Miniaturization of Electronic System and Various Characteristic lengths in low dimensional systems

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1 Miniaturization of Electronic System and Various Caracteristic lengts in low dimensional systems R. Jon Bosco Balaguru Professor Scool of Electrical & Electronics Engineering SASTRA University B. G. Jeyaprakas Assistant Professor Scool of Electrical & Electronics Engineering SASTRA University Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 15

2 Table of Content 1. MINIATURIZATION OF ELECTRONIC SYSTEM AND MOORE S LAW INTRODUCTION MERITS AND CHALLENGES OF NANOELECTRONICS BASED SYSTEM ELECTRON IN MESOSCOPIC SYSTEM...5. VARIOUS CHARACTERISTIC LENGTH IN LOW DIMENSIONAL SYSTEMS INTRODUCTION How big te eart in comparison wit sun? Wat is te size of atom and sub-atomic particle?...7. DE BROGLIE WAVELENGTH MEAN FREE PATH DIFFUSSION LENGTH SCREENING LENGTH PHASE RELAXATION LENGTH QUIZ AND ASSIGNMENT SOLUTIONS REFERENCES...15 Joint Initiative of IITs and IISc Funded by MHRD Page of 15

3 1 Miniaturization of Electronic System and Moore s law 1.1 Introduction Tis lecture provides you te basics of development in electronics device size and te carge transport in te reduced device size At present, development in electronic devices means a race for a constant decrease in te dimension of device or system. Figure sows te various electronics devices development from large size vacuum/gas filled tube to single molecule based field effect transistor wit size in sub nanomater level. We are well aware of te fact tat we live in te age of microelectronics, a word wic is derived from te size (1 µm) of a device s active zone, e.g., te cannel lengt of a field effect transistor or te tickness of a gate dielectric. However, tere are substantial indications tat we are entering anoter era, namely te age of nanotecnology. Te word nanoelectronics is again derived from te typical geometrical dimension of an electronic device, wic is in nanometer (10-9 ) scale and leads to acieve te Moore s predicated law. Te observation made in 1965 by Gordon Moore, co-founder of Intel tat te number of transistors per square inc on integrated circuits rougly doubles every year. Moore predicted tat tis trend would continue for te foreseeable future. In subsequent years, te pace slowed down a bit, but data density as doubled approximately every 18 monts. A typical electronic device of te fifties was a single device wit a dimension of 1 cm, wile te age of microelectronics began in te eigties. Figure 1 indicates te size of device as a function of time and it indicate te beginning of nanoelectronics from te year 010. Tese predictions are not restricted to nanoelectronics alone but can also be valid for materials, metods, and systems. Joint Initiative of IITs and IISc Funded by MHRD Page 3 of 15

4 Fig. 1. Device size as a function of time 1. Merits and Callenges of Nanoelectronics based system Te reduction in product size is due to srinking te size of individual electronic devices, suc as transistors. Tis reduction also, of course, can lead to improved functionality, as more devices can be packed into a given area. Tere are direct economic advantages of small device size as well, since te cost of integrated circuit cips is related to te number of cips tat can be produced per silicon wafer. Terefore, iger device density leads to more cips per wafer, and reduced cost. Also, as electronics srink, te possibility of furter incorporating electronics wit biological systems (see te adjacent figure) rapidly expands.tus, reduced electronic devices size ave advantages like ig frequency operation, superior functionality and low power consumption coupled wit lesser costs. New effects, like resonant tunneling, quantum electrical conductance, coulomb blockade, Quantum Hall effects, etc. makes low dimensional based electronic devices to ave enanced performance and practical devices like single electron transistors, quantum well lasers, Joint Initiative of IITs and IISc Funded by MHRD Page 4 of 15

5 and optical modulators, etc. were work on tese effects. Terefore, tere are many factors driving te miniaturization of electronic devices. However, in reality, tese reduced dimensions affect almost all electrical parameters like amplification, transconductance, frequency limits, power consumption, leakage currents, etc. wic we are going to see in te subsequent modules. In integrated cip, if 100 transistors were fabricated wit a size close or greater tan 100nm, ten carge transport can be treated classically. However if te same number of transistors fabricated to a size ~10nm; ten classical teories does not fold. Te devices wit a size of 10nm can be described quantum mecanically. Also, tere are significant callenges / problems in srinking conventional devices to te nanoscale suc as device fabrication, operation and eat dissipation. Let we see ow te reduced dimensional device reflects in carge transport. 1.3 Electron in Mesoscopic system Te word mesoscopic deals wit te size lying between macroscopic and te microscopic or atomic system. Te mesoscopic size usually ranges from a few nanometers to 100nm and also called as nanostructures. In terms of material dimension, nanostructures can be broadly categories as zero dimension (0D), one dimension (1D) and two dimension (D) nanomaterials. Te carge transport in tese reduced dimensions is unique wen compared wit its bulk one. For example, it is well known tat te conductance G of a macroscopic conductor is given as G A σ L (1) Were σ is te conductivity and te intrinsic property of te material, A is te area of cross-section of te material, and L is its lengt. So te conductance decreases as te cross-sectional area is reduced and it increases as te lengt of te conductor is reduced. Joint Initiative of IITs and IISc Funded by MHRD Page 5 of 15

6 One may suspect tat te conductance goes to infinitely large values as te lengt of te conductor is made extremely small. But tis is not true. Te above mentioned simple scaling law (or te so-called omicbeaviour) breaks down at mesoscopic lengt scales (sub-micrometer lengt scales). It does not become infinite, but it reaces a limiting value G c wic we will see in te subsequent module of tis course. To understand te cause of te breakdown of tis simple scaling law, one as to take into account te quantum nature of te electrons, according to wic, te electron is not a classical tiny carged particle, but a quantum mecanical wave-particle. Tis wave caracter of te electron is responsible for many analogies in te field of Anderson localization, propagation of ligt troug a random medium, and mescoscopic conduction troug a disordered sample wit static disorder. Tus, te mesoscopic lengt scales (usually ) can also be defined as lengt scales at wic te wave caracter of electrons as definite effect on te measurable pysical properties, suc as conductance. Te conductance no longer monotonically varies in a field effect transistor if gate voltage is varied but it sows `jumps' or `steps' in units of: e ( ) G c Tis is a universal caracter and independent of material. Tus wen te size of te system becomes smaller, te electrical conductance does not obey classical transport. Joint Initiative of IITs and IISc Funded by MHRD Page 6 of 15

7 Various Caracteristic in low dimensional systems.1 Introduction Tis lecture provides you te different scale for defining te dimension of a pysical system or device or materials size in low scale.1.1 How big te eart in comparison wit sun? If te radius of te eart isassumed to be a widt of an ordinary paper clip ten te radius of te sun would be rougly te eigt of a desk, and te sun would be about 100 times of eart (see te adjacent figure)..1. Wat is te size of atom and sub-atomic particle? Te typical size of an atom is about 1Å (10-10 m). Tere are electrons orbiting around te atomic nucleus aving size less tan m. Centre of atom were Joint Initiative of IITs and IISc Funded by MHRD Page 7 of 15

8 almost all te mass of an atom concentrates is called te nucleus, wic as a typical size of m. Te nucleus consists of protons and neutrons aving a size of m. Te size of an entity varies from less tan to m to km te diameter of sun. All te matter is composed of atom and ence te electrons state and energy decides te properties of te matter. Te electron in different nanostructures as wavelike properties and te beaviour depends on te sape of te structures. Tus te states of te electrons in low dimensional systems are wavelike and te transport is similar to wave propagation in te waveguides. To describe te beaviour of electrons in low dimensional system, it is convenient to define few caracteristics lengt. Tese caracteristics lengt are benc mark, below tis size, if te electrons are confined, ten te materials may as new properties tan in its bulk form. Let we see te various caracteristics lengts defined in mesoscopic systems.. de Broglie wavelengt In 194, Louis de Broglie, a Frenc pysicist, suggested tat te wave-particle dualism. He proposed tat every kind of particle as bot wave and particle properties. Figure sows te wave nature associated wit electrons in eac orbit of an atom Fig.. Te standing de Broglie wave of electrons in an atom Joint Initiative of IITs and IISc Funded by MHRD Page 8 of 15

9 Small particles like electron wit momentum p, as wavelike properties and te wavelengt associated wit it is given by de Broglie wavelengt p * m v λ ( 3) Were m* is te effective electron mass. In semiconductor, te electrons beaves dynamically and as m* instead of rest mass m o as in vacuum. In semiconductors like GaAs and InSb, were m* is lesser tan m o and it sows tat for smaller effective mass, larger te de Broglie wavelengt and is easier to observe te quantum effect in nanostructures. Hence in te semiconductor nanostructure new properties can be observed if one or two dimension is in te order of de Broglie wavelengt. If te velocity v is given to an electron by accelerating it troug a potential difference V, ten te work done on te electron is ev. Tis work done is converted into te kinetic energy of te electron, tus 1 mv ev mv mev ( 4) If we ignore te relativistic considerations, ten m * m o, te rest mass of te electron, ten using eqn (4) in (3), te wavelengt associated wit te electron λ mev 1.7 nm V λ ( 5) Joint Initiative of IITs and IISc Funded by MHRD Page 9 of 15

10 .3 Mean free pat Te lengt covered by an electron between two inelastic collisions wit impurities or ponons is termed as mean free pat. In metals, te mean free pats of electrons were in te order of nm, wile in pure semiconductors in te order of micron at mk temperatures. Below tis scale, te electron transport becomes ballistic and classical concepts i.e. te diffuse motion of electrons will be no more applicable. Te mean free pat of a single species of gas is given by ( 6) 1 l m nσ were n is te gas density and σ is te collisional cross section. Te mean free pat expression for nitrogen can be obtained approximately using ideal gas equation: N P KT nkt V P n ( 7) kt were V is te volume occupied by te gas, n is te total number of gas particles, and k is Boltzmann's constant. Te term N/V is te density, n. using equation (7) in (6), te mean free pat for gas particles is kt l m ( 8) Pσ Joint Initiative of IITs and IISc Funded by MHRD Page 10 of 15

11 Te collisional cross section σ πd, were d is te gas molecule diameter. It is also representing a measure of area (centered on te centre of mass of one particle) troug wic te particles cannot pass eac oter witout colliding. If we treat te nitrogen molecule as ard spere, ten te diameter of nitrogen molecule is twice of diameter of single nitrogen atom. For any gas particle, te diffusion lengt is in te order of nanometer..4 Diffusion lengt If excess carriers are injected into a semiconductor tey diffuse away from te point of injection. As tey diffuse tey recombine at a rate described by teir lifetime and witin a certain distance te excess carrier density falls to zero. Tis distance is called te diffusion lengt of carriers. It can also be defined as te average lengt a carrier moves between generation and recombination. In low dimensional materials of reduced size L, te electrons transport may be eiter in ballistic or diffusive type depending upon means free pat and size. If diffusion lengt is far greater tan L, te electrons follows ballistic transport i.e., transports witout scattering wit impurities of ponons; owever te surface is te only scattering centre. If l m << L, te electron transport follows diffusion process caracterised by diffusion coefficient D e. Te diffusion lengt l d is ten given by l d Dτ e ( 9) D e ktµ e ( 10) e Were τ e is te relaxation time, k Boltzmann constant, µ e mobility of electron at te temperature T. Joint Initiative of IITs and IISc Funded by MHRD Page 11 of 15

12 .5 Screening lengt Te ionized form of impurities or dopants in extrinsic semiconductors acts as scattering centre. Te free opposite carge carriers screens te dopants and ence te net electric potential due to dopant or impurities over distance is reduced. Te screening lengt l s is given as εkt e n l s ( 11) were e, ε and n were electronic carge, dielectric constant and carrier concentration. l s usually lie in te range of 10 to100 nm, and tus it indicate te attenuation of carge disturbance. Above expression sows te screening lengt for metal is less tan semiconductors. Te coulomb and screening potential can be correlated as φ SP e γ r / λ r s ( 1) were γ 1/4πε 0. As λ s tends to infinity, te screening effect disappears, giving Coulombic potential. Te screening effect discussed above is not only for an impurity or dopant, but it is related to any carge carrier perturbation..6 Pase - relaxation lengt Te pase-relaxation lengt lφ is te average distance tat an electron travels before it experiences inelastic scattering wic destroys its initial coerent state. Typical scattering events, suc as electron-ponon or electron-electron collisions, cange te energy of te electron and randomize its quantum-mecanical pase. Impurity scattering may also contribute to pase relaxation if te impurity as an internal degree of freedom Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 15

13 so tat it can cange its state. For example, magnetic impurities ave an internal spin tat fluctuates wit times. In ig mobility degenerate semiconductors, pase relaxation often occurs on a timescale τ φ wic is of te same order or sorter tan te momentumrelaxation time τ m. Ten te pase relaxation lengt is I φ v f τ φ ( 13) were V f is te Fermi velocity. In low-mobility semiconductors te momentumrelaxation time τ m can be considerably sorter tan te pase-relaxation time τ φ and diffusive motion may occur over a pase-coerent region; ten l φ Dτ φ ( 14) were D is te diffusion constant and is equal to v τ. f m 3 Quiz and Assignment 1. At lengt, materials does not obeys om s law. Electronic devices in te fifties were in te size of cm. 3. Te conductance of low dimensional sows steps in units of. 4. Te new effects like,,, and were observed in low dimensional system. 5. In low dimensional materials, electron is treated as 6. At wat condition, te transport of electron is ballistic a. L < l m b. L < l φ c. L > l m d. L > l φ 7. Wat is te diffusion coefficient of Ge at 303K, if it as a mobility of 1880 cm V 1 s 1? 8. An object wit 50cm 3 volume as mass of 50g and flies wit a velocity of 75 m/sec for 10 min. Wat is te de Broglie wavelengt associated wit it? Can we observe te wave associated it? 9. Wat is te wavelengt of an electron moving at 5.31 x 10 6 m/sec? Given: mass of electron 9.11 x kg and 6.66 x J s Joint Initiative of IITs and IISc Funded by MHRD Page 13 of 15

14 10. A Silicon P-N junction potodiode as a uniform cross sectional area of 0.04cm. Te diffusion constants of te electrons and oles at room temperature were D n 35cm /sec and D p 1.5cm /sec. Hole lifetime in N- region is τ p 100µsec; and electron life time in te P-region is τ n 35µsec. Calculate te Debye lengt for oles in N region and electrons in P region. 11. Calculate number of gas particle in a camber aving pressure of MPa at 300k. 1. Calculate te limiting resistance in low dimensional materials. 3.1 Solutions 1. mesoscopic. 1 cm e 3. G c 4. resonant tunneling, quantum electrical conductance, coulomb blockade, Quantum Hall effects 5. wave nature 6. Ans. (a) 7. µ kt D e x1.3806x10 x cm s x10 8. Te de Broglie wavelengt λ is given by, λ mv x 10 λ 3 (50 x 10 x 75) x 10 m Reason: No, since te wavelengt associated it is very small compared to te volume of te object. Hence te wave nature associated to it is not explicit. 9. de Broglie's equation is Joint Initiative of IITs and IISc Funded by MHRD Page 14 of 15

15 λ mv x 10 λ x 10 x 5.31x x A o 10 m In te N-region, L p D τ p p 1.5x x10 cm In te P-region, L n D τ n n 35x35x x10 cm 11. P 10 n kt 3 ( ) We know e Gc 1 R G c 6.65x10 x(1.60x10 4 References ) 1.9kΩ [1] J.M. Martinez- Duart, R.J.Martin-Palma, F. Agullo-Rueda, Nanotecnology for Microelectronics and Optoelectronics, Elsevier, 006. [] William D. Callister, Jr., Fundamentals of Materials Science and Engineering-An Interactive e-text, Jon Wiley & Sons, Inc., 001. [3] Guozong Cao, Nanostructures & Nanomaterials Syntesis, Properties & Applications, Imperial College Press, 004. Joint Initiative of IITs and IISc Funded by MHRD Page 15 of 15