so we will continue our discussion about ah
the conduction of carriers in metal and semiconductor
so here you see this if we apply electric
field on on a sample whether it is metal or
semiconductor then what is the what what is
the current density now i want to apply electric
magnetic field along along with this ah electric
field means electric field is there and because
of that this current is flowing through the
sample now if i apply electric field magnetic
field on it so what will happen that we want
to study
so let us take ah as as simplified ah ah um
ah model means ah just one dimensional ah
um conduction so means if i take a sample
rod type of sample or of this just sample
i think i have to is slightly or i can take
ok fine so these the these the sample ah dimension
or geometry of the sample so these the x direction
say this is the x direction this is the y
direction this is the y direction this is
the y direction this is the z direction this
is the z direction ok this y this is z ok
so length is along the x axis length is say
l and this ah this is width of the sample
width of the sample so this is say w and these
the thickness of the sample we say it is t
this is a t thickness of the sample this t
right so its its t thickness of the sample
is t width of the sample is w and length of
the sample z right so we have applied electric
field along the x direction along the x direction
ok so due to this electric field current will
flow through this along the x direction ok
so jx so i can now give this jx jx is may
be ah current density jx because of the electric
field because of the electric field so i was
even electric field this ok because of this
electric field along this so this let us say
x right electric field is given along x direction
so current is flowing in this direction so
without so before apply any magnetic field
before apply any magnetic field b equal to
zero right so this will be the current density
this will be the current density j equal to
nev so i will put one negative sign i will
put one negative sign that we have seen earlier
right and yes so ah so this ah so ah so now
if i apply magnetic field if i apply magnetic
field so what will happen so if i choose the
direction of magnetic field is along the along
the z direction along the z direction so i
have drawn like this means its ah this sample
is under uniform magnetic field so field direction
is in in the so all the whole the sample are
getting are getting expressed with this ah
magnetic field that magnitude of this field
is b z so z we are giving this direction is
along the z direction
so now now before applying magnetic field
is charge carrier was moving charge carrier
was moving right so whether it is it is so
just let us consider the metal then we can
generalize to the semiconductor so that means
electron are moving electron are moving in
which direction so if it is electric field
in in this direction so field are having this
this is having negative ah electrons are negative
charge ok so it will it will move in this
direction ok this negative x direction but
conventionally current is taken that this
is along opposite to the ah electron flow
ok
so so current is ah ok so so this is the case
now if i if i consider the effect of this
magnetic field on this on this on this charge
so now charge is moving charge has velocity
right now that moving charge will experience
force because of this lorentz force what is
lorentz force due to magnetic field lorentz
force f equal to minus e so for electric field
its there so these v cross b v cross b ok
so what will be the force direction that will
depend on the direction of v and direction
of of of b right so direction of b is along
the k direction i can write this is a k direction
unit vector k with z direction right so i
think i can i can tell here so this f of i
can write here f that for lorentz force will
act on this moving charge carrier so that
is minus e right then v cross b right
now in order geometric b is along x direction
so i can write vx along this unit vector i
and this b i can write this it is along z
direction so bz along k direction unit vector
k right so now here basically you are getting
minus e vx b z into here i cross k so i cross
k is basically minus j because k cross i equal
to j so i cross k will be minus j so this
is j and minus minus plus ok
so here some force is lorentz force will act
on the moving charge carrier because if v
zero so there is this lorentz force force
will not act on this on the charge carrier
all the magnetic field is there so in presence
of magnetic field moving carrier only get
affected field force ok so that force is is
along the z direction that force is along
the ah along the z direction right means along
the y direction along the y direction so this
is the y direction so force will act on this
in this direction
so it looks like this but problem is that
i cannot say because i have to i have to i
have to i have to this so what is the direction
of v ok so i have to take kth rid of this
v from this expression so v from here i can
get j equal to minus nev right nev so so jx
and this is vx right so jx this is v x so
vx so vx is jx by any minus sign is there
minus sign is there so this is coming as a
vx if i want to replace so vx i have to replace
by so this is e and then j x by ah minus ne
minus ne then bz so these the unit vector
along the y direction ah it is it is telling
this force along the y direction
so i can i can move to move this one and i
can write here f y so it means lorentz force
acting on the along the y direction so thats
why this j we should not be confused with
the dj with this current density ok so so
here we can see that this basically you are
getting minus sign and then ah ad is there
so i will get minus sign ah g x bz and i am
missing any electric field 
ok so so this minus sign divided by n divided
by n divided by n so that is the ah force
i got so due to this magnetic field along
the y direction along the y direction i am
getting lorentz force acting on the moving
ah carriers so minus jx ah bz by n
so here what is this this is the carrier density
the jx it is set constant value some particular
value current density we have shaped in this
sample now particular magnetic field we have
applied so for that in that case so this amount
of ah force will act on the moving charge
carrier so this now here it is negative sign
that means it this force along the negative
y direction this force along the negative
y direction so earlier it was difficult to
say because this vx we dont know this vx its
whether its a because it was of course this
vx one can take negative because it was moving
along this direction right velocity so so
that way if you consider the negative sign
of this vx so same same theory will come but
it is better to just replace ah in terms of
our known direction this is positive direction
this is positive direction z direction this
is of course density is positive so minus
sign is telling so this force is acting alone
just opposite to the ah this negative direction
of the of the along the y axis
so electron are moving electron are moving
electron are moving now these electrons are
experienced force in this direction negative
direction ok so electrons will accumulate
will will move in this direction and come
to this to this phase come to this tortoise
phase and when this electrons are moving ah
towards this phase of density of electrons
will be moved here on this space and ah divisions
of electron will be there this other phase
opposite phase right so here one can say this
its there will be positive charge accumulated
in this side and this a negative charge so
here there is no hole ok so these the electrons
are in this side so its a more negative the
other side equivalent be is more positive
ok
so so what will happen because of this what
will happen so due to this lorentz force here
this charge separation as if here so this
it will give electric field ok so this direction
of the electric field will be positive to
negative ok in this direction so this electric
field along the y direction but it is towards
negative y direction negative y direction
and we can write e y ok so so we have applied
electric field along the x direction because
of that we are getting current now because
of magnetic field along the z direction there
is a voltage there is electric field generated
in the in our sample and that is just along
the y direction ok
so this electric field is called a hall electric
field this electric field e y is called hall
electric field ey is basically called hall
electric field hall electric field right electric
field n corresponding voltage and corresponding
voltage corresponding voltage is called hall
voltage so v we write vh ok but it is along
this y direction so vh equal to basically
ey and this separation right so this is w
this is w so ew will be the voltage corresponding
to this hall electric field so this is called
hall voltage right
so so this from here i can ah so so these
electric field because of this electric field
if this electric field this electric field
does not form what will happen because of
this magnetic field magnetic field so all
charge will try to move in this direction
ok negative y direction ok but all the hall
all the time it it does not happen so there
should be some steady state and for this steady
state we need some opposite opposite force
on this on this ah on this charge so that
opposite force basically will come when these
voltage this this voltage what is called hall
voltage or hall electric field is ah is formed
here ok or generated here ok so due to this
electric field what will happen so ah it will
act on this carrier my carrier is minus e
negative charge right so this field was in
this direction and now because of this field
electron charge so they is it will to get
force so that is that is the ah that is n
ah again lorentz force right because of electric
field so e minus e into y right
so these force these force these force due
to hall voltage so it will act in this direction
this force will act in this direction right
so this force is basically it will be in this
direction positive x direction ok so so it
will try to get back the electrons again pair
towards positive direction when this this
force was trying to get the electrons in this
direction ok so so this these two has to be
balanced then only equilibrium condition steady
state will arrive so this if y its ah its
the minus j x so basically one has to analyst
so f y f y along the negative direction ok
this has to be equal to this has to be equal
to this
so now note that here ah this is the electric
field direction this e y is basically itself
is negative right so just since electric field
is in this direction so on negative charge
this force will act other direction so it
is positive direction right so so if i consider
the direction then basically this force is
positive y direction so how we are getting
positive or y direction so that force that
force this is because of these force because
of ah hall electric field hall electric field
if i right fh so fh f force due to this hall
electric field so this fh ah will be this
fh has to be this fh has to be fh equal to
my ey in this direction so i have to write
if i consider the direction minus ey and it
is acting on this acting on this ah on this
negative charge right on this negative charge
minus you have to write minus e right minus
e so this is basically eye ok eye
so thats now it is positive direction along
the y direction positive direction this is
negative direction negative y direction so
they are magnitude there so they they with
they are in opposite direction and they will
balance each other so i will i can write this
this is minus j x bz by n equal to here ey
e ok so here also now we can use this now
i dont need direction either positive or negative
so but this then i can replace this y by h
so then this y i can replace by h so eh ok
ok so this i can write eh hall electric field
so ultimately what i am getting what i am
getting so from here i am getting ah eh equal
to so i want to find out hall electric field
so i am getting hall electric field e eh eh
is hall electric field because of this force
is i dont need so hall electric field equal
to hall electric field equal to please just
divide by n so i am getting minus gx bz by
ne right so if i define if i define if i define
r h equal to minus one by ne one by ne if
i define so then this i can write this i can
write eh equal to rh jx bz ok there is the
hall so rh it is called hall coefficient it
is called hall coefficient hall coefficient
it is called the hall coefficient right hall
coefficient its called hall coefficient ok
this its ah ah will see what is this
so it is minus one by ne and if we look at
this ah ah yes so then hall voltage i can
write hall voltage equal to now this y is
basically i have replaced by h is w right
so hall voltage this here let me write v h
hall voltage equal to rh and then ah i have
to multiply it with just w so here ah jx bz
w right so that will be the hall voltage
now again we are not much familiar with jx
right we have familiar with current ix i so
i so i have to multiply with the ah ix equal
to ix equal to a area cross sectional area
into jx right what is the cross sectional
area this is the w into t that is the area
w into d that the cross section so w into
t right so jx i can replace by ix w into t
right so here you replace it here you replace
it jx by ix divided by w into t right now
this w you can remove this w then you are
getting rh ix bz by t right
now this is very important formula you know
this very important formula so if i if current
ix is flowing through these along the exact
axis ok so constant current if i set right
applying some voltage ok so this constant
current is flowing through this sample along
the x axis now if i place the sample in a
magnetic field of constant magnetic field
of of b so along the z direction so its bz
ok
so what will happen here so there will be
voltage along the y direction so i can measure
this voltage i can measure this voltage so
i have to just measure the voltage along this
y direction so i can get directly this vh
i can get directly ix i can get directly bz
i can get directly right because this i applied
these i have applied so i have applied constant
magnetic field i know the what is the ah magnitude
of this magnetic field i have apply constant
current ix what is the magnitude of this from
a meter i can know on this what is the current
is flowing through this and from this from
gauss meter right you know the gauss meter
is used for measuring the magnetic field from
gauss meter reading i can get the magnetic
field and then then this how voltage the voltage
another another voltmeter i have to put ah
across this so i can get i can measure this
voltage vh right
so t is of course from geometry one can find
out the t a thickness of the sample from geometry
one can find out so these also known ok one
can measure or either so you can what what
you can get then you can get rh you can get
rh equal to vh into t divided by ix bz right
so you can get the rh value so rh value is
nothing but one by any one by any right so
any equal to you will get any equal to so
from here you will get value of any you will
get from here so this is the ah so i can replace
this one by one by ne one by ne so negative
sign is there negative sign is there negative
sign is there right so from here i know the
arrangement so from here i can find out ne
value so this negative sign is basically taken
care by this vh ok
so when you will just ah just ah ah i think
if this side is positive terminal of this
your volt meter and this the negative terminal
of the voltmeter so along the so we have chosen
in such a way these it is along the a positive
x direction ok positive this negative so ah
all voltage you will get here you will get
here negative voltage ok you will get here
negative voltage so it is negative voltage
or positive voltage that we have to set reference
right so in this direction when i will set
this reference so when i will get the whatever
voltage that i will take as a negative because
i know this the for electron ok negative charge
so so so keeping this staple is fixed if i
get this voltage just opposite sign then it
is positive voltage i will tell and then this
value this value automatically this value
automatically whatever this rh value because
depending on the sign of this vh will get
rh either positive or negative because because
of sign of vh and then that will tell me that
whether it is ah one by minus ne or one by
plus ne so plus ne then i will tell this carrier
is whole plus ne charge easy but sine is positive
so it is whole so this conduction is because
of hole or if i get this value r is very negative
so this then i will tell its ah a charge negative
charge ah eclectic so this conduction is for
electron right this conduction is for electron
so this hole this this over whatever i have
discussed here this called hall effect very
famous ah discovery is called hall effect
hall effect and it is so important it is so
important that you can you can find out what
you can find out you can find out the you
can find out the carrier density carrier density
you can find out carrier density carrier density
ok it can be n it can be p also in case of
semiconductor p type semiconductor that will
be pe please positive charge ok and the sign
will tell you tell you about the type of carrier
whether it is ah hole or electrons
so this for in specially for semiconductor
it is very important to very important this
effect is very important and based on this
effect ah we can we can tell ah which type
of carrier is there in semiconductor either
it is hole type or or electron ok that means
whether it is n type semiconductor or it is
p type semiconductor and what is the density
of these of this carrier ah that also one
can tell ok in p type it is what is the density
of there so this important so so far whatever
we have studied so for whatever we have studied
everything we try to fire in case of electrical
property ah to understand the electrical properties
so this fee electron theory band theory ah
then in case of semiconductor in case of metal
just all theoretical calculation is to get
the density of states to get the density of
of carriers right and how to measure this
density of carrier that one should know one
should be able to measure and that can be
measured using this fantastic effect hall
effect and it is so simple it is so simple
experimentally also one can ah measure easily
so that is basically your ah this measurement
will tell you the area density it will tell
you the sign of the it will tell you the whether
it is n type semiconductor or p types in conductor
right ok
so so the yes so basically ah this the form
and another important fact is there that ah
ah j equal to yes ah another important fact
i should tell you so in general i can tell
that j in general i can tell that je equal
to nev ok without specifying the direction
nev and now ah if i define if i define if
i define that ok this just divided by electric
field divided by electric field and then multiplied
with electric field ok ok multiplied with
electric field i think its not clear so i
am dividing this expression and multiplying
this this electric field ok so this i can
write j equal to j equal to minus n and again
you forget this sign ok you forget the sign
the sign we are we using this whether it is
electron or whether it is hole
so now it just i am writing but in general
ok in general this i can write this e is as
a q you can be whole or you can be ah electron
so accordingly we can choose sign so in general
if i write this way so i will not bother about
the sign one can put sign whenever necessary
so i can write nq this v by electric field
into electric field so if i define this v
by e v by e this deep voltage deep velocity
per unit electric field if it is defined as
a as a mu it is defined as a mu mu is called
the mobility mobility of the carrier that
is important mobility of the carrier of the
carrier right mobility of the carrier that
is very important ah concept or parameter
for for studying the conductivity
so here what i can write j equal to n q mu
and then you see right ok so now so j equal
to sigma e j equal to sigma e so now sigma
another form of sigma you are getting another
form of sigma you are getting so that is sigma
another form of sigma you are getting that
sigma so j equal to so this one can write
ah get is sigma e so sigma equal to n q mu
right nq mu nq mu ok so this see this is important
because because this conductivity it does
the depend only on the on the carrier density
it depends on depends on the ah carrier mobility
also so it is just like resistance resistivity
no because the velocity depends on the ah
amount of electric field right now if i divided
by this electric field now it is independent
of electric field so its like a constant right
ok
so mobility is more ah independent parameters
like resistivity likes ah conductivity is
is different from the from the conductance
difference from the resistance so that way
this these the parameter ah mu ah is have
significance ok so so this conductivity is
is very important important ah in that sense
that it depends not only on the carrier density
n it depends on the mobility of the carrier
also now mobility is for electron it varies
in which band it is mobility of the hole it
varies in which band it is sometimes it becomes
negative sometimes like mass you know mass
sometimes become negative some sometimes become
positive whether it is in upper band or it
is the bottom of the band etcetera
so that effective mass ah you have seen so
this so we want to measure the mobility also
we want to measure the mobility also right
so to measure the mobility if i want to get
mobility so from here i cannot get mobility
from here i cannot get mobility so for that
what i need so i so here i have measured the
hall voltage and i got the got the got the
things right and what things i got i got the
carrier density i got the whether it is n
type or p type but i cannot get mobility so
to get mobility i have to measure the conductivity
i have to measure the conductivity that can
be measured conductive what is conductivity
conductivity is basically ah you know this
i equal to ah yes so so v equal to i r v equal
to i r right v equal to i r so this if i so
v by i equal to r v r and r equal to rho l
by a i think yes v equal to rho l by a so
this basically i can write yes sigma one by
sigma right
so i can find sigma so sigma i can give this
side sigma and v by i means here i by i by
v so i can do experiment where i will apply
ah voltage and measure the current ok say
in this sample ok i will apply voltage and
measure the current then i know the dimension
of the of the of the of the of my sample what
is the length what is the idea cross section
area then i will get i will get the c right
so if i measure the conductivity then i will
get the i will get the sigma i will get the
sigma i will get the sigma now if sigma is
known if sigma is not conductivity is known
then if n is known yes n is known now because
from here i get the current density from hall
measurement i can get the current density
so come here ah n is known ok then you can
find out is mu so mu you will get in terms
of sigma by n q ok
so n from hall measurement one can get sigma
you can one can get from resistivity measurement
ok resistivity measurement resistivity measurements
and then one can get the mobility so if you
measure the ah resistivity or conductivity
of your sample and if you measure the hall
voltage of your sample ok then you can get
all information on the all information about
the conduct ah or about the parameter which
participate in conduction ok so there is the
carrier or carrier density there is the ah
mobility of the carrier so there is these
two are very important and of course type
of carrier and this that also one can know
so and one can tell which type of of similar
this p type or n type or both at the earth
so that one can tell so thats the thats what
ah yes yes um yes i think another important
fact is there if we looked at this expression
if we look at this expression i just you see
so so whether i can use this formula for whether
i can use this formula for measuring for for
for measuring the magnetic field ok so whether
i can i can make device using the hall effect
what device i want to make i want to make
a device which which will act as a sensor
which can sense the ah magnetic field and
can tell me the magnitude of the magnetic
field ok or reading of of the of the of the
magnetic field ok if i want to get that type
of sensors using the hall effect so whether
it is possible or not so ah for any sensor
with the temperature sensor magnetic field
sensor or gas sensor or or whatever so it
has to so we get the signal we should get
the signal from that sensor that electrical
signal if i get electrical signal which will
be proportional to that ah parameter whether
it is temperature or this magnetic field right
or the density of of gas ok
so that ah that i should get in terms of electrical
signal and this some relation with them then
i can use that property for device purpose
so here you can see you can see if i take
a very small piece of of of of sample ok and
set the current set the current ok ix current
if i said its easy to set i will set the current
and then ah then the magnetic field whatever
magnetic field i want to measure if i put
this sensor to that ah in that magnetic field
in that magnetic field that value i dont know
but i want to know i want to know so what
will happen what will happen ah these r h
is constant of course its its its the property
of the sample right nothing to do it is this
constant and this t constant ok so so what
i will get i will only measure this this hall
voltage now this hall voltage is hall voltage
is proportional to mediate magnetic field
right and these if i keep them constant so
this will be some proportionality constant
say c some proportionality constant
so now when b is increased we are increasing
b so voltage is increasing so i can measure
the voltage and after some calibration i can
tell what is the magnetic field so this can
so this this sample using this hall effect
ah this can be used for measuring the magnetic
field so thats thats what it is there we use
this sensor in or may ah elaborative measure
the magnetic field so this is called hall
sensor hall flow hall sensor or hall flow
ok if i i have a proof arrangement with this
current and if i put in magnetic field then
i will get the ah get the voltage so it sometimes
one has to calibrate it ok then using that
calibration all the time because this calibration
if i keep this is constant so the calibration
factor is all the time it will work so ah
i can use anywhere ok so i will get i will
get magnetic field
now important is sensor for sensor it is important
that this how we are it is very important
ah that ah the sensitivity of the sensor or
regulation of the sensor what does it mean
this sensitivity of the sensor means it very
small so magnetic field is there now i have
put my sensor now i am changing the magnetic
field ok but this voltage change is not much
so it will not be sensitive to the small change
of the magnetic field ok so our proof our
device will be more sensitive if for small
change of magnetic field the change of hall
voltage will be more ok for small change of
magnetic field this change of hall voltage
will be more ok so so thats so a one has to
design the the sensor so here we have scope
you know here you see these things i kept
constant right so rh i cannot control you
know because this is the density ah i ne or
pe whatever ok so nothing to do with that
so i have only ix and t in my hand so ix i
cannot also put very high value then this
sample will be hot so i cannot put a higher
current in the system so kept it also current
a particular suitable value ok
after that this t is there you see this is
in denominator means when t will be smaller
t will be smaller so this factor this constant
will be higher this factor will be higher
if t is higher so this factor will be so this
constant will be higher or smaller it will
depend on we can control with this so if c
is higher then what does it mean c is higher
so small change of magnetic field since this
constant is higher so this vh will be higher
ok because this is the multiplication factor
right so does this if you see any time in
your lab or in ah higher education this we
are using this some proof for measuring the
magnetic field ah that is hall flow and in
that hall flow there is a small ah sample
and that sample is taken is a very thin ok
so because that will four same magnetic field
for this geometry i will get higher hall voltage
so my sensor will be sensitive so whenever
you see whenever we get some formula always
we should look at the at the parameter and
try to find out whether we can use this effect
for some any device purpose so for device
purpose where when we are going to use then
i have to think how i can make it more sensitive
ok so thats one has to look at it
so
thank you for your kind attention ah i will
stop here yeah
