together with a~Lorenzian shape of the peak, shown in Figs.~\ref{H1-abs} and \ref{frequency}, is suitable for application at high enough $B$. At lower fields one can observe a~deviation from the linearity related to excitation of magnetoplasmons.
Thus, designing a~detector one should be aware of magnetoplasmons that can be generated by the radiation. According to our analysis, it is possible to estimate the range of magnetic fields at which the separation of the magnetoplasmon from the CR position is lower than the experimental resolution (see Fig.~\ref{resolution}). On the other hand, oscillations of the resonance intensity with the electron filling factor, observed also by other groups\cite{Seidenbusch87, Thiele89, Wilson79, Heitmann86, Batke88, Tsujino98}, can limit the applicability of the detector since one can not trace the investigated signal in a~broad range of $B$~with the same sensitivity.

It is not obvious to determine by theoretical considerations the wave vector of the fundamental plasmon mode in a~given sample. The thing is that in a~real detector one always finds some metallization pads or signal-conducting paths which can trigger plasmon excitation with a~wave vector determined by a~geometrical detail of the structure. This is clearly seen in the investigated samples where under the same conditions of laser excitations, plasmons were generated according to the geometry of the grid-gate or side signal paths (in this case, paths leading to Hall voltage pads).
The energy difference between the fundamental modes in these two cases is equal to about 7~cm$^{-1}$ which is far more than the spectral resolution of measurements. Testing the same detectors at much lower frequencies, in the range 0.1 -- 0.6~THz, we found that we were able to excite plasmons with the wave vector of the fundamental mode equal to the width of the mesa, i.e., 65~$\mu$m.\cite{Bialek12} Taking all these observations into account we conclude that the spectrum of plasmons  generated  depends both on the range of the radiation frequency used and geometrical details of a~given detector design. This variety of possible plasmons generated in one sample can be considered as discouraging, but we rather see in this observation a~chance to control plasmon resonances by a~proper design of the detector geometry.
For instance, if in the ungated sample investigated the fundamental mode of the plasmons corresponds to the width of a~side path, one could propose a~device with tens of such paths positioned at the circumference of the mesa. Such a~design should lead to a~strong amplification of the plasmonic response.

Summing up, in a~Fourier spectroscopy experiments, the detectors investigated showed a~clear and narrow spectral response but the amplitude of the signal strongly depends on the magnetic field. The narrow spectral width results from the fact that only one magnetoplasmon mode was generated in the Fourier experiment. Since the variations of the amplitude of the detection signal result from basic physical properties of a~2DEG, we consider this drawback as hard to eliminate. In a~THz laser spectroscopy, the signal is spectrally much broader and dominated by a~number of plasmon resonances triggered according to the geometrical details of the detector design and the range of THz frequency radiation used. We suggest that the geometry of a~properly designed detector could stimulate generation of only one plasmon mode which could lead to a~narrow response in THz laser spectroscopy experiments.

\section{Conclusions}
Gated and ungated Hall bars were used in a~THz magnetospectroscopy experiments. Excitation with a~THz laser allowed to determine dispersion relations of plasmons. In the case of the gated sample, magnetoplasmons were generated with the wave vector of the fundamental mode equal to the period of the grid. For the ungated sample, it was determined by details of the geometry of the mesa. Fourier spectroscopy showed that the amplitude of the magnetoplasmon resonance was an oscillating function of the electron filling factor which can be described by a~model assuming an increase of the electron gas temperature. These two types of experiment gave consistent spectral results

\section{Acknowledgments}
This work was partially supported by Foundation for Polish Science grant POMOST/2010-1/8. We are grateful to dr K.~Fronc for the help in samples processing.

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