Prop Appl Carbonaceous Nanomat
Prop Appl Carbonaceous Nanomat ECE 695V
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This 12 page Class Notes was uploaded by Hulda Gaylord on Monday September 21, 2015. The Class Notes belongs to ECE 695V at University of Virginia taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/209735/ece-695v-university-of-virginia in ELECTRICAL AND COMPUTER ENGINEERING at University of Virginia.
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Date Created: 09/21/15
ECE 587687 7 FUNDAMENTALS OF NANOELECTRONICS UNIVERSITY OF VIRGINIA Spring 2009 TakeHorne Final Due Wednesday May 6th print name above Please show all your work7 and give clear answers Attach numerical code if used Usual disciplinary issues apply 7 no unauthorized assistance of any sort is allowed You can consult your textbook7 slides7 HW solutions and class notes You are allowed to look up the internet just to extract the parameter values for Problem 3 Answer all 4 problems Problem 125 Problem 225 Problem 325 Problem 425 TOTAL100 On my honor as a student I have neither giuen nor receiued unauthorized assistance on this exam sign name above Problem 1 Current conduction 25 points chamwl Consider a wire with a cut In a basis set of one orbital per wire atom it is easy to write down 2t 0 the Hamiltonian of the two atom device boxed above as H 0 2t 1 What are the self energies 212 and the broadenings P1 due to the two 1 D contacts shown Remember each must be a 2x2 matrix Keep your answers in terms of E t and ka 2 x 5 10 2 Calculate the Green s function G do not just write down the formula 7 evaluate the inverse matrix 5 3 Calculate the transmission TE traceP1GP2GT Does this make sense 82 Problem 2 Heat ow 25 points 1 In an earlier exam7 you calculated the charge current and determined the electronic con ductance quantum Let us now try to nd the thermal conductance quantum with an analogous process We impose a temperature di erence instead of a voltage di erence between two contacts so that al ag a but T1 T2 AT The thermal current is given by 1Q g dETEE 7 i mm 7 mm lt1 where the energy or thermal charge injected by each contact is proportional to E 7 a while the Fermi function difference arises from the temperature difference Assuming a ballistic channel7 let s try to calculate this thermal current Taylor expand the difference f1E 7 f2E z AT 8fE8T Given that fE 11 em where z E 7 aI BT7 simplify the leading term for a given T as a compact function of X Keep the chain rule in mind 15 2 Substituting this expression into the thermal current integral7 changing variables to z E7 ukBT7 and using one of the integrals provided at the back of the exam7 write down the thermal conductance quantum a IQAT It should be a compact ratio of a few terms only 6 3 Substituting the values k3 138 x 10 23JK and h 66 x 1073415 nd the value of a in units of wattskelvin keeping in mind that 1 watt 1Js 0 an my nanuwue In m wuxds we W1 mans on W Duh semqummame spam at h a g gt1 and see 1 m bandmcure Haws us m understand Lhasa 7 d amund them There axe ax eqmvalem axes cmespundmg m Lhe r 7 x duecunn namely Lhe 119 11g and 110 axes 5 mm L W ax m dspexsuns max 32 b mum can be mm as Em we ag2m rz 1 I 2mmd mm pmmums mung 19 10 Pm 1mmaehhe mband5 xscemetedamundlc wky0 andk0 W reman To uro zorhw 2m xmzpty qwduz the 1g and 0 Wm zntzwm 0 W and man 1c vacmm 0 M mm mm a m 1c and 9 m 1717 mamam barairdwgmm about m 39 m m WLh mass m Tm awn amt m m wmphmy 0 m banirdmgmm about 03 we mu ngl 1 By consulting the internet or your text book7 list the values of a the lattice constant of Si in Angstroms 7 NOT the Si Si bondlength7 ii k0 as a fraction of 27rot7 and the iii longitudinal and iv transverse masses ml and mt in terms of the free electron mass mo7 for silicon 4 2 a Show that four of the six equivalent E 7 k bands7 upon quantizing the kg and kz sets7 are responsible for creating the 1D subbands at k 0 in the above gure Which four are they b Why are they centered at k 0 Remember you are only looking at the conduction band7 and that the actual answer has more features than you get from this effective mass analysis7 since the atomistic simulations have many orbitals and higher bands 3 3 3 a Show that the remaining two which two create 1D E ks that are offset from k 0 above b What are these offset k values remember that the original k values need to be folded into the new Brillouin zone7 which runs from 77ra to 7ra The way to fold them is to add a reciprocal lattice vector i27ra so it is brought into this Brillouin zone Check your calculated k value against the gure above 2 4 4 what are the effective masses of the two sets of bands you just calculated Once you start with the six bulk E k bands7 and eliminate the kg contributions which are quantized constants7 the only variable is km that appears in the two sets of bands with its own effective mass This would be a combination of ml and mt in each case 25 25 5 Given the masses ml rm and the width W which creates different quantization con ditions for the six original loands7 nd the vertical energy offset between the lowest conduction band at k0 and the other two next higher conduction hands This should agree with the plot 4 Problem 4 Trap channel interactions 25 Let us try to understand how a trap can block the current through a channel and create a drop in current This is very relevant for present day devices The trap is essentially a non conducting level that communicates very weakly with the contacts small 39ys so that once a charge is injected onto it it stays there for a very long time Filling the trap however pushes the conducting level channel with larger 39ys out of the conduction window through Coulomb repulsion thus blocking the channel Our aim in this problem is to create a toy numerical model for this trap channel interaction The energy diagram below shows a conducting channel centered around 60 with equal couplings 39yl w to the contacts The channel density of states is described by a broad Lorentzian DchE Lg ME 7 so MW the level so that the channel is initially empty Let us start raising the drain electrochemical potential ag EF 1 qV by putting a negative voltage on it relative to the source which is held at al EF At some point the channel is lled by the drain and it starts conducting The Fermi energy of the channel lies below 1 What is the expected maximum current through this level Set up a toy model from chapter 1 to numerically compute the l V through this system for a voltage range between zero and 06 V ignore Coulomb interactions and self consistency Use the following parameters EF O 60 02eV 39yl 39yl 0005eV kT 0025eV Choose 101 voltage points in the voltage range between 0 and 06 V For the energy integration needed to compute the current choose 501 energy grid points between 4 to 4 eV Does your maximum computed current agree with your analytical estimate 3 5 2 Let us now include a trap7 which acts as another level with the same shape of the density of states as the channel7 but different parameters 61 04cV7 and weak couplings 39ylt 39th 000055V to the contacts Once the drain voltage is high enough to ll the trap7 you will get additional current from the trap We assume the levels themselves do not slip7 in other words7 no local Laplace or Poisson potential for either level What is the analytically expected additional current due to the trap ie7 the current contribution from the trap alone7 Set up the numerical code to include this additional density of states ie7 in your computed current7 you now have a sum of two terms each proportional to the respective density of states and the 39y combination Does your computed current from the trap agree with your analytical estimate 3 5 e 01 L112 Cam In of mm 75 s em In m wards L112 channel has a 0mm mm due map mg mp lave sees nu Laplace m Caulumb pmenua and stays xed yo m m a cunenc Lhmugh my WhaL dues m w Shaw Dltplam Wm ls happemng What demmmzs me cunenc um ls m m an huh bus mmugh Lhs 5mm 7 2
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