ELEN$6906,$HW$#3$
Spring$2014$
Homework$#3$
Quantum"of"Conductance"–"20"points"
DUE$@$Beginning$of$Class:$$Friday,$February$21"
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1)"Explain"why"Ohm’s"law"for"conductance"does"not"apply"to"low"dimensional,"ballistic"
conductors.""What"is"used"instead"of"Ohm’s"law?""(2"points)"
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2)"Consider"a"copper"nanowire.""The"resistivity"of"copper"is"ρCu"="1.68"x"10K8"ΩKm.""If"the"
nanowire"has"a"diameter"of"2"nm,"how"far"apart"would"two"contacts"have"to"be"in"order"
to"measure"a"resistance"(according"to"Ohm’s"law)"equal"to"the"quantum"of"resistance?""
If"we"assume"transport"in"this"copper"nanowire"to"be"ballistic"with"perfect"contacts"
established"at"this"calculated"length,"what"would"the"actual"resistance"be?""(2"points)"
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3)"This"problem"involves"a"channel"with"a"density"of"states,"D(E)"="D0#u(E#–#μ),"where"D0#
is"a"constant"and"u"is"the"unit"step"function.""The"channel"is"connected"to"two"contacts"
with"the"same"electrochemical"potential"μ"positioned"as"shown"in"the"below"diagram.""
An"ammeter"is"connected"to"detect"any"current"flow"between"the"contacts"through"the"
channel."(8"points)"
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i) Obtain"an"expression"for"the"current"in"this"device"if"the"temperatures"of"the"
contacts"(T1"and"T2)"are"different."
ii) Using"a"diagram"like"the"one"shown"above,"sketch"the"Fermi"functions"for"the"
distribution"of"carriers"in"the"contacts"for"the"case"of"T2">"T1.""Indicate"the"
energy"states"where"current"is"flowing"in"the"channel"DOS."
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4)"Two"people"(A"and"B)"calculate"the"currentKvoltage"characteristics"for"a"device"with"a"
single"energy"level"ε"located"0.25"eV"above"the"equilibrium"electrochemical"potential"μ"
and"obtain"two"different"results"as"shown"on"the"next"page.""(8"points)"
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1"
ELEN$6906,$HW$#3$
Spring$2014$
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Both"A"and"B"have"calculated"correctly"but"they"have"made"different"assumptions"about"
how"the"level"ε"moves"under"the"applied"bias."""
i) What"was"the"assumption"made"by"person"A"regarding"how"ε#moves?"
ii) What"was"the"assumption"made"by"person"B"regarding"how"ε#moves?"
iii) From"the"given"IKV’s,"estimate"the"value"of"γ1"assuming"it"is"equal"to"γ2."
[HINT:""I"="(q/ħ)((γ1γ2)/(γ1+γ2))]"
iv) What"is"the"corresponding"carrier"escape"rate"in"per"second?"
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2"
ELEN$6906,$HW$#3$
Spring$2014$
1)"Explain"why"Ohm’s"law"for"conductance"does"not"apply"to"low"dimensional,"ballistic"
conductors.""What"is"used"instead"of"Ohm’s"law?""(2"points)"
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Ohm’s$law$for$conductance$says$that$the$conductance$is$inversely$dependent$on$the$
length$of$a$material.$$This$would$suggest$that$the$shorter$the$length$of$a$material,$the$
higher$the$conductance$can$be$until$it$approaches$infinity$when$the$length$approaches$
zero.$$However,$quantum$mechanics$shows$that$there$is$a$maximum$conductance$(per$
mode),$known$as$the$conductance$quantum,$for$any$lowWdimensional$material$that$is$
measured$with$two$contacts$(it$is$all$contact$resistance$limited).$$
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!
3"
ELEN$6906,$HW$#3$
Spring$2014$
4)"Two"people"(A"and"B)"calculate"the"currentKvoltage"characteristics"for"a"device"with"a"
single"energy"level"ε"located"0.25"eV"above"the"equilibrium"electrochemical"potential"μ"
and"obtain"two"different"results"as"shown"on"the"next"page.""(8"points)"
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"
Both"A"and"B"have"calculated"correctly"but"they"have"made"different"assumptions"about"
how"the"level"ε"moves"under"the"applied"bias."""
i) What"was"the"assumption"made"by"person"A"regarding"how"ε#moves?"
ii) What"was"the"assumption"made"by"person"B"regarding"how"ε#moves?"
iii) From"the"given"IKV’s,"estimate"the"value"of"γ1"assuming"it"is"equal"to"γ2."
[HINT:""I"="(q/ħ)((γ1γ2)/(γ1+γ2))]"
iv) What"is"the"corresponding"carrier"escape"rate"in"per"second?"
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i)$Person$A$assumed$that$ε#moves$with$the$right$contact$by$the$entire$applied$voltage.$$
Hence,$the$following$scenario$takes$place:$(2"points)$
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ii)$Person$B$assumed$that$ε#moves$according$to$½$of$the$applied$voltage,$creating$the$
following$scenarios$for$current$flow$under$both$positive$and$negative$bias:$(2"points)$
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!
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