霍尔效应Hall Effect

U H = V A − VB .
(2)
The Hall bias is determined by the deviation of the charge carriers, which form a current through the sample, under the action of the Lorenz force: r r r FL = ±e v ∧ B , (3) r where v is the average (drift) velocity of the charge carriers moving r through the sample under the action of the field E and e is the elementary
(
)
(7’)
(
)
(8’)
(10)
where n and p are the electron and hole concentrations. Solving the r equations (6 – 10) under the condition j y = 0 ( j = ( j , 0, 0) ), we obtain:
(
)
charge ( e ≅ 1.6 ⋅ 10−19 C). The absolute value of the Hall field intensity is:
EH = UH . a (4)
r The external and the Hall electric fields produce the electric force Fel : r r r r Fel = eEt = e E + EH , (5) r where Et is the total electric field. The total force that acts on the charge
j = σ(B )E , EH = RH (B ) jB,
(11) (12)
where the conductivity σ and the Hall constant RH are given by the relations:
168
npµ nµ p µ p − µ n 2 B 2 , σ(B ) = e nµ n + pµ p 1 + 2 2 2 2 2 nµ n + pµ p + ( p − n ) µ n µ p B
*
(1)
Extrinsic semiconductors are semiconductors with impurities in which the electric conduction is done either by electrons (semiconductors with donor impurities called n-type semiconductors), or by holes (semiconductors with acceptor impurities called p-type semiconductors). ** Galvanomagnetic effects are physical phenomena that appear in substances during the interaction between the externally applied magnetic field and the charges moving through those substances.
166
Figure 1.
r Under the action of the external electric field E = (E , 0, 0 ) , through
the semiconductor sample flows a current I. Applying on the sample the r magnetic field B = (0, 0, B ) , a potential difference U H , called Hall bias, r r appears between its lateral faces, on the direction normal to both E and B :
(22)
(23)
where i is the current flowing in Ox direction and:
σ= ab cb = , crx ary
(24)
rx , ry being the sample resistance in Ox, Oy directions, respectively.
(
)
(
)
(
)
−1
(13)
2 2 2 2 pµ 2 1 p − nµ n + ( p − n )µ nµ p B . RH (B ) = ⋅ 2 2 2 e nµ n + pµ p 2 + ( p − n )2 µ n µ pB
(
)
(14)
One can observe that the conductivity monotonously decreases from: σ(0) ≡ σ0 = e nµ n + pµ p to np µ p − µ n 2 , σ(∞ ) ≡ σ∞ = σ 0 1 + 2 ( p − n ) µ nµ p while the Hall constant monotonously increases from:
µn = so that Eqs. (7), (8) become:
eτe
* me
, µp =
eτh
* mh
,
(9)
vex = −µ n E + vey B , vey = −µ n (EH − vex B ), v = 0 ez and: vhx = µ p E + vhy B , vhy = µ p (EH − vhx B ), v = 0. hz The current density is defined as: r r r j = e ( p vh − n ve ) ,
III.7. THE STUDY OF THE HALL EFFECT IN SEMICONDUCTORS
1. Work purpose The Hall effect is one of the most important effects in the determination of the parameters that characterize from the electrical point of view the semiconductor materials. The goals of the work are: - The determination of the concentration of the charge carriers (n or p) in a sample of extrinsic semiconductors*; - The determination of the Hall mobility of the charge carriers in the respective semiconductor. 2. Theory The Hall effect is a galvanomagnetic** effect, which was observed for the first time by E. H. Hall in 1880. This effect consists in the r appearance of an electric field called Hall field E H , due to the deviation of the charge carrier trajectories by an external magnetic field. We will study the Hall effect in a parallelepipedic semiconductor sample of sizes a, b, c (see Figure 1). The Hall field appears when the sample is placed under an r r external electric field E and an external magnetic field B . The Hall field r r r r r r E H is orthogonal on both E and B . The vectors E , E H and B determine a right orhogonal trihedron (Figure 1): r r r E = (E , 0, 0), EH = (0, EH , 0 ), B = (0, 0, B ).
(
)
carriers is:
r r r Ft = Fel + FL .
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(6)
The (negative) electrons and (positive) holes moving through the sample satisfy the equations: r r r dve ve e r r + = − * Et + ve ∧ B , dt τe me r r r dvh vh e r r + = * Et + vh ∧ B , dt τh mh
(
)
(7) (8)
(
)
* * , mh and τe , τh are the electrons and holes effective masses where me
and relaxation times, respectively. In steady states, the time derivatives cancel and we will define the electron and hole mobilities as:
RH (B ) = 1 µ p − µn ⋅ . eni µ p + µ n (19)
For heavily doped (extrinsic) semiconductors we have:
Байду номын сангаас
σ(B ) ≅ enµ n , RH (B ) ≅ − σ(B ) ≅ epµ p , R H (B ) ≅
1 , n >> p, en p >> n.
(20) (21)
1 , ep
From these relations, one can observe that the Hall mobility of the carriers can be defined as:
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µ H ( B ) = σ(B ) RH (B ) . For our sample geometry, we obtain from Eqs. (4) and (12): RH = U H bU H = , ajB iB
2 2 1 pµ p − nµ n = e nµ n + pµ p 2
(
)
(15)
(
)
−1
(16)
RH (0) ≡ RH 0 to:
(
)
(17)
RH (∞ ) ≡ RH∞ =
1 . e( p − n )
(18)
For intrinsic semiconductors (n = p ≡ ni ), we have σ∞ = 0 and:
One can observe that the units for the Hall constant and mobility are:
RH
IS
=
1 en
= m3C −1,
IS
µH
IS
=
v e
= m 2V −1s −1 = T −1.
IS
(25)
3. Experimental set-up
The sample set-up presented in Figure 2, is made of: - an electromagnet with weak magnetic remnant steel core, which allows a better concentration of the magnetic field lines;