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Electronics From the Bottom Up: Strategies for Teaching Nanoelectronics at the Undergraduate Level Mani Vaidyanathan, Member, IEEE
Abstract—Nanoelectronics is an emerging area of electrical and computer engineering that deals with the current–voltage behavior of atomic-scale electronic devices. As the trend toward ever smaller devices continues, there is a need to update traditional undergraduate curricula to introduce electrical engineers to the fundamentals of the field. These fundamentals encompass topics from quantum mechanics and condensed-matter physics, and they pose new teaching challenges in electronics education; specifically, unconventional ideas must be presented in a rapid and yet complete way so that engineering undergraduates can quickly yet satisfyingly absorb the key concepts, and then apply these concepts to emerging devices. This paper describes the strategies employed by the author in teaching the subject to large undergraduate classes at his institution. These strategies include the use of computer visualization, a careful introduction of quantum mechanics, and a constant demonstration of the relevance of theory by practical examples and calculations. The effectiveness of the approach is illustrated through survey results of the Universal Student Ratings of Instruction at the author’s institution and by way of typical assignment and exam questions that demonstrate the level of sophistication that students can attain in what might otherwise be viewed as a purely mathematical and esoteric subject. Index Terms—Electronics education, electronics from the bottom up, emerging electronics, nanoelectronics, solid-state electronics, teaching strategies for electronics.
I. INTRODUCTION ANOTECHNOLOGY is presently an intense area of global research and development. In Europe, the 7th Framework Program has a budget of $3.48 billion between 2007 and 2013 for research into “Nanosciences, Nanotechnologies, Materials, and New Production Technologies” [1], and in the United States, the National Nanotechnology Initiative has an estimated total budget of $1.657 billion in 2009 and a proposed budget of $1.640 billion in 20101 [2]. Similar numbers can be found in jurisdictions all over the world, including the author’s home province of Alberta, Canada, where in 2007, the government announced a $130 million commitment toward “achieving [the] goal of a $20 billion nanotechnology supply industry in Alberta by the year 2020” [3]. Such funding levels indicate the importance of incorporating nanotechnology into undergraduate university curricula.
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Manuscript received December 28, 2009; revised January 19, 2010. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The author is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: maniv@ece. ualberta.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TE.2010.2043845 1All Web sites referenced in this paper were last viewed on January 26, 2010.
Within electrical engineering, the development of small-scale electronics naturally fits into nanotechnology. Aggressive scaling according to Moore’s law has led to modern processors that contain approximately 1 billion transistors, each with a gate length that is presently around 50 nm, and each able to switch on and off about 300 billion times a second. Additionally, the 2007 International Technology Roadmap for Semiconductors (ITRS) has identified the need to address “exotic” new devices beyond conventional silicon transistors, citing the slogan “More than Moore” over and above “More Moore” as part of its roadmap [4]. Given that electronic devices are now well into the nanometer regime and continuing to shrink, and given that new devices are on the horizon, undergraduate electronics curricula need to be revamped. Traditionally, and tracing back several decades, solid-state electronics was taught with a minimum coverage of quantum mechanics and condensed-matter physics. Classical texts in electronics, such as those of Sedra and Smith [5] and Streetman and Banerjee [6], were able to describe the current–voltage behavior of long-channel MOSFETs and bipolar transistors solely with classical tools, such as the well-known drift-diffusion equation (DDE) (1) where is the electron current density, is the electron conand centration, is position, is the electronic charge, and refer to the electron mobility and diffusivity, respectively. In such classical approaches, underlying quantum phenomena were masked and embodied in collective material parameters, like the values of mobility and diffusivity appearing in (1). However, as recently pointed out by educators at Purdue University, West Lafayette, IN [7], if today’s graduates are to fully meet the challenges and opportunities [8]–[10] they will encounter in electronics during their working careers, then they will need to be familiar with a more detailed description of charge transport and electronic device phenomena than in the past; they will need to learn electronics “from the bottom up” [7], i.e., from the atomic to the macroscopic level. Such knowledge will be useful not only for specialized careers in device modeling, like those described in recent placement advertisements by IBM [11], but also for more general careers in electronics (e.g., circuits and systems design) that will require the ability to understand and exploit the new devices that will no doubt emerge over the 30-year careers of graduating students. The bottom-up approach presents an immense teaching challenge, requiring that nonintuitive and mathematical concepts from quantum mechanics and condensed-matter physics be introduced into undergraduate engineering curricula. Moreover, such material has to be introduced in a manner that satisfies three
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simultaneous and yet seemingly conflicting requirements: 1) it has to be compact so as to emphasize only the most relevant concepts that might otherwise be covered in several courses in the sciences; 2) it has to be complete so that mathematical and nonintuitive theory can be satisfyingly understood by engineering students; 3) it has to contain sufficient depth and sophistication for students to be able to apply the concepts to describe and understand real device phenomena. While universities are slowly moving to incorporate undergraduate courses in “nanoelectronics” into their programs, and while textbooks on the subject are beginning to emerge [12]–[15], well-accepted teaching methods to convey the subject successfully—and especially to address the three requirements mentioned—have not been established. The purpose of this paper is to share the author’s successful experience in delivering an introductory course in nanoelectronics to third- and fourth-year undergraduate students at the University of Alberta, Edmonton, Canada. Students enter the class with a background in electronics at the level discussed by Sedra and Smith [5]. The paper briefly describes the author’s course, discusses the author’s teaching strategies, and examines their effectiveness in motivating students to learn what they might otherwise dismiss as a purely mathematical and esoteric subject. The organization of the paper is as follows. Section II provides an overview of the course, including its aim and the topics covered. Section III discusses the three main teaching strategies employed by the author: the use of computer visualization, a careful introduction of quantum mechanics, and a constant demonstration of the relevance of theory by practical examples and calculations. Section IV shows the effectiveness of the strategies by way of feedback in the Universal Student Ratings of Instruction used by the author’s institution. Samples of typical exam and assignment questions are provided in the appendices to illustrate the level of sophistication that students are able to obtain. The conclusions of the paper are summarized in Section V. II. COURSE DESCRIPTION A. Overall Approach The “bottom-up” approach was pioneered by Supriyo Datta and Mark Lundstrom of Purdue University. The author’s course at the University of Alberta builds on the lead established by Datta’s recent and excellent text, Quantum Transport: Atom to Transistor [12], [16], which, as far as the author knows, was the first to adopt the bottom-up strategy. Two notable differences between the author’s approach and that of Datta are in the way quantum mechanics is taught and in the course “punchline” for undergraduates. Datta’s text introduces quantum mechanics at a numerical level and then builds to Green’s-function quantum transport, while the author’s course spends more time on emphasizing quantum-mechanical fundamentals (such as the meaning of the wave function and the postulates of quantum mechanics) and then ends by introducing the semiclassical Boltzmann transport equation (BTE). The BTE is used to conclude the author’s course by showing the origin of the drift-diffusion equation (1) and by deriving an equivalent circuit for the ac behavior of a ballistic nanotransistor [17]. The connection between the BTE and the macroscopic DDE helps establish relevance of the underlying quantum phenomena by connecting the course material
to that encountered by students in classical electronics texts [5], [6], and the derivation of an equivalent circuit shows how an engineering tool can ultimately be obtained “from the bottom up,” i.e., starting from the atomic level. Further details on this overall approach will be discussed throughout this paper. B. Individual Topics As with Datta’s text [12], the author’s course emphasizes the fundamentals that will be required to understand future electronic devices, regardless of their specific form, which is to be contrasted with an approach that might emphasize specific technological issues and specific devices, as would be found in a professional short course. The author’s course hence covers the following major topics, leading up to the Boltzmann equation and the transistor equivalent circuit under ballistic operation. 1) The Basics of Electron Flow in Nanodevices: Following Datta’s lead, a simple rate-equation model [12, Ch. 1] for current is used to give students an overview of all the major concepts needed to study electron flow at the nanoscale. The generic structure of a field-effect transistor is presented, and it is emphasized to students that this picture needs to be replaced by an “energy picture,” where the contacts to the transistor are described by Fermi functions and the transistor’s channel is described by its allowed energy levels or density of states. Here, the channel is modeled as a single point so that spatial variation of electron density or current in the channel can be ignored. Students acquire comfort with the energy picture, and they learn that after self-consistency is achieved with Poisson’s equation, the channel’s energy levels must lie somewhere between the two contacts’ Fermi levels in order for electrons to flow. 2) The Schrödinger Equation and Band Structure: With the importance of the channel’s energy levels established in the first part of the course, considerable time is spent on the basics of quantum mechanics and its use in determining the energy-band structure of solids. The tight-binding band structure of graphene and a Fermi-golden rule calculation of the electron mobility in GaAs are used as illustrative paradigms for the application of band-structure concepts. Considerable time is spent on the postulates of quantum mechanics and the meaning of the wave function, so students can be as comfortable as possible in performing calculations and meaningfully interpreting the results. 3) The Boltzmann Equation and Current in a Ballistic Nanotransistor: The course concludes by tying the band-structure theory back to ideas encountered at the beginning of the course and by extending the theory to discuss important macroscopic device concepts such as band-gap, the Boltzmann transport equation, and the drift-diffusion equation. The tie-in with earlier material is done by showing how to use band-structure information to obtain the density of states, which is required information for the rate-equation model from the beginning of the course. The idea of band-gap is explored by seeing how the band structure of graphene leads to subbands in carbon nanotubes (rolled sheets of graphene) and by looking at metallic versus semiconducting tubes.
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The Boltzmann transport equation is then introduced as one way to keep a detailed track of electrons in the channel when spatial variations in the channel cannot be ignored, and it is shown how this leads to the classical drift-diffusion equation. The Boltzmann-equation picture is then used to examine current under ballistic conditions and ultimately to derive an equivalent circuit for the ac operation of a ballistic transistor [17]. C. Outcome In the author’s opinion, the result of the above approach for students is the ability to comfortably do calculations that are representative of those in the field and a tie-in to more familiar macroscopic ideas, such as the drift-diffusion equation and an ac equivalent circuit. To emphasize this outcome, it is pointed out to students that while the course experience by no means makes them experts, it does allow them to have a feeling for the type of work involved in understanding state-of-the-art and emerging devices, which is shown by their ability, at the end of the course, to relate to actual job descriptions in the area. Excerpts of such descriptions, quoted directly from [11], are as follows. • Atomistic/Quantum Effects Modeling Engineer: Involves theoretical understanding of quantum effects in 22 nm scale devices; these include carrier transport, geometry, and material dependent band-structure calculations… • Technology CAD Engineer: Involves the improvement and development of [a] tool for the simulation of leading edge semiconductor devices. In particular, physical and numerical models for carrier transport influenced by e.g. channel materials… • Candidates for Atomistic Modeling and Technology CAD positions should have a strong background in semiconductor physics, quantum effects in nano-devices, numerical programming… While these descriptions illustrate the type of knowledge that students obtain from the course, it is important to reiterate that the course material will be valuable not only for the few students who might seek careers directly in advanced device modeling, but also for any who plan on working in electronics over the next few decades. III. TEACHING STRATEGIES A. Computer Visualization Assignments in the course involve both MATLAB [18] programming as well as hand calculations. The assignments themselves are structured like a computer laboratory notebook and include monographs on the course topics that extend and augment ideas from the lectures. For the MATLAB, sample code is often provided, but the students are asked to use such code only as a guide, typing their own even if it is very similar to the sample. This somewhat unorthodox approach eases the coding burden while encouraging students to gain familiarity with the equations and concepts through numerical solution. Following the MATLAB problems, simplified problems involving the same concepts are assigned to be solved by way of hand calculations.
Fig. 1. (a) Conventional schematic diagram of a nanoscale MOSFET at high drain bias. The illustrated device has a structure that is similar to a silicon-oninsulator NMOS transistor, but the diagram is not to scale. The shaded triangle in the channel region represents the surface electron density at “pinch-off” [5, Ch. 4], i.e., when the current begins to saturate. (b) Energy diagrams of the same MOSFET from a simple MATLAB simulation, drawn for a drain voltage V : V and a gate voltage V : V. (c) Drain current versus drain : V, and the voltage for the same MOSFET; the upper curve is for V lower curve is for V : V.
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The author has found that this approach of combining MATLAB with hand calculations is particularly effective in instilling the proper nonclassical visualizations, which is best illustrated through specific examples that are extensions of ones found in [12].
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Fig. 1(a) shows a conventional schematic diagram of a nanoscale MOSFET, while Fig. 1(b) shows the same device from a simplified energy perspective [12, Ch. 1]. In the graphs of Fig. 1(b), which are generated by MATLAB code, the contacts are represented on the left and right by their Fermi functions, while the channel in the center is represented by ; one additional graph is presented its density of states, underneath the others, in which the difference in the two Fermi , is superimposed on a functions, , where refers to the self-consistent potential. plot of Fig. 1(c) shows the MOSFET’s current–voltage behavior. By running the code, and seeing how the quantities in the graphs of Fig. 1(b) “move” as the drain bias is increased, students are asked to explain the MOSFET’s current–voltage curve in Fig. 1(c) by way of the equation (2) where and are rate constants that describe the coupling of the channel to the contacts [12, Ch. 1]. In this example, students obtain an alternative (energy-based) explanation for MOSFET current saturation, a phenomenon classically described by the concept of “channel pinch-off” [5, Ch. 4] and a diagram such as that in Fig. 1(a). A second example is illustrated in Fig. 2, where Fig. 2(a) shows the energy diagrams for a hypothetical two-level molecule, and Fig. 2(b) shows the current–voltage behavior. Students are asked to explain the current–voltage curve, and especially the negative-differential resistance, by observing in MATLAB how the energy levels and Fermi functions “move” in Fig. 2(a) as the applied voltage is changed. It is pointed out to students that this problem involves a simplified version of the real theory-to-experiment comparison performed in [19]. After coding and solving such MATLAB problems, students are then asked to apply the energy perspective to solve similar problems analytically, i.e., by way of their own hand drawings and hand calculations. Examples of such problems are provided in Appendix I, and students are readily and eagerly able to solve these (both in assignments and on exams) following the MATLAB exercises. B. Introduction of Quantum Mechanics A key ingredient of any course in nanoelectronics is an introduction to quantum mechanics. While the goal in an engineering course must be to teach students how to apply concepts from the discipline to do practical calculations, it is the author’s opinion that with quantum mechanics, this can only be accomplished by first allowing students fair time to deal with the mysterious and nonintuitive nature of the field. It is human nature to ask questions such as, “What does it really mean?” and “How do I picture this?” Indeed, the leading scientists of the 20th century who developed the field struggled with such questions, so it is no surprise that novice engineering students would do so. To deal with these types of questions, and to get students to the point where they believe they are doing meaningful calculations rather than mere algebra or numerics, the author has taken a two-pronged approach. The approach involves: 1) providing students with a working definition of quantum mechanics that is consistent with
Fig. 2. (a) Energy diagrams for a hypothetical molecule that exhibits negative differential resistance. (b) Current–voltage curve for the molecule.
an agnostic point of view on its deeper meanings; and 2) connecting ideas in quantum mechanics back to more familiar ideas in electrical engineering. 1) Working Definition of Quantum Mechanics: The material begins with a formal definition of the term “quantum mechanics,” which can be paraphrased from Morrison [20, p. 2] as being “nothing more than a set of postulates and the mathematical tools derived from those postulates that are used to understand nano phenomena.” Following this definition, the usual concepts of wave-particle duality and the Principle of Complementarity are covered, and a simplified set of postulates are provided. Consistent with an agnostic perspective, it is emphasized that mysterious entities such as the wave function and its governing (Schrödinger) equation are merely formal slaves that contain information students can access to make predictions about experimental observations, and that no more can be said about these strange entities than what is said by the postulates. Students are hence encouraged to learn about the tools needed to extract information from these “slaves” and to apply such tools to engineering problems of interest. It is also emphasized that students must avoid the pitfalls of trying to visualize the outcomes of their calculations in terms of everyday experience.
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These last two points are absolutely crucial for a novice engineering audience. Otherwise, students will inevitably get hung up on questions of deeper meaning or get confounded by trying to visualize the outcome of calculations in terms of “everyday” intuition, both tactics being a natural part of an engineer’s understanding strategy. While texts in physics [20, Ch.1–3] often adopt such an introductory strategy, texts in engineering tend to jump quickly to applications and calculations. In the author’s experience, students benefit immensely by the type of introduction mentioned, which gives them a chance to struggle and dispense with the same difficulties faced by anyone who first encounters quantum physics. Within the framework of a course on nanoelectronics, the introduction should be nonmathematical and can be done in the time of a few lectures. While the required time constitutes a very small fraction of a typical semester, the benefits it achieves toward student appreciation of the entire course should not be underestimated. 2) Relation to Concepts in Electrical Engineering: A notable example of relating ideas in quantum mechanics back to those in electrical engineering stems from the standard particle-in-a-box problem. In this case, students are asked to examine momentum information for the particle. Since the particle is confined to a , its momentum will not be sharp region [21], and information on the momentum is contained in the mo, which can be derived from the pomentum wave function . The exercise illustrates how “approsition wave function, priate mathematical operations” can be used to extract “desired physical information” from wave functions, a notion already asserted to students during their qualitative classroom introduction to quantum mechanics. The required operations are compared to familiar mathematical operations in electrical engineering. The starting point is to first obtain the amplitude function , which is the Fourier transform of the position wave func, just as represents the Fourier transform of a tion in electrical engineering time signal (3) (4) The inverse transform relations are equally instructive (5)
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Once the amplitude function is known, the momentum wave function itself is found through a change of variables
(7) where is the reduced Planck’s constant. Mathematically, the position wave function for the ground state can be written as (8) where (9)
By use of standard Fourier-transform tables involving the variables and [22, Ch. 3], and with the discussion surrounding (3)–(7) borne in mind, students are then able to derive the momentum wave function
(10) where and . MATLAB and the two sinc-function components in (10) are plots of provided in Fig. 3(a), and the corresponding momentum-probis shown in Fig. 3(b). With ability distribution in hand, students are asked to comment on the validity of the following statement: “For a particle in the ground state of an escape-proof box, the only possible momenta are the classical , where is the particle’s mass and values is the particle’s energy.” Even a few examples such as the above allow students to learn about the nonintuitive and unfamiliar nature of quantum phenomena through familiar engineering ideas. A sample exam question related to the above material is provided in Appendix II. While not discussed in this paper, an additional exam problem regarding the numerical solution of the Schrödinger equation is also provided. Such problems are answered well by students, and the author believes that this outcome is a result of student comfort with quantum physics resulting from teaching strategies such as those discussed here.
(6) C. Connection to Practical Examples The inverse transform relation (5) to which students are accuscarries information on the complex amtomed says that that are plitudes of an infinite sum of complex exponentials and that the sum is over all used to reconstruct the signal radian frequencies . Similarly, students are told that the inverse is an amtransform relation (6) says an analogous thing: plitude function that carries magnitude and phase information that are on an infinite sum of plane-wave, wave functions , and the sum used to reconstruct the original wave function is over all wave vectors .
Throughout the lectures and assignments, the author has found that connecting theory to practical examples to establish relevance and retain student interest is particularly important for undergraduate engineering students in a subject such as nanoelectronics, given the nonintuitive nature of the material. In the lectures, articles on emerging devices and experimental results that validate developed theory are regularly cited and discussed. Examples of articles on experiments include those of van Wees et al. [23] and Wharam et al. [24], which are discussed in class following a derivation of the quantum of conductance
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the time-dependent Schrödinger equation) and then leads to experimentally verifiable results. Sample exam and assignment questions that illustrate the level of sophistication that students are then able to attain in the application of theory are provided in Appendix III. The provided samples are concerned with the band structure of graphene and carbon nanotubes and with electron flow in the channel of a ballistic transistor. IV. TEACHING EFFECTIVENESS To illustrate that it is possible to use strategies such as those mentioned to deliver a well-received undergraduate course in the area, results are presented from the Universal Student Ratings of Instruction (USRI), which is used by the author’s institution as one measure of teaching effectiveness. For every class, the USRI consists of 10 mandated questions, five of which are about the course content, and the remaining five of which are about the instructor. Of relevance to this paper are the five regarding course content. Results for these five questions are discussed in this section, along with written comments from students, which can anonymously and optionally be provided as part of the USRI evaluation. A. USRI Scores on Course Content
Fig. 3. (a) Momentum wave function 8(p) from (10), along with the two sinc-function components, 8 (p) (1=2) (L= h) sinc [(L=2h )(p + p )] and 8 (p) (1=2) (L=h) sinc [(L=2h)(p p )]. (b) Corresponding momentum-probability distribution, 8(p) . The results have not been normalized.
j
j
0
, and the work of Wildöer et al. [25], which is discussed following a detailed and mathematical treatment of the band structure of carbon nanotubes. In both cases, it is emphasized to students that the papers provide striking experimental verification of the very results developed in class. Examples of articles on emerging devices are [26]–[29]; such articles serve to show that the fundamentals in the course do lead to practical electronic applications, and they also motivate the derivation of an equivalent circuit for a ballistic transistor [17] that is performed at the end of the course. In the assignments, a similar strategy is employed. A notable example is a homework problem that is assigned following the course material on band structure, where the students read a short monograph on modeling collisions through Fermi’s golden rule and are then led through a calculation of the ionized impurity mobility in GaAs. Upon obtaining the final result, students are asked to plot the analytical values along with the experimental values from Stillman et al. [30], and they are astonished to see the remarkable agreement. Such examples provide crucial satisfaction for students by letting them experience a calculation that starts at the atomic level (e.g., with
The author has taught the nanoelectronics class for the past four years, during the Winter semesters of 2006–2009, inclusive, at his institution. Class sizes during these years were 37, 44, 38, and 71 students, respectively, and enrollment for the Winter 2010 term is expected to be approximately 85 students. The USRI results for the nanoelectronics class are presented in Table I. Possible answers to each question range from 1 (Strongly Disagree) to 5 (Strongly Agree), as detailed in the caption of Table I, and the reported overall score for each question represents a calculated and effective median of possible responses; a calculation of the median is required since the five-point rating scale constrains responses to a set of discrete values, whereas the underlying attribute being measured is really continuous. For comparison, the ranks of scores in other classes are presented in Tables II and III; in these tables, the “Tukey Fence” represents a reasonable lower limit beyond which a score could be considered an outlier; the upper fence is usually calculated to be above 5.0 and hence is not reported. Details on the method of calculation of USRI scores and the Tukey Fence, as well as all other information on the USRI, can be found at the University of Alberta’s USRI Web site [31]. Table I shows the yearly scores for the nanoelectronics class, as well as a four-year average of these scores. These results should be interpreted by comparison to the reference data in Tables II and III. Table II shows the ranks of scores for classes in the Faculty of Engineering from the 2000–2001 academic year to the 2006–2007 academic year, and Table III shows the same for all university classes from the 2005–2006 academic year to the 2008–2009 academic year. Ranks of scores only from the Faculty of Engineering beyond 2006–2007 are not available, but the data in Table II were collected from over 4000 classes and hence provide a more than adequate baseline for comparison. Several important points become evident upon examination of the data in Tables I–III. • In each year, the scores in the nanoelectronics class rank significantly higher than the 75th percentile of scores for
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TABLE I YEARLY SCORES FOR THE NANOELECTRONICS CLASS FROM 2006–2009, WHERE 1 = Strongly Disagree, 2 = Disagree, 3 = Neutral, 4 = Agree, AND
5 = Strongly Agree
TABLE II RANKS OF SCORES FROM ALL CLASSES IN THE FACULTY OF ENGINEERING, 2000/01–2006/07
TABLE III RANKS OF SCORES FROM ALL UNIVERSITY CLASSES, 2005/06–2008/09
the Faculty of Engineering and even the 75th percentile of all university classes. This outcome provides one measure of the success of the teaching strategies discussed in Section III. • It is instructive to note the improvement in the scores for the nanoelectronics class in Table I from 2006 to 2007. This coincides with the author learning from his experience in the first year (2006) and more deliberately implementing the strategies outlined in Section III in subsequent years (2007–2009). • By using the average scores in the last column of Table I as a measure of the success of the nanoelectronics class over four years, and by comparing these to the ranks of scores in Tables II and III, the author hopes to demonstrate that it is possible to meet the three simultaneous yet seemingly conflicting requirements outlined in the Introduction to create a well-received course in a challenging field. Furthermore, the sample questions provided in the appendices illustrate that this can be achieved while bringing students to a significant level of sophistication in the subject matter. B. Student Comments on Course Content Student comments regarding the course content are consistent with the USRI scores in Table I and have been almost unanimously positive. Excerpts are provided below and are reproduced verbatim without correction for spelling, punctuation, or grammar. Some students remarked that the teaching approach was indeed effective in introducing an uninitiated audience to the subject.
• … this turned out to be a very interesting course. The “ground-up” approach allowed even complete QM newbies to grasp the concepts relatively easily. • I also like the style of the course, in that it does not take for granted that we know any quantum mechanics and teaches us accordingly. • I have never seen this material before, but I feel like I am walking away with a very firm understanding of nanoelectronics. • [The instructor] turned what could have been a very confusing and difficult course into something accessible and understandable. Other students remarked that they wanted even more exposure to the subject and often referred to a (future) follow-up course. These types of response are consistent with the relatively high ratings for USRI Question 3 received by the nanoelectronics class, which can be seen by comparing the data for Question 3 in Table I to the baseline values in Tables II and III. • This was an excellent course … It’s a shame that more courses like this are not offered. Electrical Engineering lacks courses that deal with the quantum mechanical world … This course should be offered in a 2 part course as to get more insight into this field. • This may have been the best course I’ve taken, I just wish the rest of the nano-electric and cmpe courses were going already. • I am sad that the would-be follow-up course, EE 454, does not exist yet in that form or I would assuredly take it. This field of study is highly engaging … • Amazing course, definately needs a follow-up. Much more practical than I was expecting.
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One student wished for a faster pace of coverage and held the opinion that other students were struggling with material that was not difficult but “simply foreign.” • Fantastic course. I look forward to 454. I wish a bit more material was covered. I realize that many students were/are having difficulty, but do not believe the material to be more difficult to understand—simply foreign. Perhaps a seminar would help those struggling and allow for room to touch on effective mass, Fermi’s Golden Rule, [multi-wall nanotubes], etc. The assignments in the course were well-received, with the caveat that some students felt that the monograph form (where substantial background material is provided on more difficult problems) made them lengthy, a shortcoming that will be addressed in the future by splitting the existing assignments into several shorter ones. • The homework assignments were excellent in furthering my understanding of the concepts … • … assignments took a long time but felt I learned a lot from them. • Assignments were long, but very helpful in understanding the material. • I really enjoyed your homework assignments. They might have been long, but I feel that they have greatly helped me to fully understand the concepts. The use of Matlab is a good idea, as hand calculations would be needlessly tedious and long … or little insight would be gained from it … I hope further courses will be offered on this subject. • I really liked the assignments. Unlike other classes, once completed, I feel I really know the assignment material. • The assignments were actually enjoyable and helped so much with understanding the material. In addition to the above feedback, it is worth mentioning one more comment about the course content that was provided in a student letter; while the feedback in this case is not anonymous, it is included here as a useful summary of how the teaching approach discussed in this paper might help to reach and motivate students and thus enhance their understanding and learning experience in a difficult subject. • [The] innovative teaching style for such a course involved MATLAB examples where a student can visualize events happening in the nano-realm, homework questions designed as labs where a student follows instructions to practice the essential mathematics required for nanotechnology while simultaneously absorbing the key concepts without being overwhelmed by details, and teaching the topics in an order which first grabs the attention of students, then forces them to apply their mind to solve the hardest parts by virtue of sheer interest, and hence lets them discover the uncomplicated truths of science.
V. CONCLUSION The following conclusions can be drawn from the author’s experience in teaching nanoelectronics to undergraduates at his institution. 1) A successful undergraduate course in nanoelectronics should meet several simultaneous yet seemingly conflicting requirements: It has to be compact, covering
topics normally covered in several courses in the sciences; it has to be complete, so that mathematical and nonintuitive theory can be satisfyingly understood; and it must contain sufficient depth and sophistication so that learned concepts can be applied to real device phenomena. 2) A possible strategy to meet these requirements includes the effective use of computer visualization, a careful treatment of quantum mechanics, and a constant connection of theory to practical applications. 3) The effectiveness of such methods are supported by scores and written feedback in the Universal Student Ratings of Instruction at the author’s institution. It is hoped that the author’s successful experience described in this paper will encourage others in the area of solid-state electronics to implement similar courses at their institutions, and that it will contribute positively to the new “bottom-up” [7] paradigm in electronics education.
APPENDIX I SAMPLE PROBLEMS INVOLVING HAND CALCULATIONS FOLLOWING COMPUTER VISUALIZATION (1) A nanoscale MOSFET has states per eV eV and for eV. for A calculation reveals that the self-consistent potential eV when , V, and is V. The equilibrium Fermi level is located at . , , , and (a) Sketch , on four separate graphs, assuming . For each graph, use the veroperation at tical axis to represent , with a scale that runs from eV to 0.6 eV, and include appropriate numbers on the horizontal axis. Shade the area under graph that determines the the current. eV, then find the value of the (b) If mA. current . Answer: and , and at sufficiently (c) At the specified high drain voltages, the self-consistent potential eV. Find the corresaturates to a value of sponding saturation current for the MOSFET. Answer: 2.4 mA. (2) A nanoscale MOSFET is cooled to 0 K in order to investigate two possible channel materials. Voltages V and V are applied. The gate voltage is not measured, but it is known that the self-consistent and the equilibrium Fermi level is potential is . Both channel materials have a constant density of states, per eV for , and a coupling eV. However, channel material coefficient eV for and for , A has for and while channel material B has eV for . Calculate the current for both channel materials, and justify your approach. State which material should be chosen for maximum current.
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Answer: For A, mA. For B, Material A should be chosen.
mA.
APPENDIX II SAMPLE PROBLEMS ON QUANTUM MECHANICS (1) A nanoelectronics engineer wants to investigate an electron governed by a Gaussian wave function
where is a small length in a system that contains the is a constant chosen to ensure proper electron and normalization. A communications engineer gives you the following Fourier-transform relation for a Gaussian time signal
where is a constant. Find an expression for the momentum wave function , and state its value when . Answer: . (2) The method of finite differences is used to solve the Schrödinger equation for an electron in a molecule. It is found that the th and neighboring components of a numerical eigenvector can be written as
where is the lattice spacing and , rad/s being with being the mass and the molecule’s “SHO frequency.” If the electron’s potential energy can be approximated , then find the corresponding numerical eigenby value when is small. Express your answer in the units of eV. for small . Hint: Remember that eV. Answer: APPENDIX III SAMPLE PROBLEMS ON THE APPLICATION OF THEORY (1) An approximate - relation for graphene near the point in -space is
(a) Starting from this - relation, and only for the band of energies, find an expression for the corre, and compute the sponding density of states at eV. Use numerical value of nm, , and eV, and assume a sheet of dimensions nm . Hint: To begin, let . states/eV. Answer: (b) Use the - relation quoted in part (a) to obtain an expression for the bandgap (difference between the
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and solutions at ) of a zigzag nanotube in terms of the roll-up parameter and an integer choice , and then calculate the bandgap for and . State your answer in eV. Hint: for Recall the circumferential vector is a zigzag nanotube. Answer: 0.16 eV. (2) Consider the semiclassical description of a ballistic nanotransistor with a one-dimensional parabolic conduction subband. Presume that the source and drain Fermi and lie above the levels , and that . With the aid band minimum of a carefully drawn diagram, obtain an expression for at the source-drain channel conductance . Show your reasoning and use only the semiclassical equations for current. . Answer:
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[23] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon, “Quantized conductance of point contacts in a two-dimensional electron gas,” Phys. Rev. Lett., vol. 60, pp. 848–850, Feb. 1988. [24] D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, “One-dimensional transport and the quantisation of the ballistic resistance,” J. Phys. C, Solid-State Phys., vol. 21, pp. L209–L214, 1988. [25] J. W. G. Wildöer, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C. Dekker, “Electronic structure of atomically resolved carbon nanotubes,” Nature, vol. 391, pp. 59–61, Jan. 1998. [26] Y.-M. Lin, K. A. Jenkins, A. Valdes-Garcia, J. P. Small, D. B. Farmer, and P. Avouris, “Operation of graphene transistors at gigahertz frequencies,” Nano Lett., vol. 9, no. 1, pp. 422–426, 2009. [27] C. Rutherglen and P. Burke, “Carbon nanotube radio,” Nano Lett., vol. 7, no. 11, pp. 3296–3299, 2007. [28] Z. Chen, J. Appenzeller, Y.-M. Lin, J. Sippel-Oakley, A. G. Rinzler, J. Tang, S. J. Wind, P. M. Solomon, and P. Avouris, “An integrated logic circuit assembled on a single carbon nanotube,” Science, vol. 311, p. 1735, Mar. 2006.
[29] G. F. Close, S. Yasuda, B. Paul, S. Fujita, and H.-S. P. Wong, “A 1 GHz integrated circuit with carbon nanotube interconnects and silicon transistors,” Nano Lett., vol. 8, no. 2, pp. 706–709, 2008. [30] G. E. Stillman, C. M. Wolfe, and J. O. Dimmock, “Hall coefficient factor for polar mode scattering in n-type GaAs,” J. Phys. Chem. Solids, vol. 31, pp. 1199–1204, 1970. [31] “Universal Student Ratings of Instruction (USRI),” University of Alberta [Online]. Available: https://www.aict.ualberta.ca/units/client-services/tsqs/usri Mani Vaidyanathan (M’99) received the Ph.D. degree in electrical engineering from the University of British Columbia, Vancouver, Canada, in 1999. He is presently an Assistant Professor with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada. His research interests are in the modeling, simulation, and understanding of electronic devices for future technologies, with a present focus on the radio-frequency performance of carbon-nanotube and graphene-based devices. Dr. Vaidyanathan is an inaugural recipient of the University of Alberta’s Provost’s Award for Early Achievement of Excellence in Undergraduate Teaching.
